# Maple integration test file: "1 Algebraic functions\1.2 Trinomial products\1.2.1 Quadratic\1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.txt"

lst:=[

# Integrands of the form (g x)^m (d+e x)^n (a+b x+c x^2)^p

# Integrands of the form (g x)^m (d+e x)^n (a+b x+c x^2)^p when b=0 and c d^2+a e^2=0

# Integrands of the form x^m (d+e x)^1 (d^2 - e^2 x^2)^(p/2)

# p>0
[x^2*(d+e*x)*(d^2-e^2*x^2)^(1/2),x,10,-1/3*d^2*(d^2-e^2*x^2)^(3/2)/e^3-1/4*d*x*(d^2-e^2*x^2)^(3/2)/e^2+1/5*(d^2-e^2*x^2)^(5/2)/e^3+1/8*d^5*arctan(e*x/sqrt(d^2-e^2*x^2))/e^3+1/8*d^3*x*sqrt(d^2-e^2*x^2)/e^2],
[x^4*(d+e*x)*(d^2-e^2*x^2)^(3/2),x,8,1/64*d^5*x*(d^2-e^2*x^2)^(3/2)/e^4-4/63*d^2*x^2*(d^2-e^2*x^2)^(5/2)/e^3-1/8*d*x^3*(d^2-e^2*x^2)^(5/2)/e^2-1/9*x^4*(d^2-e^2*x^2)^(5/2)/e-1/5040*d^3*(128*d+315*e*x)*(d^2-e^2*x^2)^(5/2)/e^5+3/128*d^9*arctan(e*x/sqrt(d^2-e^2*x^2))/e^5+3/128*d^7*x*sqrt(d^2-e^2*x^2)/e^4],
[x^3*(d+e*x)*(d^2-e^2*x^2)^(3/2),x,7,1/64*d^4*x*(d^2-e^2*x^2)^(3/2)/e^3-1/7*d*x^2*(d^2-e^2*x^2)^(5/2)/e^2-1/8*x^3*(d^2-e^2*x^2)^(5/2)/e-1/560*d^2*(32*d+35*e*x)*(d^2-e^2*x^2)^(5/2)/e^4+3/128*d^8*arctan(e*x/sqrt(d^2-e^2*x^2))/e^4+3/128*d^6*x*sqrt(d^2-e^2*x^2)/e^3],
[x^2*(d+e*x)*(d^2-e^2*x^2)^(3/2),x,12,1/24*d^3*x*(d^2-e^2*x^2)^(3/2)/e^2-1/5*d^2*(d^2-e^2*x^2)^(5/2)/e^3-1/6*d*x*(d^2-e^2*x^2)^(5/2)/e^2+1/7*(d^2-e^2*x^2)^(7/2)/e^3+1/16*d^7*arctan(e*x/sqrt(d^2-e^2*x^2))/e^3+1/16*d^5*x*sqrt(d^2-e^2*x^2)/e^2],
[x*(d+e*x)*(d^2-e^2*x^2)^(3/2),x,5,1/24*d^2*x*(d^2-e^2*x^2)^(3/2)/e-1/30*(6*d+5*e*x)*(d^2-e^2*x^2)^(5/2)/e^2+1/16*d^6*arctan(e*x/sqrt(d^2-e^2*x^2))/e^2+1/16*d^4*x*sqrt(d^2-e^2*x^2)/e],
[x*(d+e*x)*(d^2-e^2*x^2)^(3/2),x,5,1/24*d^2*x*(d^2-e^2*x^2)^(3/2)/e-1/30*(6*d+5*e*x)*(d^2-e^2*x^2)^(5/2)/e^2+1/16*d^6*arctan(e*x/sqrt(d^2-e^2*x^2))/e^2+1/16*d^4*x*sqrt(d^2-e^2*x^2)/e],
[(d+e*x)*(d^2-e^2*x^2)^(3/2)/x,x,8,1/12*(4*d+3*e*x)*(d^2-e^2*x^2)^(3/2)+3/8*d^4*arctan(e*x/sqrt(d^2-e^2*x^2))-d^4*arctanh(sqrt(d^2-e^2*x^2)/d)+1/8*d^2*(8*d+3*e*x)*sqrt(d^2-e^2*x^2)],
[(d+e*x)*(d^2-e^2*x^2)^(3/2)/x^2,x,8,-1/3*(3*d-e*x)*(d^2-e^2*x^2)^(3/2)/x-3/2*d^3*e*arctan(e*x/sqrt(d^2-e^2*x^2))-d^3*e*arctanh(sqrt(d^2-e^2*x^2)/d)+1/2*d*e*(2*d-3*e*x)*sqrt(d^2-e^2*x^2)],
[(d+e*x)*(d^2-e^2*x^2)^(3/2)/x^3,x,8,-1/2*(d-e*x)*(d^2-e^2*x^2)^(3/2)/x^2-3/2*d^2*e^2*arctan(e*x/sqrt(d^2-e^2*x^2))+3/2*d^2*e^2*arctanh(sqrt(d^2-e^2*x^2)/d)-3/2*d*e*(d+e*x)*sqrt(d^2-e^2*x^2)/x],
[(d+e*x)*(d^2-e^2*x^2)^(3/2)/x^4,x,8,-1/6*(2*d+3*e*x)*(d^2-e^2*x^2)^(3/2)/x^3+d*e^3*arctan(e*x/sqrt(d^2-e^2*x^2))+3/2*d*e^3*arctanh(sqrt(d^2-e^2*x^2)/d)+1/2*e^2*(2*d-3*e*x)*sqrt(d^2-e^2*x^2)/x],
[(d+e*x)*(d^2-e^2*x^2)^(3/2)/x^5,x,8,-1/12*(3*d+4*e*x)*(d^2-e^2*x^2)^(3/2)/x^4+e^4*arctan(e*x/sqrt(d^2-e^2*x^2))-3/8*e^4*arctanh(sqrt(d^2-e^2*x^2)/d)+1/8*e^2*(3*d+8*e*x)*sqrt(d^2-e^2*x^2)/x^2],
[(d+e*x)*(d^2-e^2*x^2)^(3/2)/x^6,x,6,-1/4*e*(d^2-e^2*x^2)^(3/2)/x^4-1/5*(d^2-e^2*x^2)^(5/2)/(d*x^5)-3/8*e^5*arctanh(sqrt(d^2-e^2*x^2)/d)/d+3/8*e^3*sqrt(d^2-e^2*x^2)/x^2],
[(d+e*x)*(d^2-e^2*x^2)^(3/2)/x^7,x,7,-1/24*e^2*(d^2-e^2*x^2)^(3/2)/(d*x^4)-1/6*(d^2-e^2*x^2)^(5/2)/(d*x^6)-1/5*e*(d^2-e^2*x^2)^(5/2)/(d^2*x^5)-1/16*e^6*arctanh(sqrt(d^2-e^2*x^2)/d)/d^2+1/16*e^4*sqrt(d^2-e^2*x^2)/(d*x^2)],
[(d+e*x)*(d^2-e^2*x^2)^(3/2)/x^8,x,8,-1/24*e^3*(d^2-e^2*x^2)^(3/2)/(d^2*x^4)-1/7*(d^2-e^2*x^2)^(5/2)/(d*x^7)-1/6*e*(d^2-e^2*x^2)^(5/2)/(d^2*x^6)-2/35*e^2*(d^2-e^2*x^2)^(5/2)/(d^3*x^5)-1/16*e^7*arctanh(sqrt(d^2-e^2*x^2)/d)/d^3+1/16*e^5*sqrt(d^2-e^2*x^2)/(d^2*x^2)],
[(d+e*x)*(d^2-e^2*x^2)^(3/2)/x^9,x,9,-1/64*e^4*(d^2-e^2*x^2)^(3/2)/(d^3*x^4)-1/8*(d^2-e^2*x^2)^(5/2)/(d*x^8)-1/7*e*(d^2-e^2*x^2)^(5/2)/(d^2*x^7)-1/16*e^2*(d^2-e^2*x^2)^(5/2)/(d^3*x^6)-2/35*e^3*(d^2-e^2*x^2)^(5/2)/(d^4*x^5)-3/128*e^8*arctanh(sqrt(d^2-e^2*x^2)/d)/d^4+3/128*e^6*sqrt(d^2-e^2*x^2)/(d^3*x^2)],

