###########################################################################
######### MAPLE Software written by Daniel LAZARD #########################
###########################################################################
######### 	LIP6, Universite' Paris VI,
######### 	case 168, 4 pl. Jussieu, 75252 Paris Cedex 05
######### 	Tel:(33/0) 1 44 27 62 40  Fax:(33/0) 1 44 27 40 42
#########	e-mail Daniel.Lazard@lip6.fr
###########################################################################
# VERSION 3 (March 2002)  #################################################
###########################################################################
# This software should only be used under the conditions of GNU licence
###########################################################################
# Version 3 differs from preceding versions in the following way :
# 	1/ More examples at the end
#	2/ Better interface : 	
# 		test of irreducibility included
# 		no need to give the name of the variable for
# 			polynomials with numeric coefficient
# 		use of a special sqrt function for polynomials depending on 
# 	 		parameters (badly managed by Maple sqrt)
#	3/ Option for getting all the roots
#	4/ Designed for Maple V.5
###########################################################################
# SPECIFICATION: Test if an irreducible polynom of degree 5 is solvable
#	by radicals and if it is provide a solution.
#
###########################################################################
# USAGE:	quintic(polynom, variable);
#	or	quintic(polynom); 		 if nops(indets(polynom))=1
#	or	quintic(polynom, variable,`all`);     for getting all roots
###########################################################################
list_of_examples:=['fa','fc','y1','y2','y3','y4','y4t','y5','y6','y7','y8','y9','y10','y11','y12','y13','y14','y15','y16','y17','y18','y19','y20','f31','f41','f61','f71','h2','hh2','h12','g0','g1','g2','g3','g4'];

quintic:= proc() 
	local 
#	global
	x,
	f,ff,d,i,t,p,q,r,s,R,i_4,i_5,i_6,i_7,i_8,
	DD,EE,F,G,eps,T,U,H,II,J,K,Ts,Q1,P1,P41,
	P42,P31,P32,P33,P34,P21,P22,P23,P24,P4,P3,P2,om1,om2,om3,om4
	;

	f:=args[1];

	if(nargs = 1) then 
	     if (nops(indets(f)) > 1) then 
		ERROR(`There are more than one variable`);
	     fi;
	     x:=op(indets(f));
        else x:=args[2];
        fi;

	f:=collect(f,x);
	if degree(f,x)<>5 then ERROR(`The degree should be 5`)
	fi;
	
	ff:=factors(numer(normal(f)))[2];
	for p in ff do 
		d:=degree(p[1],x);
		if d=5 then f:=p[1]
		elif d > 0 then 
			ERROR(`The polynomial is not irreducible`)
		fi;
	od;
	