# p<0
[x^2*(d+e*x)/(d^2-e^2*x^2)^(1/2),x,8,1/3*(d^2-e^2*x^2)^(3/2)/e^3+1/2*d^3*arctan(e*x/sqrt(d^2-e^2*x^2))/e^3-d^2*sqrt(d^2-e^2*x^2)/e^3-1/2*d*x*sqrt(d^2-e^2*x^2)/e^2],
[x^2*(d+e*x)/(d^2-e^2*x^2)^(3/2),x,5,-d*arctan(e*x/sqrt(d^2-e^2*x^2))/e^3+d*(d+e*x)/(e^3*sqrt(d^2-e^2*x^2))+sqrt(d^2-e^2*x^2)/e^3],
[x^2*(d+e*x)/(d^2-e^2*x^2)^(5/2),x,3,1/3*x^2*(d+e*x)/(d*e*(d^2-e^2*x^2)^(3/2))+(-2/3)/(e^3*sqrt(d^2-e^2*x^2))],
[x^7*(d+e*x)/(d^2-e^2*x^2)^(7/2),x,6,1/5*x^6*(d+e*x)/(e^2*(d^2-e^2*x^2)^(5/2))-1/15*x^4*(6*d+7*e*x)/(e^4*(d^2-e^2*x^2)^(3/2))-7/2*d^2*arctan(e*x/sqrt(d^2-e^2*x^2))/e^8+1/15*x^2*(24*d+35*e*x)/(e^6*sqrt(d^2-e^2*x^2))+1/10*(32*d+35*e*x)*sqrt(d^2-e^2*x^2)/e^8],
[x^6*(d+e*x)/(d^2-e^2*x^2)^(7/2),x,6,1/5*x^5*(d+e*x)/(e^2*(d^2-e^2*x^2)^(5/2))-1/15*x^3*(5*d+6*e*x)/(e^4*(d^2-e^2*x^2)^(3/2))-d*arctan(e*x/sqrt(d^2-e^2*x^2))/e^7+1/5*x*(5*d+8*e*x)/(e^6*sqrt(d^2-e^2*x^2))+16/5*sqrt(d^2-e^2*x^2)/e^7],
[x^5*(d+e*x)/(d^2-e^2*x^2)^(7/2),x,5,1/5*x^4*(d+e*x)/(e^2*(d^2-e^2*x^2)^(5/2))-1/15*x^2*(4*d+5*e*x)/(e^4*(d^2-e^2*x^2)^(3/2))-arctan(e*x/sqrt(d^2-e^2*x^2))/e^6+1/15*(8*d+15*e*x)/(e^6*sqrt(d^2-e^2*x^2))],
[x^4*(d+e*x)/(d^2-e^2*x^2)^(7/2),x,4,1/5*x^4*(d+e*x)/(d*e*(d^2-e^2*x^2)^(5/2))-4/15*d^2/(e^5*(d^2-e^2*x^2)^(3/2))+4/5/(e^5*sqrt(d^2-e^2*x^2))],
[x^3*(d+e*x)/(d^2-e^2*x^2)^(7/2),x,3,1/5*x^2*(d+e*x)/(e^2*(d^2-e^2*x^2)^(5/2))+1/15*(-2*d-3*e*x)/(e^4*(d^2-e^2*x^2)^(3/2))+1/5*x/(d^2*e^3*sqrt(d^2-e^2*x^2))],
[x^2*(d+e*x)/(d^2-e^2*x^2)^(7/2),x,3,1/5*x^2*(d+e*x)/(d*e*(d^2-e^2*x^2)^(5/2))-2/15*(d-e*x)/(d*e^3*(d^2-e^2*x^2)^(3/2))-2/15*x/(d^3*e^2*sqrt(d^2-e^2*x^2))],
[x*(d+e*x)/(d^2-e^2*x^2)^(7/2),x,3,1/5*(d+e*x)/(e^2*(d^2-e^2*x^2)^(5/2))-1/15*x/(d^2*e*(d^2-e^2*x^2)^(3/2))-2/15*x/(d^4*e*sqrt(d^2-e^2*x^2))],
[(d+e*x)/(d^2-e^2*x^2)^(7/2),x,3,1/5*(d+e*x)/(d*e*(d^2-e^2*x^2)^(5/2))+4/15*x/(d^3*(d^2-e^2*x^2)^(3/2))+8/15*x/(d^5*sqrt(d^2-e^2*x^2))],
[(d+e*x)/(x*(d^2-e^2*x^2)^(7/2)),x,7,1/5*(d+e*x)/(d^2*(d^2-e^2*x^2)^(5/2))+1/15*(5*d+4*e*x)/(d^4*(d^2-e^2*x^2)^(3/2))-arctanh(sqrt(d^2-e^2*x^2)/d)/d^6+1/15*(15*d+8*e*x)/(d^6*sqrt(d^2-e^2*x^2))],
[(d+e*x)/(x^2*(d^2-e^2*x^2)^(7/2)),x,7,1/5*(d+e*x)/(d^2*x*(d^2-e^2*x^2)^(5/2))+1/15*(6*d+5*e*x)/(d^4*x*(d^2-e^2*x^2)^(3/2))-e*arctanh(sqrt(d^2-e^2*x^2)/d)/d^7+1/5*(8*d+5*e*x)/(d^6*x*sqrt(d^2-e^2*x^2))-16/5*sqrt(d^2-e^2*x^2)/(d^7*x)],
[(d+e*x)/(x^3*(d^2-e^2*x^2)^(7/2)),x,8,1/5*(d+e*x)/(d^2*x^2*(d^2-e^2*x^2)^(5/2))+1/15*(7*d+6*e*x)/(d^4*x^2*(d^2-e^2*x^2)^(3/2))-7/2*e^2*arctanh(sqrt(d^2-e^2*x^2)/d)/d^8+1/15*(35*d+24*e*x)/(d^6*x^2*sqrt(d^2-e^2*x^2))-7/2*sqrt(d^2-e^2*x^2)/(d^7*x^2)-16/5*e*sqrt(d^2-e^2*x^2)/(d^8*x)],
[x^2*(d+e*x)/(d^2-e^2*x^2)^(9/2),x,4,1/7*x^2*(d+e*x)/(d*e*(d^2-e^2*x^2)^(7/2))-2/35*(d-2*e*x)/(d*e^3*(d^2-e^2*x^2)^(5/2))-4/105*x/(d^3*e^2*(d^2-e^2*x^2)^(3/2))-8/105*x/(d^5*e^2*sqrt(d^2-e^2*x^2))],
[x^2*(d+e*x)/(d^2-e^2*x^2)^(11/2),x,5,1/9*x^2*(d+e*x)/(d*e*(d^2-e^2*x^2)^(9/2))-2/63*(d-3*e*x)/(d*e^3*(d^2-e^2*x^2)^(7/2))-2/105*x/(d^3*e^2*(d^2-e^2*x^2)^(5/2))-8/315*x/(d^5*e^2*(d^2-e^2*x^2)^(3/2))-16/315*x/(d^7*e^2*sqrt(d^2-e^2*x^2))],
[x^2*(1-a*x)/(1-a^2*x^2)^(3/2),x,4,-arcsin(a*x)/a^3+(-1+a*x)/(a^3*sqrt(1-a^2*x^2))-sqrt(1-a^2*x^2)/a^3],

# Integrands of the form x^m (d+e x)^2 (d^2 - e^2 x^2)^(p/2)

# p>0

# p<0
[x^4*(d+e*x)^2/sqrt(d^2-e^2*x^2),x,7,11/16*d^6*arctan(e*x/sqrt(d^2-e^2*x^2))/e^5-8/15*d^3*x^2*sqrt(d^2-e^2*x^2)/e^3-11/24*d^2*x^3*sqrt(d^2-e^2*x^2)/e^2-2/5*d*x^4*sqrt(d^2-e^2*x^2)/e-1/6*x^5*sqrt(d^2-e^2*x^2)-1/240*d^4*(256*d+165*e*x)*sqrt(d^2-e^2*x^2)/e^5],
[x^3*(d+e*x)^2/sqrt(d^2-e^2*x^2),x,6,3/4*d^5*arctan(e*x/sqrt(d^2-e^2*x^2))/e^4-3/5*d^2*x^2*sqrt(d^2-e^2*x^2)/e^2-1/2*d*x^3*sqrt(d^2-e^2*x^2)/e-1/5*x^4*sqrt(d^2-e^2*x^2)-3/20*d^3*(8*d+5*e*x)*sqrt(d^2-e^2*x^2)/e^4],
[x^2*(d+e*x)^2/sqrt(d^2-e^2*x^2),x,5,7/8*d^4*arctan(e*x/sqrt(d^2-e^2*x^2))/e^3-2/3*d*x^2*sqrt(d^2-e^2*x^2)/e-1/4*x^3*sqrt(d^2-e^2*x^2)-1/24*d^2*(32*d+21*e*x)*sqrt(d^2-e^2*x^2)/e^3],
[x*(d+e*x)^2/sqrt(d^2-e^2*x^2),x,4,d^3*arctan(e*x/sqrt(d^2-e^2*x^2))/e^2-1/3*x^2*sqrt(d^2-e^2*x^2)-1/3*d*(5*d+3*e*x)*sqrt(d^2-e^2*x^2)/e^2],
[(d+e*x)^2/sqrt(d^2-e^2*x^2),x,4,3/2*d^2*arctan(e*x/sqrt(d^2-e^2*x^2))/e-3/2*d*sqrt(d^2-e^2*x^2)/e-1/2*(d+e*x)*sqrt(d^2-e^2*x^2)/e],
[(d+e*x)^2/(x*sqrt(d^2-e^2*x^2)),x,7,2*d*arctan(e*x/sqrt(d^2-e^2*x^2))-d*arctanh(sqrt(d^2-e^2*x^2)/d)-sqrt(d^2-e^2*x^2)],
[(d+e*x)^2/(x^2*sqrt(d^2-e^2*x^2)),x,7,e*arctan(e*x/sqrt(d^2-e^2*x^2))-2*e*arctanh(sqrt(d^2-e^2*x^2)/d)-sqrt(d^2-e^2*x^2)/x],
[(d+e*x)^2/(x^3*sqrt(d^2-e^2*x^2)),x,5,-3/2*e^2*arctanh(sqrt(d^2-e^2*x^2)/d)/d-1/2*sqrt(d^2-e^2*x^2)/x^2-2*e*sqrt(d^2-e^2*x^2)/(d*x)],
[(d+e*x)^2/(x^4*sqrt(d^2-e^2*x^2)),x,6,-e^3*arctanh(sqrt(d^2-e^2*x^2)/d)/d^2-1/3*sqrt(d^2-e^2*x^2)/x^3-e*sqrt(d^2-e^2*x^2)/(d*x^2)-5/3*e^2*sqrt(d^2-e^2*x^2)/(d^2*x)],
[(d+e*x)^2/(x^5*sqrt(d^2-e^2*x^2)),x,7,-7/8*e^4*arctanh(sqrt(d^2-e^2*x^2)/d)/d^3-1/4*sqrt(d^2-e^2*x^2)/x^4-2/3*e*sqrt(d^2-e^2*x^2)/(d*x^3)-7/8*e^2*sqrt(d^2-e^2*x^2)/(d^2*x^2)-4/3*e^3*sqrt(d^2-e^2*x^2)/(d^3*x)],
[(d+e*x)^2/(x^6*sqrt(d^2-e^2*x^2)),x,8,-3/4*e^5*arctanh(sqrt(d^2-e^2*x^2)/d)/d^4-1/5*sqrt(d^2-e^2*x^2)/x^5-1/2*e*sqrt(d^2-e^2*x^2)/(d*x^4)-3/5*e^2*sqrt(d^2-e^2*x^2)/(d^2*x^3)-3/4*e^3*sqrt(d^2-e^2*x^2)/(d^3*x^2)-6/5*e^4*sqrt(d^2-e^2*x^2)/(d^4*x)],
[x^5*(d+e*x)^2/(d^2-e^2*x^2)^(7/2),x,6,1/5*d^4*(d+e*x)^2/(e^6*(d^2-e^2*x^2)^(5/2))-22/15*d^3*(d+e*x)/(e^6*(d^2-e^2*x^2)^(3/2))-2*d*arctan(e*x/sqrt(d^2-e^2*x^2))/e^6+2/15*d*(30*d+23*e*x)/(e^6*sqrt(d^2-e^2*x^2))+sqrt(d^2-e^2*x^2)/e^6],
[x^4*(d+e*x)^2/(d^2-e^2*x^2)^(7/2),x,6,1/5*d^3*(d+e*x)^2/(e^5*(d^2-e^2*x^2)^(5/2))-17/15*d^2*(d+e*x)/(e^5*(d^2-e^2*x^2)^(3/2))-arctan(e*x/sqrt(d^2-e^2*x^2))/e^5+2/15*(15*d+13*e*x)/(e^5*sqrt(d^2-e^2*x^2))],
[x^3*(d+e*x)^2/(d^2-e^2*x^2)^(7/2),x,3,1/5*d^2*(d+e*x)^2/(e^4*(d^2-e^2*x^2)^(5/2))-4/5*d*(d+e*x)/(e^4*(d^2-e^2*x^2)^(3/2))+1/5*(5*d+2*e*x)/(d*e^4*sqrt(d^2-e^2*x^2))],
[x^2*(d+e*x)^2/(d^2-e^2*x^2)^(7/2),x,3,1/5*d*(d+e*x)^2/(e^3*(d^2-e^2*x^2)^(5/2))-7/15*(d+e*x)/(e^3*(d^2-e^2*x^2)^(3/2))+1/15*x/(d^2*e^2*sqrt(d^2-e^2*x^2))],
[x*(d+e*x)^2/(d^2-e^2*x^2)^(7/2),x,3,1/5*(d+e*x)^2/(e^2*(d^2-e^2*x^2)^(5/2))-2/15*(d+e*x)/(d*e^2*(d^2-e^2*x^2)^(3/2))-4/15*x/(d^3*e*sqrt(d^2-e^2*x^2))],
[(d+e*x)^2/(d^2-e^2*x^2)^(7/2),x,3,2/5*(d+e*x)/(e*(d^2-e^2*x^2)^(5/2))+1/5*x/(d^2*(d^2-e^2*x^2)^(3/2))+2/5*x/(d^4*sqrt(d^2-e^2*x^2))],
[(d+e*x)^2/(x*(d^2-e^2*x^2)^(7/2)),x,7,2/5*(d+e*x)/(d*(d^2-e^2*x^2)^(5/2))+1/15*(5*d+8*e*x)/(d^3*(d^2-e^2*x^2)^(3/2))-arctanh(sqrt(d^2-e^2*x^2)/d)/d^5+1/15*(15*d+16*e*x)/(d^5*sqrt(d^2-e^2*x^2))],
[(d+e*x)^2/(x^2*(d^2-e^2*x^2)^(7/2)),x,7,2/5*e*(d+e*x)/(d^2*(d^2-e^2*x^2)^(5/2))+1/15*e*(10*d+13*e*x)/(d^4*(d^2-e^2*x^2)^(3/2))-2*e*arctanh(sqrt(d^2-e^2*x^2)/d)/d^6+1/15*e*(30*d+41*e*x)/(d^6*sqrt(d^2-e^2*x^2))-sqrt(d^2-e^2*x^2)/(d^6*x)],
[(d+e*x)^2/(x^3*(d^2-e^2*x^2)^(7/2)),x,8,2/5*e^2*(d+e*x)/(d^3*(d^2-e^2*x^2)^(5/2))+1/5*e^2*(5*d+6*e*x)/(d^5*(d^2-e^2*x^2)^(3/2))-9/2*e^2*arctanh(sqrt(d^2-e^2*x^2)/d)/d^7+2/5*e^2*(10*d+11*e*x)/(d^7*sqrt(d^2-e^2*x^2))-1/2*sqrt(d^2-e^2*x^2)/(d^6*x^2)-2*e*sqrt(d^2-e^2*x^2)/(d^7*x)],
[(d+e*x)^2/(x^4*(d^2-e^2*x^2)^(7/2)),x,9,2/5*e^3*(d+e*x)/(d^4*(d^2-e^2*x^2)^(5/2))+1/15*e^3*(20*d+23*e*x)/(d^6*(d^2-e^2*x^2)^(3/2))-7*e^3*arctanh(sqrt(d^2-e^2*x^2)/d)/d^8+2/15*e^3*(45*d+53*e*x)/(d^8*sqrt(d^2-e^2*x^2))-1/3*sqrt(d^2-e^2*x^2)/(d^6*x^3)-e*sqrt(d^2-e^2*x^2)/(d^7*x^2)-14/3*e^2*sqrt(d^2-e^2*x^2)/(d^8*x)],
[x^3*(1+x)^2/sqrt(1-x^2),x,5,3/4*arcsin(x)-3/5*x^2*sqrt(1-x^2)-1/2*x^3*sqrt(1-x^2)-1/5*x^4*sqrt(1-x^2)-3/20*(8+5*x)*sqrt(1-x^2)],
[x^2*(1+x)^2/sqrt(1-x^2),x,4,7/8*arcsin(x)-2/3*x^2*sqrt(1-x^2)-1/4*x^3*sqrt(1-x^2)-1/24*(32+21*x)*sqrt(1-x^2)],
[x*(1+x)^2/sqrt(1-x^2),x,3,arcsin(x)-1/3*x^2*sqrt(1-x^2)-1/3*(5+3*x)*sqrt(1-x^2)],
[(1+x)^2/sqrt(1-x^2),x,3,3/2*arcsin(x)-3/2*sqrt(1-x^2)-1/2*(1+x)*sqrt(1-x^2)],
[(1+x)^2/(x*sqrt(1-x^2)),x,6,2*arcsin(x)-arctanh(sqrt(1-x^2))-sqrt(1-x^2)],
[(1+x)^2/(x^2*sqrt(1-x^2)),x,6,arcsin(x)-2*arctanh(sqrt(1-x^2))-sqrt(1-x^2)/x],
[(1+x)^2/(x^3*sqrt(1-x^2)),x,5,-3/2*arctanh(sqrt(1-x^2))-1/2*sqrt(1-x^2)/x^2-2*sqrt(1-x^2)/x],
[(1+x)^2/(x^4*sqrt(1-x^2)),x,6,-arctanh(sqrt(1-x^2))-1/3*sqrt(1-x^2)/x^3-sqrt(1-x^2)/x^2-5/3*sqrt(1-x^2)/x],
[(1+x)^2/(x^5*sqrt(1-x^2)),x,7,-7/8*arctanh(sqrt(1-x^2))-1/4*sqrt(1-x^2)/x^4-2/3*sqrt(1-x^2)/x^3-7/8*sqrt(1-x^2)/x^2-4/3*sqrt(1-x^2)/x],
[(1+x)^2/(x^6*sqrt(1-x^2)),x,8,-3/4*arctanh(sqrt(1-x^2))-1/5*sqrt(1-x^2)/x^5-1/2*sqrt(1-x^2)/x^4-3/5*sqrt(1-x^2)/x^3-3/4*sqrt(1-x^2)/x^2-6/5*sqrt(1-x^2)/x],