	t:=coeff(f,x^4)/coeff(f,x^5)/5;
	f:=collect(
		subs(x=x-t,f/coeff(f,x^5))
		,x);
	p:=coeff(f,x,3);
	q:=coeff(f,x,2);
	r:=coeff(f,x,1);
	s:=coeff(f,x,0);
R:=   x^6
   +8*x^5*r
   +(-6*p^2*r+2*p*q^2-50*q*s+40*r^2)*x^4
   +(-15*p^2*q*s-40*p^2*r^2+21*p*q^2*r+125*p*s^2-2*q^4-400*q*r*s+160*r^3)*x^3
   +(9*p^4*r^2-6*p^3*q^2*r+p^2*q^4+90*p^2*q*r*s-136*p^2*r^3-50*p*q^3*s
     +76*p*q^2*r^2+500*p*r*s^2-8*q^4*r+625*q^2*s^2-1400*q*r^2*s+400*r^4)*x^2
   +(-108*p^5*s^2+117*p^4*q*r*s+32*p^4*r^3-31*p^3*q^3*s-51*p^3*q^2*r^2
     +525*p^3*r*s^2+19*p^2*q^4*r-325*p^2*q^2*s^2+260*p^2*q*r^2*s
     -256*p^2*r^4-2*p*q^6+105*p*q^3*r*s+76*p*q^2*r^3+625*p*q*s^3
     -500*p*r^2*s^2-58*q^5*s+3*q^4*r^2+2750*q^2*r*s^2-2400*q*r^3*s
     +512*r^5-3125*s^4)*x
   -27*p^7*s^2+18*p^6*q*r*s-4*p^6*r^3-4*p^5*q^3*s+p^5*q^2*r^2-99*p^5*r*s^2
     -150*p^4*q^2*s^2+196*p^4*q*r^2*s+48*p^4*r^4+12*p^3*q^3*r*s
     -128*p^3*q^2*r^3+1200*p^3*r^2*s^2-12*p^2*q^5*s+65*p^2*q^4*r^2
     -725*p^2*q^2*r*s^2-160*p^2*q*r^3*s-192*p^2*r^5+3125*p^2*s^4-13*p*q^6*r
     -125*p*q^4*s^2+590*p*q^3*r^2*s-16*p*q^2*r^4-1250*p*q*r*s^3
     -2000*p*r^3*s^2+q^8-124*q^5*r*s+17*q^4*r^3+3250*q^2*r^2*s^2
     -1600*q*r^4*s+256*r^6-9375*r*s^4:
R:=factor(R):
	if not type(R,`*`) then RETURN(`Not solvable by radicals`)
	fi;
	for i in R do if degree(i,x)=1 then i_4:=solve (i,x) fi od;
	if i_4='i_4' then RETURN (`Not solvable by radicals`)
	fi;
i_5:=
((135*p^5*s-90*p^4*q*r+25*p^3*q^3-1500*p^3*r*s+875*p^2*q^2*s+1000*p^2*q
*r^2-925*p*q^3*r+3125*p*q*s^2+180*q^5-1250*q^2*r*s-31250*s^3)*i_4^5+(-81
*p^7*s+27*p^6*q*r-6*p^5*q^3+2160*p^5*r*s-900*p^4*q^2*s-840*p^4*q*r^2+
430*p^3*q^3*r-1875*p^3*q*s^2-14000*p^3*r^2*s-43*p^2*q^5+12875*p^2*q^2
*r*s+6000*p^2*q*r^3-28125*p^2*s^3-2500*p*q^4*s-5800*p*q^3*r^2+37500*p*q*
r*s^2+1230*q^5*r-6250*q^3*s^2-7500*q^2*r^2*s-187500*r*s^3)*i_4^4+(-999*
p^7*r*s+189*p^6*q^2*s+558*p^6*q*r^2-287*p^5*q^3*r-5850*p^5*q*s^2+16680
*p^5*r^2*s+39*p^4*q^5-6050*p^4*q^2*r*s-8720*p^4*q*r^3-28125*p^4*s^3-
25*p^3*q^4*s+8045*p^3*q^3*r^2+66875*p^3*q*r*s^2-62000*p^3*r^3*s-2512*p
^2*q^5*r-38750*p^2*q^3*s^2+8750*p^2*q^2*r^2*s+28000*p^2*q*r^4+106250*
p^2*r*s^3+282*p*q^7+34375*p*q^4*r*s-26900*p*q^3*r^3-246875*p*q^2*s^3+
162500*p*q*r^2*s^2-9300*q^6*s+5765*q^5*r^2+37500*q^3*r*s^2-35000*q^2*r
^3*s+1484375*q*s^4-875000*r^2*s^3)*i_4^3+(405*p^9*r*s-135*p^8*q^2*s-135
*p^8*q*r^2+75*p^7*q^3*r+1485*p^7*q*s^2-11880*p^7*r^2*s-10*p^6*q^5+
9198*p^6*q^2*r*s+4380*p^6*q*r^3+27000*p^6*s^3-1927*p^5*q^4*s-4181*p^5*
q^3*r^2-85950*p^5*q*r*s^2+105040*p^5*r^3*s+1302*p^4*q^5*r+28375*p^4*q
^3*s^2-86025*p^4*q^2*r^2*s-41360*p^4*q*r^4-140625*p^4*r*s^3-127*p^3*q
^7+36575*p^3*q^4*r*s+50200*p^3*q^3*r^3+121875*p^3*q^2*s^3+630000*p^3*
q*r^2*s^2-256000*p^3*r^4*s-6920*p^2*q^6*s-24345*p^2*q^5*r^2-598125*p^
2*q^3*r*s^2-29000*p^2*q^2*r^3*s+104000*p^2*q*r^5+2156250*p^2*q*s^4+
375000*p^2*r^2*s^3+5299*p*q^7*r+119125*p*q^5*s^2+218125*p*q^4*r^2*s-
101200*p*q^3*r^4-2556250*p*q^2*r*s^3+700000*p*q*r^3*s^2-4296875*p*s^5-
396*q^9-58600*q^6*r*s+21970*q^5*r^3+350000*q^4*s^3+250000*q^3*r^2*s^2-
130000*q^2*r^4*s+9218750*q*r*s^4-3250000*r^3*s^3)*i_4^2+(1296*p^9*q*s^2
+1836*p^9*r^2*s-1728*p^8*q^2*r*s-792*p^8*q*r^3-14175*p^8*s^3+276*p^7*q
^4*s+748*p^7*q^3*r^2+6750*p^7*q*r*s^2-41424*p^7*r^3*s-216*p^6*q^5*r+
11340*p^6*q^3*s^2+37074*p^6*q^2*r^2*s+16288*p^6*q*r^4+371250*p^6*r*s^
3+20*p^5*q^7-16330*p^5*q^4*r*s-19672*p^5*q^3*r^3-108000*p^5*q^2*s^3-
498600*p^5*q*r^2*s^2+288320*p^5*r^4*s+1893*p^4*q^6*s+9417*p^4*q^5*r^2
+169250*p^4*q^3*r*s^2-144800*p^4*q^2*r^3*s-100480*p^4*q*r^5+1434375*p^
4*q*s^4-1500000*p^4*r^2*s^3-1938*p^3*q^7*r+33100*p^3*q^5*s^2+12600*p^
3*q^4*r^2*s+121520*p^3*q^3*r^4-12500*p^3*q^2*r*s^3+2070000*p^3*q*r^3*
s^2-608000*p^3*r^5*s-3750000*p^3*s^5+145*p^2*q^9-5150*p^2*q^6*r*s-
56962*p^2*q^5*r^3+178125*p^2*q^4*s^3-1382500*p^2*q^3*r^2*s^2-104000*p
^2*q^2*r^4*s+192000*p^2*q*r^6+4218750*p^2*q*r*s^4+1200000*p^2*r^3*s^3
-530*p*q^8*s+12812*p*q^7*r^2-116250*p*q^5*r*s^2+587500*p*q^4*r^3*s-
193600*p*q^3*r^5+3781250*p*q^3*s^4-8325000*p*q^2*r^2*s^3+1800000*p*q*r^
4*s^2-7812500*p*r*s^5-1050*q^9*r+139750*q^7*s^2-179425*q^6*r^2*s+46160*q
^5*r^4+593750*q^4*r*s^3+600000*q^3*r^3*s^2-240000*q^2*r^5*s-16015625*q
^2*s^5+23750000*q*r^2*s^4-6000000*r^4*s^3)*i_4+486*p^11*q*s^2-324*p^11
*r^2*s-108*p^10*q^2*r*s+108*p^10*q*r^3-13365*p^10*s^3+36*p^9*q^4*s-42*
p^9*q^3*r^2+4158*p^9*q*r*s^2+9072*p^9*r^3*s+4*p^8*q^5*r+4680*p^8*q^3
*s^2-11583*p^8*q^2*r^2*s-2784*p^8*q*r^4+200475*p^8*r*s^3+1496*p^7*q^4
*r*s+3976*p^7*q^3*r^3-165375*p^7*q^2*s^3-85140*p^7*q*r^2*s^2-89024*p^
7*r^4*s+54*p^6*q^6*s-1570*p^6*q^5*r^2+84095*p^6*q^3*r*s^2+166268*p^6*
q^2*r^3*s+25088*p^6*q*r^5+506250*p^6*q*s^4-760500*p^6*r^2*s^3+248*p^5
*q^7*r+8310*p^5*q^5*s^2-127222*p^5*q^4*r^2*s-43120*p^5*q^3*r^4+
1018500*p^5*q^2*r*s^3-295200*p^5*q*r^3*s^2+359680*p^5*r^5*s-843750*p^5
*s^5-14*p^4*q^9+30289*p^4*q^6*r*s+32903*p^4*q^5*r^3-518750*p^4*q^4*s
^3-135750*p^4*q^3*r^2*s^2-292000*p^4*q^2*r^4*s-94720*p^4*q*r^6-290625
*p^4*q*r*s^4+400000*p^4*r^3*s^3-2635*p^3*q^8*s-11198*p^3*q^7*r^2+
215800*p^3*q^5*r*s^2+279200*p^3*q^4*r^3*s+75520*p^3*q^3*r^5+1078125*p
^3*q^3*s^4-3450000*p^3*q^2*r^2*s^3+1560000*p^3*q*r^4*s^2-512000*p^3*
r^6*s-5000000*p^3*r*s^5+1754*p^2*q^9*r+3875*p^2*q^7*s^2-245245*p^2*q^
6*r^2*s-2452*p^2*q^5*r^4+2712500*p^2*q^4*r*s^3-145000*p^2*q^3*r^3*s^
2-200000*p^2*q^2*r^5*s-6640625*p^2*q^2*s^5+128000*p^2*q*r^7-812500*p^2
*q*r^2*s^4+2200000*p^2*r^4*s^3-102*p*q^11+73260*p*q^8*r*s-6366*p*q^7*r
^3-725000*p*q^6*s^3-1145875*p*q^5*r^2*s^2+680000*p*q^4*r^4*s-134400*p*q
^3*r^6+11500000*p*q^3*r*s^4-8400000*p*q^2*r^3*s^3+1600000*p*q*r^5*s^2-
29296875*p*q*s^6+25000000*p*r^2*s^5-8160*q^10*s+1189*q^9*r^2+421750*q^7*
r*s^2-225100*q^6*r^3*s+36640*q^5*r^5-1359375*q^5*s^4-118750*q^4*r^2*s
^3+600000*q^3*r^4*s^2-160000*q^2*r^6*s-34375000*q^2*r*s^5+22500000*q*r
^3*s^4-4000000*r^5*s^3+97656250*s^7)/(729*p^10*s^2-486*p^9*q*r*s+108*p
^8*q^3*s+81*p^8*q^2*r^2-18225*p^8*r*s^2-36*p^7*q^4*r+12150*p^7*q^2*s
^2+10800*p^7*q*r^2*s+4*p^6*q^6-9720*p^6*q^3*r*s-1800*p^6*q^2*r^3+
135000*p^6*r^2*s^2+1584*p^5*q^5*s+2015*p^5*q^4*r^2-175500*p^5*q^2*r*s
^2-60000*p^5*q*r^3*s+928125*p^5*s^4-623*p^4*q^6*r+60000*p^4*q^4*s^2+
92500*p^4*q^3*r^2*s+10000*p^4*q^2*r^4-1012500*p^4*q*r*s^3-250000*p^4*r
^3*s^2+59*p^3*q^8-45050*p^3*q^5*r*s-17500*p^3*q^4*r^3+225000*p^3*q^3