# Integrands of the form x^m (d+e x)^3 (d^2 - e^2 x^2)^(p/2)

# p>0
[(d+e*x)^3*sqrt(d^2-e^2*x^2)/x^5,x,9,-1/4*d*(d^2-e^2*x^2)^(3/2)/x^4-e*(d^2-e^2*x^2)^(3/2)/x^3-e^4*arctan(e*x/sqrt(d^2-e^2*x^2))+13/8*e^4*arctanh(sqrt(d^2-e^2*x^2)/d)-1/8*e^2*(13*d+8*e*x)*sqrt(d^2-e^2*x^2)/x^2],
[x^5*(d+e*x)^3*(d^2-e^2*x^2)^(5/2),x,12,35/3072*d^10*x*(d^2-e^2*x^2)^(3/2)/e^5+7/768*d^8*x*(d^2-e^2*x^2)^(5/2)/e^5-124/1287*d^5*x^2*(d^2-e^2*x^2)^(7/2)/e^4-7/48*d^4*x^3*(d^2-e^2*x^2)^(7/2)/e^3-31/143*d^3*x^4*(d^2-e^2*x^2)^(7/2)/e^2-7/24*d^2*x^5*(d^2-e^2*x^2)^(7/2)/e-3/13*d*x^6*(d^2-e^2*x^2)^(7/2)-1/14*e*x^7*(d^2-e^2*x^2)^(7/2)-1/1153152*d^6*(31744*d+63063*e*x)*(d^2-e^2*x^2)^(7/2)/e^6+35/2048*d^14*arctan(e*x/sqrt(d^2-e^2*x^2))/e^6+35/2048*d^12*x*sqrt(d^2-e^2*x^2)/e^5],
[x^4*(d+e*x)^3*(d^2-e^2*x^2)^(5/2),x,11,9/512*d^9*x*(d^2-e^2*x^2)^(3/2)/e^4+9/640*d^7*x*(d^2-e^2*x^2)^(5/2)/e^4-20/143*d^4*x^2*(d^2-e^2*x^2)^(7/2)/e^3-9/40*d^3*x^3*(d^2-e^2*x^2)^(7/2)/e^2-45/143*d^2*x^4*(d^2-e^2*x^2)^(7/2)/e-1/4*d*x^5*(d^2-e^2*x^2)^(7/2)-1/13*e*x^6*(d^2-e^2*x^2)^(7/2)-1/320320*d^5*(12800*d+27027*e*x)*(d^2-e^2*x^2)^(7/2)/e^5+27/1024*d^13*arctan(e*x/sqrt(d^2-e^2*x^2))/e^5+27/1024*d^11*x*sqrt(d^2-e^2*x^2)/e^4],
[x^3*(d+e*x)^3*(d^2-e^2*x^2)^(5/2),x,10,41/1536*d^8*x*(d^2-e^2*x^2)^(3/2)/e^3+41/1920*d^6*x*(d^2-e^2*x^2)^(5/2)/e^3-23/99*d^3*x^2*(d^2-e^2*x^2)^(7/2)/e^2-41/120*d^2*x^3*(d^2-e^2*x^2)^(7/2)/e-3/11*d*x^4*(d^2-e^2*x^2)^(7/2)-1/12*e*x^5*(d^2-e^2*x^2)^(7/2)-1/221760*d^4*(14720*d+28413*e*x)*(d^2-e^2*x^2)^(7/2)/e^4+41/1024*d^12*arctan(e*x/sqrt(d^2-e^2*x^2))/e^4+41/1024*d^10*x*sqrt(d^2-e^2*x^2)/e^3],
[x^2*(d+e*x)^3*(d^2-e^2*x^2)^(5/2),x,9,19/384*d^7*x*(d^2-e^2*x^2)^(3/2)/e^2+19/480*d^5*x*(d^2-e^2*x^2)^(5/2)/e^2-37/99*d^2*x^2*(d^2-e^2*x^2)^(7/2)/e-3/10*d*x^3*(d^2-e^2*x^2)^(7/2)-1/11*e*x^4*(d^2-e^2*x^2)^(7/2)-1/55440*d^3*(5920*d+13167*e*x)*(d^2-e^2*x^2)^(7/2)/e^3+19/256*d^11*arctan(e*x/sqrt(d^2-e^2*x^2))/e^3+19/256*d^9*x*sqrt(d^2-e^2*x^2)/e^2],
[x*(d+e*x)^3*(d^2-e^2*x^2)^(5/2),x,9,11/128*d^6*x*(d^2-e^2*x^2)^(3/2)/e+11/160*d^4*x*(d^2-e^2*x^2)^(5/2)/e-33/560*d^3*(d^2-e^2*x^2)^(7/2)/e^2-11/240*d^2*(d+e*x)*(d^2-e^2*x^2)^(7/2)/e^2-1/30*d*(d+e*x)^2*(d^2-e^2*x^2)^(7/2)/e^2-1/10*(d+e*x)^3*(d^2-e^2*x^2)^(7/2)/e^2+33/256*d^10*arctan(e*x/sqrt(d^2-e^2*x^2))/e^2+33/256*d^8*x*sqrt(d^2-e^2*x^2)/e],
[(d+e*x)^3*(d^2-e^2*x^2)^(5/2),x,8,55/192*d^5*x*(d^2-e^2*x^2)^(3/2)+11/48*d^3*x*(d^2-e^2*x^2)^(5/2)-11/56*d^2*(d^2-e^2*x^2)^(7/2)/e-11/72*d*(d+e*x)*(d^2-e^2*x^2)^(7/2)/e-1/9*(d+e*x)^2*(d^2-e^2*x^2)^(7/2)/e+55/128*d^9*arctan(e*x/sqrt(d^2-e^2*x^2))/e+55/128*d^7*x*sqrt(d^2-e^2*x^2)],
[(d+e*x)^3*(d^2-e^2*x^2)^(5/2)/x,x,11,1/192*d^4*(64*d+125*e*x)*(d^2-e^2*x^2)^(3/2)+1/240*d^2*(48*d+125*e*x)*(d^2-e^2*x^2)^(5/2)-3/7*d*(d^2-e^2*x^2)^(7/2)-1/8*e*x*(d^2-e^2*x^2)^(7/2)+125/128*d^8*arctan(e*x/sqrt(d^2-e^2*x^2))-d^8*arctanh(sqrt(d^2-e^2*x^2)/d)+1/128*d^6*(128*d+125*e*x)*sqrt(d^2-e^2*x^2)],
[(d+e*x)^3*(d^2-e^2*x^2)^(5/2)/x^2,x,11,1/8*d^3*e*(8*d-5*e*x)*(d^2-e^2*x^2)^(3/2)+1/10*d*e*(6*d-5*e*x)*(d^2-e^2*x^2)^(5/2)-1/7*e*(d^2-e^2*x^2)^(7/2)-d*(d^2-e^2*x^2)^(7/2)/x-15/16*d^7*e*arctan(e*x/sqrt(d^2-e^2*x^2))-3*d^7*e*arctanh(sqrt(d^2-e^2*x^2)/d)+3/16*d^5*e*(16*d-5*e*x)*sqrt(d^2-e^2*x^2)],
[(d+e*x)^3*(d^2-e^2*x^2)^(5/2)/x^3,x,11,1/24*d^2*e^2*(4*d-85*e*x)*(d^2-e^2*x^2)^(3/2)+1/30*e^2*(3*d-85*e*x)*(d^2-e^2*x^2)^(5/2)-1/2*d*(d^2-e^2*x^2)^(7/2)/x^2-3*e*(d^2-e^2*x^2)^(7/2)/x-85/16*d^6*e^2*arctan(e*x/sqrt(d^2-e^2*x^2))-1/2*d^6*e^2*arctanh(sqrt(d^2-e^2*x^2)/d)+1/16*d^4*e^2*(8*d-85*e*x)*sqrt(d^2-e^2*x^2)],
[(d+e*x)^3*(d^2-e^2*x^2)^(5/2)/x^4,x,11,-1/12*d*e^3*(26*d+25*e*x)*(d^2-e^2*x^2)^(3/2)-1/30*e^2*(50*d+39*e*x)*(d^2-e^2*x^2)^(5/2)/x-1/3*d*(d^2-e^2*x^2)^(7/2)/x^3-3/2*e*(d^2-e^2*x^2)^(7/2)/x^2-25/8*d^5*e^3*arctan(e*x/sqrt(d^2-e^2*x^2))+13/2*d^5*e^3*arctanh(sqrt(d^2-e^2*x^2)/d)-1/8*d^3*e^3*(52*d+25*e*x)*sqrt(d^2-e^2*x^2)],
[(d+e*x)^3*(d^2-e^2*x^2)^(5/2)/x^5,x,11,15/8*d*e^3*(2*d-e*x)*(d^2-e^2*x^2)^(3/2)/x-3/8*e^2*(3*d+2*e*x)*(d^2-e^2*x^2)^(5/2)/x^2-1/4*d*(d^2-e^2*x^2)^(7/2)/x^4-e*(d^2-e^2*x^2)^(7/2)/x^3+45/8*d^4*e^4*arctan(e*x/sqrt(d^2-e^2*x^2))+45/8*d^4*e^4*arctanh(sqrt(d^2-e^2*x^2)/d)-45/8*d^2*e^4*(d-e*x)*sqrt(d^2-e^2*x^2)],
[(d+e*x)^3*(d^2-e^2*x^2)^(5/2)/x^6,x,11,1/24*d*e^3*(25*d-52*e*x)*(d^2-e^2*x^2)^(3/2)/x^2-1/60*e^2*(52*d+25*e*x)*(d^2-e^2*x^2)^(5/2)/x^3-1/5*d*(d^2-e^2*x^2)^(7/2)/x^5-3/4*e*(d^2-e^2*x^2)^(7/2)/x^4+13/2*d^3*e^5*arctan(e*x/sqrt(d^2-e^2*x^2))-25/8*d^3*e^5*arctanh(sqrt(d^2-e^2*x^2)/d)+1/8*d^2*e^4*(52*d+25*e*x)*sqrt(d^2-e^2*x^2)/x],
[(d+e*x)^3*(d^2-e^2*x^2)^(5/2)/x^7,x,11,1/48*d*e^3*(8*d+85*e*x)*(d^2-e^2*x^2)^(3/2)/x^3-1/120*e^2*(85*d+12*e*x)*(d^2-e^2*x^2)^(5/2)/x^4-1/6*d*(d^2-e^2*x^2)^(7/2)/x^6-3/5*e*(d^2-e^2*x^2)^(7/2)/x^5-1/2*d^2*e^6*arctan(e*x/sqrt(d^2-e^2*x^2))-85/16*d^2*e^6*arctanh(sqrt(d^2-e^2*x^2)/d)-1/16*d*e^5*(8*d-85*e*x)*sqrt(d^2-e^2*x^2)/x],
[(d+e*x)^3*(d^2-e^2*x^2)^(5/2)/x^8,x,11,1/16*e^4*(16*d+5*e*x)*(d^2-e^2*x^2)^(3/2)/x^3-1/40*e^2*(24*d+5*e*x)*(d^2-e^2*x^2)^(5/2)/x^5-1/7*d*(d^2-e^2*x^2)^(7/2)/x^7-1/2*e*(d^2-e^2*x^2)^(7/2)/x^6-3*d*e^7*arctan(e*x/sqrt(d^2-e^2*x^2))-15/16*d*e^7*arctanh(sqrt(d^2-e^2*x^2)/d)-3/16*e^6*(16*d-5*e*x)*sqrt(d^2-e^2*x^2)/x],
[(d+e*x)^3*(d^2-e^2*x^2)^(5/2)/x^9,x,11,1/192*e^4*(125*d+64*e*x)*(d^2-e^2*x^2)^(3/2)/x^4-1/240*e^2*(125*d+48*e*x)*(d^2-e^2*x^2)^(5/2)/x^6-1/8*d*(d^2-e^2*x^2)^(7/2)/x^8-3/7*e*(d^2-e^2*x^2)^(7/2)/x^7-e^8*arctan(e*x/sqrt(d^2-e^2*x^2))+125/128*e^8*arctanh(sqrt(d^2-e^2*x^2)/d)-1/128*e^6*(125*d+128*e*x)*sqrt(d^2-e^2*x^2)/x^2],
[(d+e*x)^3*(d^2-e^2*x^2)^(5/2)/x^10,x,9,55/192*e^5*(d^2-e^2*x^2)^(3/2)/x^4-11/48*e^3*(d^2-e^2*x^2)^(5/2)/x^6-1/9*d*(d^2-e^2*x^2)^(7/2)/x^9-3/8*e*(d^2-e^2*x^2)^(7/2)/x^8-29/63*e^2*(d^2-e^2*x^2)^(7/2)/(d*x^7)+55/128*e^9*arctanh(sqrt(d^2-e^2*x^2)/d)/d-55/128*e^7*sqrt(d^2-e^2*x^2)/x^2],
[(d+e*x)^3*(d^2-e^2*x^2)^(5/2)/x^11,x,10,11/128*e^6*(d^2-e^2*x^2)^(3/2)/(d*x^4)-11/160*e^4*(d^2-e^2*x^2)^(5/2)/(d*x^6)-1/10*d*(d^2-e^2*x^2)^(7/2)/x^10-1/3*e*(d^2-e^2*x^2)^(7/2)/x^9-33/80*e^2*(d^2-e^2*x^2)^(7/2)/(d*x^8)-5/21*e^3*(d^2-e^2*x^2)^(7/2)/(d^2*x^7)+33/256*e^10*arctanh(sqrt(d^2-e^2*x^2)/d)/d^2-33/256*e^8*sqrt(d^2-e^2*x^2)/(d*x^2)],
[(d+e*x)^3*(d^2-e^2*x^2)^(5/2)/x^12,x,11,19/384*e^7*(d^2-e^2*x^2)^(3/2)/(d^2*x^4)-19/480*e^5*(d^2-e^2*x^2)^(5/2)/(d^2*x^6)-1/11*d*(d^2-e^2*x^2)^(7/2)/x^11-3/10*e*(d^2-e^2*x^2)^(7/2)/x^10-37/99*e^2*(d^2-e^2*x^2)^(7/2)/(d*x^9)-19/80*e^3*(d^2-e^2*x^2)^(7/2)/(d^2*x^8)-74/693*e^4*(d^2-e^2*x^2)^(7/2)/(d^3*x^7)+19/256*e^11*arctanh(sqrt(d^2-e^2*x^2)/d)/d^3-19/256*e^9*sqrt(d^2-e^2*x^2)/(d^2*x^2)],