*s^3+850000*p^3*q^2*r^2*s^2-2812500*p^3*r*s^4+5700*p^2*q^7*s+10825*p^
2*q^6*r^2-478125*p^2*q^4*r*s^2-50000*p^2*q^3*r^3*s+1875000*p^2*q^2*s
^4+1250000*p^2*q*r^2*s^3-2610*p*q^8*r+93750*p*q^6*s^2+32500*p*q^5*r^2*
s-1187500*p*q^3*r*s^3+216*q^10-9000*q^7*r*s+625*q^6*r^3+175000*q^5*s^3+
15625*q^4*r^2*s^2-390625*q^2*r*s^4-9765625*s^6):
######################################################################
i_6:= 
((-15*p^3*q^2*r-5625*p^3*s^2+5*p^2*q^4+4000*p^2*q*r*s-1000*p*q^3*s-500*
p*q^2*r^2+12500*p*r*s^2+150*q^4*r-6250*q^2*s^2)*i_4^5+(-27*p^6*q*s+9*p
^5*q^2*r+2025*p^5*s^2-2*p^4*q^4-900*p^4*q*r*s+10*p^3*q^2*r^2-41250*p
^3*r*s^2+40*p^2*q^4*r+6250*p^2*q^2*s^2+24000*p^2*q*r^2*s-6*p*q^6-6625
*p*q^3*r*s-3000*p*q^2*r^3-9375*p*q*s^3+75000*p*r^2*s^2-300*q^5*s+1025*q
^4*r^2-37500*q^2*r*s^2-78125*s^4)*i_4^4+(-810*p^7*s^2+432*p^6*q*r*s-
132*p^5*q^3*s+21*p^5*q^2*r^2+47475*p^5*r*s^2-14*p^4*q^4*r-16125*p^4*q
^2*s^2-29600*p^4*q*r^2*s+3*p^3*q^6+17500*p^3*q^3*r*s+3480*p^3*q^2*r^
3+262500*p^3*q*s^3-275000*p^3*r^2*s^2-2780*p^2*q^5*s-2495*p^2*q^4*r^2
-78750*p^2*q^2*r*s^2+112000*p^2*q*r^3*s-46875*p^2*s^4+581*p*q^6*r+26875
*p*q^4*s^2-8000*p*q^3*r^2*s-14000*p*q^2*r^4-600000*p*q*r*s^3+350000*p*r
^3*s^2-36*q^8-7700*q^5*r*s+4700*q^4*r^3+287500*q^3*s^3-175000*q^2*r^2
*s^2-312500*r*s^4)*i_4^3+(243*p^9*s^2-27*p^8*q*r*s-9*p^7*q^3*s-18*p^7*
q^2*r^2-15120*p^7*r*s^2+13*p^6*q^4*r+6885*p^6*q^2*s^2+6840*p^6*q*r^2
*s-2*p^5*q^6-3604*p^5*q^3*r*s-410*p^5*q^2*r^3-6750*p^5*q*s^3+254250*p
^5*r^2*s^2+279*p^4*q^5*s+123*p^4*q^4*r^2-183750*p^4*q^2*r*s^2-134000
*p^4*q*r^3*s-759375*p^4*s^4+48*p^3*q^6*r+42125*p^3*q^4*s^2+114375*p^3
*q^3*r^2*s+15440*p^3*q^2*r^4+2390625*p^3*q*r*s^3-1110000*p^3*r^3*s^2-
11*p^2*q^8-31335*p^2*q^5*r*s-14405*p^2*q^4*r^3-340625*p^2*q^3*s^3-
682500*p^2*q^2*r^2*s^2+416000*p^2*q*r^4*s+2015625*p^2*r*s^4+1930*p*q^7
*s+4440*p*q^6*r^2+190625*p*q^4*r*s^2+28500*p*q^3*r^3*s-52000*p*q^2*r^5-
578125*p*q^2*s^4-3812500*p*q*r^2*s^3+1300000*p*r^4*s^2-402*q^8*r+27250*q
^6*s^2-54625*q^5*r^2*s+18100*q^4*r^4+1825000*q^3*r*s^3-650000*q^2*r^3
*s^2+3515625*q*s^5-1562500*r^2*s^4)*i_4^2+(-729*p^11*s^2+486*p^10*q*r*s
-108*p^9*q^3*s-81*p^9*q^2*r^2+21465*p^9*r*s^2+36*p^8*q^4*r-12663*p^8*
q^2*s^2-12528*p^8*q*r^2*s-4*p^7*q^6+10374*p^7*q^3*r*s+1956*p^7*q^2*r
^3+1350*p^7*q*s^3-247860*p^7*r^2*s^2-1652*p^6*q^5*s-2084*p^6*q^4*r^2
+253800*p^6*q^2*r*s^2+121312*p^6*q*r^3*s-540000*p^6*s^4+631*p^5*q^6*r-
68640*p^5*q^4*s^2-148962*p^5*q^3*r^2*s-17504*p^5*q^2*r^4-348750*p^5*q
*r*s^3+1189600*p^5*r^3*s^2-59*p^4*q^8+59750*p^4*q^5*r*s+25846*p^4*q^4
*r^3+171250*p^4*q^3*s^3-843000*p^4*q^2*r^2*s^2-409600*p^4*q*r^4*s+
281250*p^4*r*s^4-6974*p^3*q^7*s-13951*p^3*q^6*r^2+277875*p^3*q^4*r*s^
2+269000*p^3*q^3*r^3*s+51520*p^3*q^2*r^5-3412500*p^3*q^2*s^4+7475000*p
^3*q*r^2*s^3-2760000*p^3*r^4*s^2+3114*p^2*q^8*r-43075*p^2*q^6*s^2-
38930*p^2*q^5*r^2*s-45800*p^2*q^4*r^4-800000*p^2*q^3*r*s^3-1360000*p^
2*q^2*r^3*s^2+768000*p^2*q*r^5*s+2031250*p^2*q*s^5+3625000*p^2*r^2*s^
4-246*p*q^10-8800*p*q^7*r*s+13921*p*q^6*r^3-57500*p*q^5*s^3+283125*p*q^4
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133750*q^6*r*s^2-148200*q^5*r^3*s+36800*q^4*r^5-2609375*q^4*s^4+4600000
*q^3*r^2*s^3-1200000*q^2*r^4*s^2+15625000*q*r*s^5-5000000*r^3*s^4)*i_4
-972*p^11*r*s^2+891*p^10*q^2*s^2+540*p^10*q*r^2*s-666*p^9*q^3*r*s-72*p
^9*q^2*r^3+2025*p^9*q*s^3+31968*p^9*r^2*s^2+120*p^8*q^5*s+115*p^8*q
^4*r^2-44604*p^8*q^2*r*s^2-16848*p^8*q*r^3*s+455625*p^8*s^4-40*p^7*q
^6*r+15990*p^7*q^4*s^2+26905*p^7*q^3*r^2*s+2344*p^7*q^2*r^4-660825*p
^7*q*r*s^3-339720*p^7*r^3*s^2+4*p^6*q^8-13700*p^6*q^5*r*s-4345*p^6*q
^4*r^3+166875*p^6*q^3*s^3+921060*p^6*q^2*r^2*s^2+153152*p^6*q*r^4*s-
3391875*p^6*r*s^4+1842*p^5*q^7*s+2857*p^5*q^6*r^2-529295*p^5*q^4*r*s^
2-359464*p^5*q^3*r^3*s-21344*p^5*q^2*r^5+3487500*p^5*q^2*s^4+1227000*p
^5*q*r^2*s^3+1509600*p^5*r^4*s^2-719*p^4*q^8*r+103200*p^4*q^6*s^2+
259094*p^4*q^5*r^2*s+51520*p^4*q^4*r^4-3623750*p^4*q^3*r*s^3-1460000*p
^4*q^2*r^3*s^2-505600*p^4*q*r^5*s-2812500*p^4*q*s^5+9600000*p^4*r^2*s
^4+61*p^3*q^10-79167*p^3*q^7*r*s-44460*p^3*q^6*r^3+751250*p^3*q^5*s^
3+2277000*p^3*q^4*r^2*s^2+148000*p^3*q^3*r^4*s+68480*p^3*q^2*r^6-
15581250*p^3*q^2*r*s^4+4300000*p^3*q*r^3*s^3-2800000*p^3*r^5*s^2+
11718750*p^3*s^6+7895*p^2*q^9*s+18352*p^2*q^8*r^2-1050100*p^2*q^6*r*s
^2-63260*p^2*q^5*r^3*s-41920*p^2*q^4*r^5+5031250*p^2*q^4*s^4+4800000*
p^2*q^3*r^2*s^3-720000*p^2*q^2*r^4*s^2+512000*p^2*q*r^6*s-4062500*p^
2*q*r*s^5-9000000*p^2*r^3*s^4-3515*p*q^10*r+164375*p*q^8*s^2+47855*p*q^
7*r^2*s+6826*p*q^6*r^4-3553750*p*q^5*r*s^3+156250*p*q^4*r^3*s^2+152000*
p*q^3*r^5*s-1953125*p*q^3*s^5-64000*p*q^2*r^7+20593750*p*q^2*r^2*s^4-
8600000*p*q*r^4*s^3+1600000*p*r^6*s^2-35156250*p*r*s^6+252*q^12-12910*q^
9*r*s+568*q^8*r^3+381250*q^7*s^3+224750*q^6*r^2*s^2-147200*q^5*r^4*s+
27200*q^4*r^6-7437500*q^4*r*s^4+4200000*q^3*r^3*s^3-800000*q^2*r^5*s^
2-19531250*q^2*s^6+25000000*q*r^2*s^5-5000000*r^4*s^4)/(729*p^10*s^2-
486*p^9*q*r*s+108*p^8*q^3*s+81*p^8*q^2*r^2-18225*p^8*r*s^2-36*p^7*q^4
*r+12150*p^7*q^2*s^2+10800*p^7*q*r^2*s+4*p^6*q^6-9720*p^6*q^3*r*s-1800
*p^6*q^2*r^3+135000*p^6*r^2*s^2+1584*p^5*q^5*s+2015*p^5*q^4*r^2-
175500*p^5*q^2*r*s^2-60000*p^5*q*r^3*s+928125*p^5*s^4-623*p^4*q^6*r+
60000*p^4*q^4*s^2+92500*p^4*q^3*r^2*s+10000*p^4*q^2*r^4-1012500*p^4*q
*r*s^3-250000*p^4*r^3*s^2+59*p^3*q^8-45050*p^3*q^5*r*s-17500*p^3*q^4*
r^3+225000*p^3*q^3*s^3+850000*p^3*q^2*r^2*s^2-2812500*p^3*r*s^4+5700*
p^2*q^7*s+10825*p^2*q^6*r^2-478125*p^2*q^4*r*s^2-50000*p^2*q^3*r^3*s
+1875000*p^2*q^2*s^4+1250000*p^2*q*r^2*s^3-2610*p*q^8*r+93750*p*q^6*s^
2+32500*p*q^5*r^2*s-1187500*p*q^3*r*s^3+216*q^10-9000*q^7*r*s+625*q^6*r
^3+175000*q^5*s^3+15625*q^4*r^2*s^2-390625*q^2*r*s^4-9765625*s^6):
######################################################################
i_7:=
((81*p^6*s-27*p^5*q*r+6*p^4*q^3-1350*p^4*r*s+900*p^3*q^2*s+300*p^3*q*r
^2-295*p^2*q^3*r-3750*p^2*q*s^2+5000*p^2*r^2*s+48*p*q^5-3625*p*q^2*r*s
+28125*p*s^3+1500*q^4*s-250*q^3*r^2-6250*q*r*s^2)*i_4^5+(351*p^6*r*s+54*
p^5*q^2*s-72*p^5*q*r^2-22*p^4*q^3*r+1125*p^4*q*s^2-6600*p^4*r^2*s+9*p
^3*q^5+3200*p^3*q^2*r*s+800*p^3*q*r^3+28125*p^3*s^3+525*p^2*q^4*s-345
*p^2*q^3*r^2-48125*p^2*q*r*s^2+30000*p^2*r^3*s-292*p*q^5*r+7500*p*q^3*
s^2-18000*p*q^2*r^2*s+137500*p*r*s^3+72*q^7+7750*q^4*r*s-1500*q^3*r^3+
18750*q^2*s^3-37500*q*r^2*s^2)*i_4^4+(-405*p^8*r*s+135*p^7*q^2*s+135*p
^7*q*r^2-75*p^6*q^3*r-4320*p^6*q*s^2+8478*p^6*r^2*s+10*p^5*q^5-4959*p
^5*q^2*r*s-1896*p^5*q*r^3-10125*p^5*s^3+1150*p^4*q^4*s+1886*p^4*q^3*r
^2+102375*p^4*q*r*s^2-56800*p^4*r^3*s-689*p^3*q^5*r-54500*p^3*q^3*s^2
+23025*p^3*q^2*r^2*s+4400*p^3*q*r^4+9375*p^3*r*s^3+80*p^2*q^7+6225*p^
2*q^4*r*s-810*p^2*q^3*r^3+203125*p^2*q^2*s^3-438750*p^2*q*r^2*s^2+
140000*p^2*r^4*s-310*p*q^6*s-1751*p*q^5*r^2+215625*p*q^3*r*s^2-86500*p*q
^2*r^3*s-1343750*p*q*s^4+662500*p*r^2*s^3+348*q^7*r-76000*q^5*s^2+48875
*q^4*r^2*s-7000*q^3*r^4+387500*q^2*r*s^3-175000*q*r^3*s^2+390625*s^5)*
i_4^3+(-1053*p^8*q*s^2-1431*p^8*r^2*s+1161*p^7*q^2*r*s+252*p^7*q*r^3+
14175*p^7*s^3-150*p^6*q^4*s-151*p^6*q^3*r^2-12690*p^6*q*r*s^2+33324*p
^6*r^3*s+9*p^5*q^5*r-11205*p^5*q^3*s^2-29595*p^5*q^2*r^2*s-3808*p^5*
q*r^4-357750*p^5*r*s^3+3*p^4*q^7+15500*p^4*q^4*r*s+3059*p^4*q^3*r^3+
185625*p^4*q^2*s^3+583875*p^4*q*r^2*s^2-240400*p^4*r^4*s-1989*p^3*q^6
*s-1108*p^3*q^5*r^2-318500*p^3*q^3*r*s^2+154600*p^3*q^2*r^3*s+11200*p
^3*q*r^5-2193750*p^3*q*s^4+750000*p^3*r^2*s^3+35*p^2*q^7*r-15975*p^2*
q^5*s^2-53625*p^2*q^4*r^2*s+5320*p^2*q^3*r^4+2128125*p^2*q^2*r*s^3-
2265000*p^2*q*r^3*s^2+520000*p^2*r^5*s+3750000*p^2*s^5+24*p*q^9+41855*p
*q^6*r*s-11533*p*q^5*r^3-159375*p*q^4*s^3+958750*p*q^3*r^2*s^2-302000*p
*q^2*r^4*s-7765625*p*q*r*s^4+2300000*p*r^3*s^3-6540*q^8*s+2110*q^7*r^2-
382000*q^5*r*s^2+204750*q^4*r^3*s-26000*q^3*r^5-843750*q^3*s^4+2250000*
q^2*r^2*s^3-650000*q*r^4*s^2+781250*r*s^5)*i_4^2+(-486*p^10*q*s^2+324*
p^10*r^2*s+108*p^9*q^2*r*s-108*p^9*q*r^3-1215*p^9*s^3-36*p^8*q^4*s+42
*p^8*q^3*r^2+22005*p^8*q*r*s^2-8424*p^8*r^3*s-4*p^7*q^5*r-12078*p^7*q
^3*s^2-4104*p^7*q^2*r^2*s+1488*p^7*q*r^4+60750*p^7*r*s^3+6886*p^6*q^
4*r*s+74*p^6*q^3*r^3+28350*p^6*q^2*s^3-336870*p^6*q*r^2*s^2+89312*p^6
*r^4*s-1167*p^5*q^6*s-1052*p^5*q^5*r^2+276850*p^5*q^3*r*s^2+29256*p^5
*q^2*r^3*s-2624*p^5*q*r^5-371250*p^5*q*s^4-837000*p^5*r^2*s^3+375*p^4
*q^7*r-79470*p^4*q^5*s^2-111475*p^4*q^4*r^2*s-18880*p^4*q^3*r^4-
401250*p^4*q^2*r*s^3+2188000*p^4*q*r^3*s^2-483200*p^4*r^5*s+3796875*p^
4*s^5-36*p^3*q^9+57830*p^3*q^6*r*s+30174*p^3*q^5*r^3+549375*p^3*q^4*s
^3-1502000*p^3*q^3*r^2*s^2+270000*p^3*q^2*r^4*s-6400*p^3*q*r^6-
7781250*p^3*q*r*s^4+2350000*p^3*r^3*s^3-7419*p^2*q^8*s-16732*p^2*q^7*r
^2+258625*p^2*q^5*r*s^2-155150*p^2*q^4*r^3*s+58560*p^2*q^3*r^5-
2193750*p^2*q^3*s^4+10012500*p^2*q^2*r^2*s^3-6160000*p^2*q*r^4*s^2+
960000*p^2*r^6*s+781250*p^2*r*s^5+3730*p*q^9*r-68700*p*q^7*s^2+119665*p*
q^6*r^2*s-42804*p*q^5*r^4-3087500*p*q^4*r*s^3+2575000*p*q^3*r^3*s^2-
456000*p*q^2*r^5*s+18046875*p*q^2*s^5-16812500*p*q*r^2*s^4+3400000*p*r^4
*s^3-288*q^11-14950*q^8*r*s+6193*q^7*r^3+836250*q^6*s^3-1078500*q^5*r^
2*s^2+398000*q^4*r^4*s-48000*q^3*r^6-7421875*q^3*r*s^4+6200000*q^2*r^3
*s^3-1200000*q*r^5*s^2-11718750*q*s^6+6250000*r^2*s^5)*i_4-8019*p^11*s^
3+9477*p^10*q*r*s^2+108*p^10*r^3*s-2430*p^9*q^3*s^2-3645*p^9*q^2*r^2*
s+144*p^9*q*r^4+170505*p^9*r*s^3+1800*p^8*q^4*r*s+227*p^8*q^3*r^3-
114750*p^8*q^2*s^3-155763*p^8*q*r^2*s^2-5664*p^8*r^4*s-220*p^7*q^6*s-
210*p^7*q^5*r^2+162006*p^7*q^3*r*s^2+50008*p^7*q^2*r^3*s-2752*p^7*q*r
^5+371250*p^7*q*s^4-1082700*p^7*r^2*s^3+52*p^6*q^7*r-31544*p^6*q^5*s
^2-65917*p^6*q^4*r^2*s+2812*p^6*q^3*r^4+974925*p^6*q^2*r*s^3+438460*p
^6*q*r^3*s^2+80128*p^6*r^5*s-4303125*p^6*s^5-4*p^5*q^9+23895*p^5*q^6
*r*s+1775*p^5*q^5*r^3-353750*p^5*q^4*s^3-796230*p^5*q^3*r^2*s^2-
110144*p^5*q^2*r^4*s+15104*p^5*q*r^6+4989375*p^5*q*r*s^4+2426000*p^5*r
^3*s^3-2565*p^4*q^8*s-1809*p^4*q^7*r^2+522040*p^4*q^5*r*s^2+167775*p
^4*q^4*r^3*s-60864*p^4*q^3*r^5+15625*p^4*q^3*s^4-9333750*p^4*q^2*r^
2*s^3+1804000*p^4*q*r^4*s^2-396800*p^4*r^6*s+24984375*p^4*r*s^5+435*p^
3*q^9*r-91625*p^3*q^7*s^2-168449*p^3*q^6*r^2*s+68324*p^3*q^5*r^4+
3779375*p^3*q^4*r*s^3+939000*p^3*q^3*r^3*s^2-41600*p^3*q^2*r^5*s-
6640625*p^3*q^2*s^5-25600*p^3*q*r^7-18037500*p^3*q*r^2*s^4-1800000*p^3
*r^4*s^3-32*p^2*q^11+58238*p^2*q^8*r*s-28679*p^2*q^7*r^3-408125*p^2*q
^6*s^3-1157475*p^2*q^5*r^2*s^2-16900*p^2*q^4*r^4*s+85440*p^2*q^3*r^
6-221875*p^2*q^3*r*s^4+20075000*p^2*q^2*r^3*s^3-6120000*p^2*q*r^5*s^2
+21484375*p^2*q*s^6+640000*p^2*r^7*s-41562500*p^2*r^2*s^5-6086*p*q^10*s
+4869*p*q^9*r^2+329150*p*q^7*r*s^2+29305*p*q^6*r^3*s-39776*p*q^5*r^5+
778125*p*q^5*s^4-9218750*p*q^4*r^2*s^3+2920000*p*q^3*r^4*s^2-224000*p*q
^2*r^6*s+36953125*p*q^2*r*s^5-5500000*p*q*r^3*s^4+1600000*p*r^5*s^3-
39062500*p*s^7-276*q^11*r-42500*q^9*s^2+2385*q^8*r^2*s+4078*q^7*r^4+
2033750*q^6*r*s^3-1199500*q^5*r^3*s^2+292000*q^4*r^5*s+1953125*q^4*s^5
-32000*q^3*r^7-14281250*q^3*r^2*s^4+6200000*q^2*r^4*s^3-800000*q*r^6*s
^2-15625000*q*r*s^6+12500000*r^3*s^5)/(729*p^10*s^2-486*p^9*q*r*s+108*p
^8*q^3*s+81*p^8*q^2*r^2-18225*p^8*r*s^2-36*p^7*q^4*r+12150*p^7*q^2*s
^2+10800*p^7*q*r^2*s+4*p^6*q^6-9720*p^6*q^3*r*s-1800*p^6*q^2*r^3+
135000*p^6*r^2*s^2+1584*p^5*q^5*s+2015*p^5*q^4*r^2-175500*p^5*q^2*r*s
^2-60000*p^5*q*r^3*s+928125*p^5*s^4-623*p^4*q^6*r+60000*p^4*q^4*s^2+
92500*p^4*q^3*r^2*s+10000*p^4*q^2*r^4-1012500*p^4*q*r*s^3-250000*p^4*r
^3*s^2+59*p^3*q^8-45050*p^3*q^5*r*s-17500*p^3*q^4*r^3+225000*p^3*q^3
*s^3+850000*p^3*q^2*r^2*s^2-2812500*p^3*r*s^4+5700*p^2*q^7*s+10825*p^
2*q^6*r^2-478125*p^2*q^4*r*s^2-50000*p^2*q^3*r^3*s+1875000*p^2*q^2*s
^4+1250000*p^2*q*r^2*s^3-2610*p*q^8*r+93750*p*q^6*s^2+32500*p*q^5*r^2*
s-1187500*p*q^3*r*s^3+216*q^10-9000*q^7*r*s+625*q^6*r^3+175000*q^5*s^3+
15625*q^4*r^2*s^2-390625*q^2*r*s^4-9765625*s^6):
######################################################################
i_8:=
((-108*p^5*q*s+66*p^4*q^2*r-2250*p^4*s^2-18*p^3*q^4+2800*p^3*q*r*s-1100
*p^2*q^3*s-1000*p^2*q^2*r^2+5000*p^2*r*s^2+800*p*q^4*r-5000*p*q^2*s^2
-144*q^6+1000*q^3*r*s+25000*q*s^3)*i_4^5+(54*p^7*q*s-18*p^6*q^2*r+810*p
^6*s^2+4*p^5*q^4-2088*p^5*q*r*s+720*p^4*q^3*s+676*p^4*q^2*r^2-16500*p
^4*r*s^2-328*p^3*q^4*r+4000*p^3*q^2*s^2+20800*p^3*q*r^2*s+32*p^2*q^6
-12950*p^2*q^3*r*s-6000*p^2*q^2*r^3+18750*p^2*q*s^3+30000*p^2*r^2*s^2
+1880*p*q^5*s+5050*p*q^4*r^2-45000*p*q^2*r*s^2-31250*p*s^4-984*q^6*r+
5000*q^4*s^2+6000*q^3*r^2*s+150000*q*r*s^3)*i_4^4+(-324*p^8*s^2+972*p^
7*q*r*s-204*p^6*q^3*s-438*p^6*q^2*r^2+18990*p^6*r*s^2+224*p^5*q^4*r-
1770*p^5*q^2*s^2-25184*p^5*q*r^2*s-30*p^4*q^6+11840*p^4*q^3*r*s+8368*p
^4*q^2*r^3+127500*p^4*q*s^3-110000*p^4*r^2*s^2-1092*p^3*q^5*s-7434*p