# p<0
[x^5*(d+e*x)^3/(d^2-e^2*x^2)^(7/2),x,7,1/5*d^4*(d+e*x)^3/(e^6*(d^2-e^2*x^2)^(5/2))-23/15*d^3*(d+e*x)^2/(e^6*(d^2-e^2*x^2)^(3/2))-13/2*d^2*arctan(e*x/sqrt(d^2-e^2*x^2))/e^6+127/15*d^2*(d+e*x)/(e^6*sqrt(d^2-e^2*x^2))+3*d*sqrt(d^2-e^2*x^2)/e^6+1/2*x*sqrt(d^2-e^2*x^2)/e^5],
[x^4*(d+e*x)^3/(d^2-e^2*x^2)^(7/2),x,6,1/5*d^3*(d+e*x)^3/(e^5*(d^2-e^2*x^2)^(5/2))-6/5*d^2*(d+e*x)^2/(e^5*(d^2-e^2*x^2)^(3/2))-3*d*arctan(e*x/sqrt(d^2-e^2*x^2))/e^5+24/5*d*(d+e*x)/(e^5*sqrt(d^2-e^2*x^2))+sqrt(d^2-e^2*x^2)/e^5],
[x^3*(d+e*x)^3/(d^2-e^2*x^2)^(7/2),x,5,1/5*d^2*(d+e*x)^3/(e^4*(d^2-e^2*x^2)^(5/2))-13/15*d*(d+e*x)^2/(e^4*(d^2-e^2*x^2)^(3/2))-arctan(e*x/sqrt(d^2-e^2*x^2))/e^4+32/15*(d+e*x)/(e^4*sqrt(d^2-e^2*x^2))],
[x^2*(d+e*x)^3/(d^2-e^2*x^2)^(7/2),x,3,1/5*d*(d+e*x)^3/(e^3*(d^2-e^2*x^2)^(5/2))-8/15*(d+e*x)^2/(e^3*(d^2-e^2*x^2)^(3/2))+7/15*(d+e*x)/(d*e^3*sqrt(d^2-e^2*x^2))],
[x*(d+e*x)^3/(d^2-e^2*x^2)^(7/2),x,3,1/5*(d+e*x)^3/(e^2*(d^2-e^2*x^2)^(5/2))-2/5*(d+e*x)/(e^2*(d^2-e^2*x^2)^(3/2))-1/5*x/(d^2*e*sqrt(d^2-e^2*x^2))],
[(d+e*x)^3/(d^2-e^2*x^2)^(7/2),x,4,1/5*sqrt(d^2-e^2*x^2)/(d*e*(d-e*x)^3)+2/15*sqrt(d^2-e^2*x^2)/(d^2*e*(d-e*x)^2)+2/15*sqrt(d^2-e^2*x^2)/(d^3*e*(d-e*x))],
[(d+e*x)^3/(x*(d^2-e^2*x^2)^(7/2)),x,7,4/5*(d+e*x)/(d^2-e^2*x^2)^(5/2)+1/15*(5*d+11*e*x)/(d^2*(d^2-e^2*x^2)^(3/2))-arctanh(sqrt(d^2-e^2*x^2)/d)/d^4+1/15*(15*d+22*e*x)/(d^4*sqrt(d^2-e^2*x^2))],
[(d+e*x)^3/(x^2*(d^2-e^2*x^2)^(7/2)),x,7,4/5*e*(d+e*x)/(d*(d^2-e^2*x^2)^(5/2))+1/5*e*(5*d+7*e*x)/(d^3*(d^2-e^2*x^2)^(3/2))-3*e*arctanh(sqrt(d^2-e^2*x^2)/d)/d^5+1/5*e*(15*d+19*e*x)/(d^5*sqrt(d^2-e^2*x^2))-sqrt(d^2-e^2*x^2)/(d^5*x)],
[(d+e*x)^3/(x^3*(d^2-e^2*x^2)^(7/2)),x,8,4/5*e^2*(d+e*x)/(d^2*(d^2-e^2*x^2)^(5/2))+1/15*e^2*(25*d+31*e*x)/(d^4*(d^2-e^2*x^2)^(3/2))-13/2*e^2*arctanh(sqrt(d^2-e^2*x^2)/d)/d^6+1/15*e^2*(90*d+107*e*x)/(d^6*sqrt(d^2-e^2*x^2))-1/2*sqrt(d^2-e^2*x^2)/(d^5*x^2)-3*e*sqrt(d^2-e^2*x^2)/(d^6*x)],