^3*q^4*r^2-85000*p^3*q^2*r*s^2+94400*p^3*q*r^3*s-18750*p^3*s^4+2242*p
^2*q^6*r+41750*p^2*q^4*s^2-10200*p^2*q^3*r^2*s-28000*p^2*q^2*r^4-
325000*p^2*q*r*s^3+140000*p^2*r^3*s^2-240*p*q^8-30580*p*q^5*r*s+23400*p*
q^4*r^3+312500*p*q^3*s^3-200000*p*q^2*r^2*s^2-125000*p*r*s^4+7440*q^7*
s-4612*q^6*r^2-30000*q^4*r*s^2+28000*q^3*r^3*s-1187500*q^2*s^4+700000*q
*r^2*s^3)*i_4^3+(243*p^10*s^2-432*p^9*q*r*s+126*p^8*q^3*s+117*p^8*q^2
*r^2-9693*p^8*r*s^2-62*p^7*q^4*r+3996*p^7*q^2*s^2+14400*p^7*q*r^2*s+8
*p^6*q^6-10744*p^6*q^3*r*s-4028*p^6*q^2*r^3-24300*p^6*q*s^3+128700*p^
6*r^2*s^2+1970*p^5*q^5*s+3797*p^5*q^4*r^2-39840*p^5*q^2*r*s^2-149632*
p^5*q*r^3*s-118125*p^5*s^4-1147*p^4*q^6*r+6150*p^4*q^4*s^2+133070*p^4
*q^3*r^2*s+41264*p^4*q^2*r^4+866250*p^4*q*r*s^3-494000*p^4*r^3*s^2+
109*p^3*q^8-50804*p^3*q^5*r*s-49422*p^3*q^4*r^3-188750*p^3*q^3*s^3-
607000*p^3*q^2*r^2*s^2+371200*p^3*q*r^4*s+243750*p^3*r*s^4+7448*p^2*q
^7*s+23417*p^2*q^6*r^2+459125*p^2*q^4*r*s^2+24600*p^2*q^3*r^3*s-
104000*p^2*q^2*r^5-1581250*p^2*q^2*s^4-1575000*p^2*q*r^2*s^3+520000*p
^2*r^4*s^2-4922*p*q^8*r-65650*p*q^6*s^2-189850*p*q^5*r^2*s+88200*p*q^4
*r^4+2537500*p*q^3*r*s^3-820000*p*q^2*r^3*s^2+4843750*p*q*s^5-625000*p*r
^2*s^4+360*q^10+45080*q^7*r*s-17451*q^6*r^3-245000*q^5*s^3-196875*q^4*
r^2*s^2+104000*q^3*r^4*s-7453125*q^2*r*s^4+2600000*q*r^3*s^3-1953125*s
^6)*i_4^2+(1296*p^10*r*s^2-1242*p^9*q^2*s^2-2160*p^9*q*r^2*s+1644*p^8
*q^3*r*s+696*p^8*q^2*r^3+11880*p^8*q*s^3-45144*p^8*r^2*s^2-248*p^7*q
^5*s-626*p^7*q^4*r^2+25920*p^7*q^2*r*s^2+57664*p^7*q*r^3*s+155250*p^7
*s^4+176*p^6*q^6*r-12528*p^6*q^4*s^2-52244*p^6*q^3*r^2*s-16032*p^6*q
^2*r^4-841500*p^6*q*r*s^3+375840*p^6*r^3*s^2-16*p^5*q^8+18944*p^5*q^
5*r*s+19076*p^5*q^4*r^3+244900*p^5*q^3*s^3+401680*p^5*q^2*r^2*s^2-
394496*p^5*q*r^4*s-1012500*p^5*r*s^4-2024*p^4*q^7*s-8784*p^4*q^6*r^2-
215500*p^4*q^4*r*s^2+203440*p^4*q^3*r^3*s+100992*p^4*q^2*r^5-1762500*p
^4*q^2*s^4+4690000*p^4*q*r^2*s^3-1104000*p^4*r^4*s^2+1752*p^3*q^8*r-
6210*p^3*q^6*s^2-12652*p^3*q^5*r^2*s-115536*p^3*q^4*r^4-785000*p^3*q
^3*r*s^3-2200000*p^3*q^2*r^3*s^2+793600*p^3*q*r^5*s+3812500*p^3*q*s^5
+1450000*p^3*r^2*s^4-128*p^2*q^10-3000*p^2*q^7*r*s+51388*p^2*q^6*r^3-
95500*p^2*q^5*s^3+1225500*p^2*q^4*r^2*s^2+134400*p^2*q^3*r^4*s-192000
*p^2*q^2*r^6-1687500*p^2*q^2*r*s^4-4800000*p^2*q*r^3*s^3+960000*p^2*r
^5*s^2-4687500*p^2*s^6+1336*p*q^9*s-10830*p*q^8*r^2+146500*p*q^6*r*s^2
-529280*p*q^5*r^3*s+169600*p*q^4*r^5-4068750*p*q^4*s^4+8500000*p*q^3*r^
2*s^3-1920000*p*q^2*r^4*s^2+12500000*p*q*r*s^5-2000000*p*r^3*s^4+840*q^
10*r-111800*q^8*s^2+143540*q^7*r^2*s-36928*q^6*r^4-475000*q^5*r*s^3-
480000*q^4*r^3*s^2+192000*q^3*r^5*s+12812500*q^3*s^5-19000000*q^2*r^2*
s^4+4800000*q*r^4*s^3)*i_4+486*p^12*r*s^2-324*p^11*q^2*s^2-108*p^11*q*
r^2*s+144*p^10*q^3*r*s-18*p^10*q^2*r^3+10044*p^10*q*s^3-9666*p^10*r^2
*s^2-24*p^9*q^5*s+4*p^9*q^4*r^2+1674*p^9*q^2*r*s^2-648*p^9*q*r^3*s+
182250*p^9*s^4-2424*p^8*q^4*s^2+3796*p^8*q^3*r^2*s+940*p^8*q^2*r^4-
388260*p^8*q*r*s^3+40692*p^8*r^3*s^2-816*p^7*q^5*r*s-1752*p^7*q^4*r^3
+174750*p^7*q^3*s^3+140616*p^7*q^2*r^2*s^2+51840*p^7*q*r^4*s-243000*p
^7*r*s^4+52*p^6*q^7*s+842*p^6*q^6*r^2-117354*p^6*q^4*r*s^2-130424*p^
6*q^3*r^3*s-15168*p^6*q^2*r^5+618750*p^6*q^2*s^4-385800*p^6*q*r^2*s^
3+195840*p^6*r^4*s^2-166*p^5*q^8*r+7464*p^5*q^6*s^2+109058*p^5*q^5*r
^2*s+28492*p^5*q^4*r^4-1238300*p^5*q^3*r*s^3+1032560*p^5*q^2*r^3*s^2
-441984*p^5*q*r^5*s-2306250*p^5*q*s^5-277500*p^5*r^2*s^4+12*p^4*q^10-
29792*p^4*q^7*r*s-23618*p^4*q^6*r^3+505500*p^4*q^5*s^3-127350*p^4*q^4
*r^2*s^2+138800*p^4*q^3*r^4*s+95168*p^4*q^2*r^6-600000*p^4*q^2*r*s^4
+4210000*p^4*q*r^3*s^3-920000*p^4*r^5*s^2+4687500*p^4*s^6+2868*p^3*q^
9*s+8790*p^3*q^8*r^2-198830*p^3*q^6*r*s^2-118624*p^3*q^5*r^3*s-63184*p
^3*q^4*r^5-50000*p^3*q^4*s^4+875000*p^3*q^3*r^2*s^3-2216000*p^3*q^2
*r^4*s^2+614400*p^3*q*r^6*s+8000000*p^3*q*r*s^5-1350000*p^3*r^3*s^4-
1506*p^2*q^10*r+13750*p^2*q^8*s^2+165328*p^2*q^7*r^2*s-3218*p^2*q^6*r
^4-1950250*p^2*q^5*r*s^3+679750*p^2*q^4*r^3*s^2+260800*p^2*q^3*r^5*s
+781250*p^2*q^3*s^5-128000*p^2*q^2*r^7+4418750*p^2*q^2*r^2*s^4-
6200000*p^2*q*r^4*s^3+640000*p^2*r^6*s^2-25781250*p^2*r*s^6+96*p*q^12-
54952*p*q^9*r*s+7158*p*q^8*r^3+475000*p*q^7*s^3+860350*p*q^6*r^2*s^2-
628880*p*q^5*r^4*s+118400*p*q^4*r^6-9643750*p*q^4*r*s^4+9350000*p*q^3*r
^3*s^3-1600000*p*q^2*r^5*s^2+19531250*p*q^2*s^6-10000000*p*q*r^2*s^5-
2000000*p*r^4*s^4+6096*q^11*s-1124*q^10*r^2-319400*q^8*r*s^2+186030*q^7
*r^3*s-29812*q^6*r^5+737500*q^6*s^4-76250*q^5*r^2*s^3-492500*q^4*r^4*
s^2+128000*q^3*r^6*s+28281250*q^3*r*s^5-17687500*q^2*r^3*s^4+3200000*q*
r^5*s^3-58593750*q*s^7+7812500*r^2*s^6)/(729*p^10*s^2-486*p^9*q*r*s+108
*p^8*q^3*s+81*p^8*q^2*r^2-18225*p^8*r*s^2-36*p^7*q^4*r+12150*p^7*q^2
*s^2+10800*p^7*q*r^2*s+4*p^6*q^6-9720*p^6*q^3*r*s-1800*p^6*q^2*r^3+
135000*p^6*r^2*s^2+1584*p^5*q^5*s+2015*p^5*q^4*r^2-175500*p^5*q^2*r*s
^2-60000*p^5*q*r^3*s+928125*p^5*s^4-623*p^4*q^6*r+60000*p^4*q^4*s^2+
92500*p^4*q^3*r^2*s+10000*p^4*q^2*r^4-1012500*p^4*q*r*s^3-250000*p^4*r
^3*s^2+59*p^3*q^8-45050*p^3*q^5*r*s-17500*p^3*q^4*r^3+225000*p^3*q^3
*s^3+850000*p^3*q^2*r^2*s^2-2812500*p^3*r*s^4+5700*p^2*q^7*s+10825*p^
2*q^6*r^2-478125*p^2*q^4*r*s^2-50000*p^2*q^3*r^3*s+1875000*p^2*q^2*s
^4+1250000*p^2*q*r^2*s^3-2610*p*q^8*r+93750*p*q^6*s^2+32500*p*q^5*r^2*
s-1187500*p*q^3*r*s^3+216*q^10-9000*q^7*r*s+625*q^6*r^3+175000*q^5*s^3+
15625*q^4*r^2*s^2-390625*q^2*r*s^4-9765625*s^6):
#############################################################
DD:=simplify(
40*i_8*p-120*i_7*q+(-24*p^2+100*r)*i_6+(88*p*q-300*s)*i_5+(-24*p^3+100*p*r+24
*q^2)*i_4-80*p^3*r+40*p^2*q^2-480*p*q*s+160*p*r^2+332*q^2*r+125*s^2):
EE:=simplify(
(3*p^2+20*r)*i_6+(-p*q-50*s)*i_5+(3*p^3+12*p*r+3*q^2)*i_4+4*p^3*r-3*p^2*q
^2+40*p*q*s+16*p*r^2-21*q^2*r+125*s^2):
#############################################################
#############################################################
if EE=0 then 
print	(`Congratulation !`);
print	(`You probably got the first example showing that`);
print	(`this algorithm is not proved on a field containing sqrt(-1).`);
print	(`PLEASE send this example to Daniel.Lazard@lip6.fr.`);
print	(`Thanks.`);
	ERROR(`exceptional case for the algorithm`);
fi;
#############################################################
#############################################################