# Integrands of the form x^m / (d+e x)^1 (d^2 - e^2 x^2)^(p/2)

# p>0
[x^4*sqrt(d^2-e^2*x^2)/(d+e*x),x,7,3/8*d^5*arctan(e*x/sqrt(d^2-e^2*x^2))/e^5+4/15*d^2*x^2*sqrt(d^2-e^2*x^2)/e^3-1/4*d*x^3*sqrt(d^2-e^2*x^2)/e^2+1/5*x^4*sqrt(d^2-e^2*x^2)/e+1/120*d^3*(64*d-45*e*x)*sqrt(d^2-e^2*x^2)/e^5],
[x^3*sqrt(d^2-e^2*x^2)/(d+e*x),x,6,-3/8*d^4*arctan(e*x/sqrt(d^2-e^2*x^2))/e^4-1/3*d*x^2*sqrt(d^2-e^2*x^2)/e^2+1/4*x^3*sqrt(d^2-e^2*x^2)/e-1/24*d^2*(16*d-9*e*x)*sqrt(d^2-e^2*x^2)/e^4],
[x^2*sqrt(d^2-e^2*x^2)/(d+e*x),x,6,-1/3*(d^2-e^2*x^2)^(3/2)/e^3+1/2*d^3*arctan(e*x/sqrt(d^2-e^2*x^2))/e^3+1/2*d*(2*d-e*x)*sqrt(d^2-e^2*x^2)/e^3],
[x*sqrt(d^2-e^2*x^2)/(d+e*x),x,4,-1/2*d^2*arctan(e*x/sqrt(d^2-e^2*x^2))/e^2-1/2*(2*d-e*x)*sqrt(d^2-e^2*x^2)/e^2],
[sqrt(d^2-e^2*x^2)/(d+e*x),x,3,d*arctan(e*x/sqrt(d^2-e^2*x^2))/e+sqrt(d^2-e^2*x^2)/e],
[sqrt(d^2-e^2*x^2)/(x*(d+e*x)),x,7,-arctan(e*x/sqrt(d^2-e^2*x^2))-arctanh(sqrt(d^2-e^2*x^2)/d)],
[sqrt(d^2-e^2*x^2)/(x^2*(d+e*x)),x,5,e*arctanh(sqrt(d^2-e^2*x^2)/d)/d-sqrt(d^2-e^2*x^2)/(d*x)],
[sqrt(d^2-e^2*x^2)/(x^3*(d+e*x)),x,6,-1/2*e^2*arctanh(sqrt(d^2-e^2*x^2)/d)/d^2-1/2*sqrt(d^2-e^2*x^2)/(d*x^2)+e*sqrt(d^2-e^2*x^2)/(d^2*x)],
[sqrt(d^2-e^2*x^2)/(x^4*(d+e*x)),x,7,1/2*e^3*arctanh(sqrt(d^2-e^2*x^2)/d)/d^3-1/3*sqrt(d^2-e^2*x^2)/(d*x^3)+1/2*e*sqrt(d^2-e^2*x^2)/(d^2*x^2)-2/3*e^2*sqrt(d^2-e^2*x^2)/(d^3*x)],
[sqrt(d^2-e^2*x^2)/(x^5*(d+e*x)),x,8,-3/8*e^4*arctanh(sqrt(d^2-e^2*x^2)/d)/d^4-1/4*sqrt(d^2-e^2*x^2)/(d*x^4)+1/3*e*sqrt(d^2-e^2*x^2)/(d^2*x^3)-3/8*e^2*sqrt(d^2-e^2*x^2)/(d^3*x^2)+2/3*e^3*sqrt(d^2-e^2*x^2)/(d^4*x)],
[x^2*(d^2-e^2*x^2)^(3/2)/(d+e*x),x,7,1/12*d*(4*d-3*e*x)*(d^2-e^2*x^2)^(3/2)/e^3-1/5*(d^2-e^2*x^2)^(5/2)/e^3+1/8*d^5*arctan(e*x/sqrt(d^2-e^2*x^2))/e^3+1/8*d^3*x*sqrt(d^2-e^2*x^2)/e^2],
[x^4*(d^2-e^2*x^2)^(5/2)/(d+e*x),x,9,1/64*d^5*x*(d^2-e^2*x^2)^(3/2)/e^4+4/63*d^2*x^2*(d^2-e^2*x^2)^(5/2)/e^3-1/8*d*x^3*(d^2-e^2*x^2)^(5/2)/e^2+1/9*x^4*(d^2-e^2*x^2)^(5/2)/e+1/5040*d^3*(128*d-315*e*x)*(d^2-e^2*x^2)^(5/2)/e^5+3/128*d^9*arctan(e*x/sqrt(d^2-e^2*x^2))/e^5+3/128*d^7*x*sqrt(d^2-e^2*x^2)/e^4],
[x^3*(d^2-e^2*x^2)^(5/2)/(d+e*x),x,8,-1/64*d^4*x*(d^2-e^2*x^2)^(3/2)/e^3-1/7*d*x^2*(d^2-e^2*x^2)^(5/2)/e^2+1/8*x^3*(d^2-e^2*x^2)^(5/2)/e-1/560*d^2*(32*d-35*e*x)*(d^2-e^2*x^2)^(5/2)/e^4-3/128*d^8*arctan(e*x/sqrt(d^2-e^2*x^2))/e^4-3/128*d^6*x*sqrt(d^2-e^2*x^2)/e^3],
[x^2*(d^2-e^2*x^2)^(5/2)/(d+e*x),x,8,1/24*d^3*x*(d^2-e^2*x^2)^(3/2)/e^2+1/30*d*(6*d-5*e*x)*(d^2-e^2*x^2)^(5/2)/e^3-1/7*(d^2-e^2*x^2)^(7/2)/e^3+1/16*d^7*arctan(e*x/sqrt(d^2-e^2*x^2))/e^3+1/16*d^5*x*sqrt(d^2-e^2*x^2)/e^2],
[x*(d^2-e^2*x^2)^(5/2)/(d+e*x),x,6,-1/24*d^2*x*(d^2-e^2*x^2)^(3/2)/e-1/30*(6*d-5*e*x)*(d^2-e^2*x^2)^(5/2)/e^2-1/16*d^6*arctan(e*x/sqrt(d^2-e^2*x^2))/e^2-1/16*d^4*x*sqrt(d^2-e^2*x^2)/e],
[(d^2-e^2*x^2)^(5/2)/(d+e*x),x,5,1/4*d*x*(d^2-e^2*x^2)^(3/2)+1/5*(d^2-e^2*x^2)^(5/2)/e+3/8*d^5*arctan(e*x/sqrt(d^2-e^2*x^2))/e+3/8*d^3*x*sqrt(d^2-e^2*x^2)],
[(d^2-e^2*x^2)^(5/2)/(x*(d+e*x)),x,9,1/12*(4*d-3*e*x)*(d^2-e^2*x^2)^(3/2)-3/8*d^4*arctan(e*x/sqrt(d^2-e^2*x^2))-d^4*arctanh(sqrt(d^2-e^2*x^2)/d)+1/8*d^2*(8*d-3*e*x)*sqrt(d^2-e^2*x^2)],
[(d^2-e^2*x^2)^(5/2)/(x^2*(d+e*x)),x,9,-1/3*(3*d+e*x)*(d^2-e^2*x^2)^(3/2)/x-3/2*d^3*e*arctan(e*x/sqrt(d^2-e^2*x^2))+d^3*e*arctanh(sqrt(d^2-e^2*x^2)/d)-1/2*d*e*(2*d+3*e*x)*sqrt(d^2-e^2*x^2)],
[(d^2-e^2*x^2)^(5/2)/(x^3*(d+e*x)),x,9,-1/2*(d+e*x)*(d^2-e^2*x^2)^(3/2)/x^2+3/2*d^2*e^2*arctan(e*x/sqrt(d^2-e^2*x^2))+3/2*d^2*e^2*arctanh(sqrt(d^2-e^2*x^2)/d)+3/2*d*e*(d-e*x)*sqrt(d^2-e^2*x^2)/x],
[(d^2-e^2*x^2)^(5/2)/(x^4*(d+e*x)),x,9,-1/6*(2*d-3*e*x)*(d^2-e^2*x^2)^(3/2)/x^3+d*e^3*arctan(e*x/sqrt(d^2-e^2*x^2))-3/2*d*e^3*arctanh(sqrt(d^2-e^2*x^2)/d)+1/2*e^2*(2*d+3*e*x)*sqrt(d^2-e^2*x^2)/x],
[(d^2-e^2*x^2)^(5/2)/(x^5*(d+e*x)),x,9,-1/12*(3*d-4*e*x)*(d^2-e^2*x^2)^(3/2)/x^4-e^4*arctan(e*x/sqrt(d^2-e^2*x^2))-3/8*e^4*arctanh(sqrt(d^2-e^2*x^2)/d)+1/8*e^2*(3*d-8*e*x)*sqrt(d^2-e^2*x^2)/x^2],
[(d^2-e^2*x^2)^(5/2)/(x^6*(d+e*x)),x,7,1/4*e*(d^2-e^2*x^2)^(3/2)/x^4-1/5*(d^2-e^2*x^2)^(5/2)/(d*x^5)+3/8*e^5*arctanh(sqrt(d^2-e^2*x^2)/d)/d-3/8*e^3*sqrt(d^2-e^2*x^2)/x^2],
[(d^2-e^2*x^2)^(5/2)/(x^7*(d+e*x)),x,8,-1/24*e^2*(d^2-e^2*x^2)^(3/2)/(d*x^4)-1/6*(d^2-e^2*x^2)^(5/2)/(d*x^6)+1/5*e*(d^2-e^2*x^2)^(5/2)/(d^2*x^5)-1/16*e^6*arctanh(sqrt(d^2-e^2*x^2)/d)/d^2+1/16*e^4*sqrt(d^2-e^2*x^2)/(d*x^2)],
[(d^2-e^2*x^2)^(5/2)/(x^8*(d+e*x)),x,9,1/24*e^3*(d^2-e^2*x^2)^(3/2)/(d^2*x^4)-1/7*(d^2-e^2*x^2)^(5/2)/(d*x^7)+1/6*e*(d^2-e^2*x^2)^(5/2)/(d^2*x^6)-2/35*e^2*(d^2-e^2*x^2)^(5/2)/(d^3*x^5)+1/16*e^7*arctanh(sqrt(d^2-e^2*x^2)/d)/d^3-1/16*e^5*sqrt(d^2-e^2*x^2)/(d^2*x^2)],
[(d^2-e^2*x^2)^(5/2)/(x^9*(d+e*x)),x,10,-1/64*e^4*(d^2-e^2*x^2)^(3/2)/(d^3*x^4)-1/8*(d^2-e^2*x^2)^(5/2)/(d*x^8)+1/7*e*(d^2-e^2*x^2)^(5/2)/(d^2*x^7)-1/16*e^2*(d^2-e^2*x^2)^(5/2)/(d^3*x^6)+2/35*e^3*(d^2-e^2*x^2)^(5/2)/(d^4*x^5)-3/128*e^8*arctanh(sqrt(d^2-e^2*x^2)/d)/d^4+3/128*e^6*sqrt(d^2-e^2*x^2)/(d^3*x^2)],
[x*sqrt(1-x^2)/(1+x),x,3,-1/2*arcsin(x)-1/2*(2-x)*sqrt(1-x^2)],
[(1-a^2*x^2)^(3/2)/(x^2*(1-a*x)),x,7,-a*arcsin(a*x)-a*arctanh(sqrt(1-a^2*x^2))-(1-a*x)*sqrt(1-a^2*x^2)/x],