F:=simplify(
(-65*p^2*q+875*p*s-550*q*r)*i_8+(-58*p^2*r+41*p*q^2-275*q*s+440*r^2)*i_7+(
85*p^3*q-520*p^2*s-298*p*q*r+366*q^3+2100*r*s)*i_6+(4*p^3*r-73*p^2*q^2+
2095*p*q*s-56*p*r^2-748*q^2*r-4875*s^2)*i_5+(85*p^4*q-418*p^3*s-440*p^2*q
*r+419*p*q^3+1590*p*r*s-1040*q^2*s+524*q*r^2)*i_4-12*p^5*s+158*p^4*q*r-85*
p^3*q^3-1462*p^3*r*s-159*p^2*q^2*s+142*p^2*q*r^2+896*p*q^3*r+175*p*q*s
^2+2900*p*r^2*s-402*q^5-1925*q^2*r*s-448*q*r^3-1875*s^3):
G:=simplify(
(-35*p^2*q-250*p*s-200*q*r)*i_8+(-22*p^2*r+19*p*q^2+650*q*s-40*r^2)*i_7+(15
*p^3*q+195*p^2*s+68*p*q*r-6*q^3-1100*r*s)*i_6+(-4*p^3*r-27*p^2*q^2-270*p*
q*s+96*p*r^2-182*q^2*r+3000*s^2)*i_5+(15*p^4*q+213*p^3*s+50*p^2*q*r+p*q^
3-940*p*r*s+515*q^2*s-184*q*r^2)*i_4+12*p^5*s+42*p^4*q*r-15*p^3*q^3+492*p
^3*r*s-156*p^2*q^2*s+358*p^2*q*r^2-246*p*q^3*r+2825*p*q*s^2-1400*p*r^2*
s+42*q^5+550*q^2*r*s-232*q*r^3-1250*s^3):
#############################################################
eps:=sqrt(factor(5*DD),symbolic);
T:=sqrt(factor(5/2*(EE+F/eps)),symbolic);
if T=0 then U:=sqrt(factor(5/2*(EE-F/eps)),symbolic) 
	else U:=simplify(5*G/T)/eps fi;
#############################################################
H:=simplify(25*(2*i_5-p*q-5*s));
II:=simplify(25*(
40*i_8*p-70*i_7*q+(-24*p^2+100*r)*i_6+(68*p*q-300*s)*i_5+(-24*p^3+100*p*r-46*
q^2)*i_4-80*p^3*r+20*p^2*q^2-255*p*q*s+160*p*r^2-28*q^2*r+125*s^2));
J:=simplify(
-25*i_8*p-25*i_7*q+(-9*p^2-60*r)*i_6+(-7*p*q+525*s)*i_5+(-p^3-96*p*r+11*q^2)
*i_4+50*p^3*r-7*p^2*q^2-145*p*q*s-308*p*r^2+128*q^2*r-1000*s^2);
K:=simplify(
-125*i_8*p+75*i_7*q+(67*p^2-420*r)*i_6+(-109*p*q+1175*s)*i_5+(63*p^3-412*p*r+
27*q^2)*i_4+210*p^3*r-79*p^2*q^2-415*p*q*s-676*p*r^2+496*q^2*r-750*s^2);
#############################################################
Q1:=5/4*(H+II/eps+(T*J+U*K)/EE);
if Q1=0 then T:=-T;U:=-U; Q1:=5/4*(H+II/eps+(T*J+U*K)/EE) fi;
if Q1=0 then eps:=-eps;Ts:=T;T:=U;U:=-Ts;Q1:=5/4*(H+II/eps+(T*J+U*K)/EE) fi;
if Q1=0 then T:=-T;U:=-U;Q1:=5/4*(H+II/eps+(T*J+U*K)/EE) fi;
P1:=Q1^(1/5);
#############################################################
P41:=-5*p;
P42:=simplify(50*i_7-20*i_5*p-70*i_4*q-20*p^2*q+225*p*s-360*q*r);
P31:=-25*q;
P32:=simplify(-250*i_8+50*i_6*p-550*i_5*q+50*i_4*p^2+500*p^2*r
   +50*p*q^2-875*q*s-1000*r^2);
P33:=simplify(175*i_8-20*i_6*p+115*i_5*q+(-30*p^2+60*r)*i_4
   -290*p^2*r+70*p*q^2-525*q*s+380*r^2);
P34:= simplify(25*i_8-110*i_6*p+70*i_5*q+(-90*p^2+80*r)*i_4
   -170*p^2*r+110*p*q^2-700*q*s+340*r^2);
P21:= simplify(15*i_4+10*p^2-80*r);
P22:=simplify(-250*i_6*q+(200*p^2-1250*r)*i_5+(-50*p*q-625*s)*i_4
   +200*p^3*q-500*p^2*s-650*p*q*r+1750*q^3+1250*r*s);
P23:=simplify(-100*i_7*p-25*i_6*q+100*i_5*r+(-75*p*q+375*s)*i_4
   +650*p^2*s-650*p*q*r+175*q^3-1000*r*s);
P24:=simplify(75*i_7*p-450*i_6*q+550*i_5*r+(-350*p*q+500*s)*i_4
   +450*p^2*s-825*p*q*r+525*q^3+750*r*s);
#############################################################
P4 := P41/2+P42/2/eps;
P3 := P31/4 + P32/4/eps+(P33*T+P34*U)/10/EE;
P2 := P21/4 + P22/4/eps+(P23*T+P24*U)/10/EE;

if ((nargs < 3) or not (args[3]=`all`)) then 
	RETURN ((P1 + P2/P1^3 + P3/P1^2 + P4/P1)/5 - t)
	fi;
om1:=-1/4-1/4*5^(1/2)+1/4*(-10+2*5^(1/2))^(1/2);
om2:=-1/4+1/4*5^(1/2)-1/8*(-10+2*5^(1/2))^(1/2)
	-1/8*(-10+2*5^(1/2))^(1/2)*5^(1/2);
om3:=-1/4+1/4*5^(1/2)+1/8*(-10+2*5^(1/2))^(1/2)
	+1/8*(-10+2*5^(1/2))^(1/2)*5^(1/2);
om4:=-1/4-1/4*5^(1/2)-1/4*(-10+2*5^(1/2))^(1/2);

	(P1 + P2/P1^3 + P3/P1^2 + P4/P1)/5 - t,
	(om1*P1 + om2*P2/P1^3 + om3*P3/P1^2 + om4*P4/P1)/5 - t,
	(om2*P1 + om4*P2/P1^3 + om1*P3/P1^2 + om3*P4/P1)/5 - t,
	(om3*P1 + om1*P2/P1^3 + om4*P3/P1^2 + om2*P4/P1)/5 - t,
	(om4*P1 + om3*P2/P1^3 + om2*P3/P1^2 + om1*P4/P1)/5 - t;
	end:
##############################################################
####EXAMPLES
##############################################################
# All the following NUMERICAL equations have been tested by the Maple program:
#
#	equation;
#	quintic(%,x);
#	subs(x=%,%%);
#	evala(simplify(%)); #which gives the result 0
#############################################################
# Two exceptions : h9, y7 and y18 (Young 6, 7 and 18) which are reducibles
#############################################################
# To do : Make Maple able to verify symbolic examples
#############################################################
#Note : Maple V.2 was not able to do this verification !!!
#############################################################

# Examples labelled #Young n are those of G. Paxton Young paper of 1888
	
fa:=x^5-a:
fc:=x^5-312:

y1:=x^5+3*x^2+2*x-1:       		 		#Young 1
y2:=x^5-10*x^3-20*x^2-1505*x-7412:       		#Young 2
y3:=x^5+625/4*x+3750:					#Young 3
y4:=x^5+x^4-4*x^3-3*x^2+3*x+1:				# 2*cos(2*k*Pi/11)
y4t:=x^5-22/5*x^3-11/25*x^2+462/125*x+979/3125: 	#Young 4
y5:=x^5+20*x^3+20*x^2+30*x+10:		 		#Young 5
#	solution a-a^2+a^3-a^4  avec a=2^(1/5)
y6:=x^5+320*x^2-1000*x+4288:		 		#Young 6 
# y6 is reductible : -8 is a root
y7:=(x/10)^5+40*(x/10)^2-69*(x/10)+108:			#Young 7
# 	y7 is reductible: -40 is a root (and 2 factors of degree 2)
y8:=x^5-20*x^3+250*x-400:				#Young 8
y9:=x^5-5*x^3+85/8*x-13/2:				#Young 9
y10:=x^5+20/17*x+21/17:					#Young 10
y11:=x^5-4/13*x+29/65:					#Young 11
y12:=x^5+10/13*x+3/13:					#Young 12
y13:= x^5+110*(5*x^3+60*x^2+800*x+8320):		#Young 13
y14 := x^5-20 *x^3 -80 *x^2 -150 *x -656  :		#Young 14
y15 := x^5 -40 *x^3 +160 *x^2 +1000 *x -5888  :		#Young 15
y16 :=(x/2)^5-50*(x/2)^3-600*(x/2)^2-2000*(x/2)-11200 :	#Young 16
y17 := x^5+110*(5 *x^3 + 20*x^2 -360 *x +800) :		#Young 17
y18 := x^5- 20 *x^3 +320 *x^2 +540 *x + 6368 :		#Young 18
#	y18 is reducible (-8 is a root)
y19 := x^5-20 *x^3 -160 *x^2 -420 *x -8928  :		#Young 19
#	y19 is reducible (8 is a root)
y20 := x^5-20 *x^3 +170 *x + 208 :			#Young 20

f31:=x^5+x^4-12*x^3-21*x^2+x+5:				# sum (E^(6^i*2*Pi/31),i=0..5
f41:=x^5+x^4-16*x^3+5*x^2+21*x-9 :			# sum (E^(3^i*2*Pi/41),i=0..7
f61:=x^5+x^4-24*x^3-17*x^2+41*x-13 :			# sum (E^(21^i*2*Pi/61),i=0..11
f71:=x^5+x^4-28*x^3+37*x^2+25*x+1 :			# sum (E^(23^i*2*Pi/71),i=0..13

h2:=x^5+15/16*x+11/8:
#	solution  (a-a^2+a^(-2)+a^(-1))/2 	avec a=(1+sqrt(2))^(1/5)
hh2:=x^5+15*x+44:
#	solution  a-a^2+a^(-2)+a^(-1) 		avec a=(1+sqrt(2))^(1/5)
h12:=x^5-4*x^4+4*x^3-5*x^2+12*x-1 :


# y1 := x^5+ *x^3 + *x^2 + *x +  :		#Young

g0:=(m^2+l^10)*x^5+5*l*(3*l^5-4*m)*x-4*(11*l^5+2*m):
	# 
g1:= (m-1)^4*(m^2-6*m+25)*x^5 + 5^5*m*l^4*x + 5^5*m*l^5:
g2:=(m^2+16)*x^5 + 5*l^4*(3-m)*x + l^5*(22+m):
g3:=(m^2+1)*x^5+5*l^4*(3-4*m)*x-4*l^5*(11+2*m):
	# g0 to g3 are parametrization of the solvable equations of  
	# the form x^5+r*x+s.
	#
	# An irreducible equation of this form is solvable by radicals iff
	# there are l and m in the base field s.t. 
	# 	r=5*l*(3*l^5-4*m)/(m^2+l^10)
	# and 	s=-4*(11*l^5+2*m)/(m^2+l^10)
	# (This correspond to equation g0)
	#
	# The other parametrizations are equivalent except that some equations
 	# correspond to infinite values of l and/or m : This is the case for 
	# the irreducible equations x^5-s  in the parametrizations
	# g1, g3 and g3
	#
	# One passes from g2 to g3 by the substitution m=4*m, l=2*l
	# One passes from g0 to g3 by the substitution m=m/l^5, l=1/l
	# One passes from g1 to g3 by the substitution l=(m-1)*l/5 followed 
	# 	by the homography on m  mapping the roots of m^2-6*m+25 
 	# 	to the roots of m^2+1 and 0 to infinity
g4:=subs(	
	_c=(_b*(a+4*m)-_p*(a-4*m)-a^2*m)/2,
	_b=l*(4*m^2+a^2)-5*_p-2*m^2,
	_p=(l^2*(4*m^2+a^2)-m^2)/4,
	x^5-5*_p*(2*x^3 + a*x^2 + _b*x)-_p*_c
	):