# p<0
[x^4/((d+e*x)*sqrt(d^2-e^2*x^2)),x,6,-3/2*d^3*arctan(e*x/sqrt(d^2-e^2*x^2))/e^5+x^3*(d-e*x)/(e^2*sqrt(d^2-e^2*x^2))-4/3*x^2*sqrt(d^2-e^2*x^2)/e^3-1/6*d*(16*d-9*e*x)*sqrt(d^2-e^2*x^2)/e^5],
[x^3/((d+e*x)*sqrt(d^2-e^2*x^2)),x,5,3/2*d^2*arctan(e*x/sqrt(d^2-e^2*x^2))/e^4+x^2*(d-e*x)/(e^2*sqrt(d^2-e^2*x^2))+1/2*(4*d-3*e*x)*sqrt(d^2-e^2*x^2)/e^4],
[x^2/((d+e*x)*sqrt(d^2-e^2*x^2)),x,5,-d*arctan(e*x/sqrt(d^2-e^2*x^2))/e^3-sqrt(d^2-e^2*x^2)/e^3-d*sqrt(d^2-e^2*x^2)/(e^3*(d+e*x))],
[x/((d+e*x)*sqrt(d^2-e^2*x^2)),x,3,arctan(e*x/sqrt(d^2-e^2*x^2))/e^2+sqrt(d^2-e^2*x^2)/(e^2*(d+e*x))],
[1/((d+e*x)*sqrt(d^2-e^2*x^2)),x,1,-sqrt(d^2-e^2*x^2)/(d*e*(d+e*x))],
[1/(x*(d+e*x)*sqrt(d^2-e^2*x^2)),x,5,-arctanh(sqrt(d^2-e^2*x^2)/d)/d^2+sqrt(d^2-e^2*x^2)/(d^2*(d+e*x))],
[1/(x^2*(d+e*x)*sqrt(d^2-e^2*x^2)),x,5,e*arctanh(sqrt(d^2-e^2*x^2)/d)/d^3-2*sqrt(d^2-e^2*x^2)/(d^3*x)+sqrt(d^2-e^2*x^2)/(d^2*x*(d+e*x))],
[1/(x^3*(d+e*x)*sqrt(d^2-e^2*x^2)),x,6,-3/2*e^2*arctanh(sqrt(d^2-e^2*x^2)/d)/d^4-3/2*sqrt(d^2-e^2*x^2)/(d^3*x^2)+2*e*sqrt(d^2-e^2*x^2)/(d^4*x)+sqrt(d^2-e^2*x^2)/(d^2*x^2*(d+e*x))],
[x^5/((d+e*x)*(d^2-e^2*x^2)^(3/2)),x,6,1/3*x^4*(d-e*x)/(e^2*(d^2-e^2*x^2)^(3/2))-5/2*d^2*arctan(e*x/sqrt(d^2-e^2*x^2))/e^6-1/3*x^2*(4*d-5*e*x)/(e^4*sqrt(d^2-e^2*x^2))-1/6*(16*d-15*e*x)*sqrt(d^2-e^2*x^2)/e^6],
[x^4/((d+e*x)*(d^2-e^2*x^2)^(3/2)),x,6,1/3*x^3*(d-e*x)/(e^2*(d^2-e^2*x^2)^(3/2))+d*arctan(e*x/sqrt(d^2-e^2*x^2))/e^5-1/3*x*(3*d-4*e*x)/(e^4*sqrt(d^2-e^2*x^2))+8/3*sqrt(d^2-e^2*x^2)/e^5],
[x^3/((d+e*x)*(d^2-e^2*x^2)^(3/2)),x,5,1/3*x^2*(d-e*x)/(e^2*(d^2-e^2*x^2)^(3/2))-arctan(e*x/sqrt(d^2-e^2*x^2))/e^4+1/3*(-2*d+3*e*x)/(e^4*sqrt(d^2-e^2*x^2))],
[x^2/((d+e*x)*(d^2-e^2*x^2)^(3/2)),x,3,2/3/(e^3*sqrt(d^2-e^2*x^2))-1/3*x^2/(d*e*(d+e*x)*sqrt(d^2-e^2*x^2))],
[x/((d+e*x)*(d^2-e^2*x^2)^(3/2)),x,2,1/3*x/(d^2*e*sqrt(d^2-e^2*x^2))+1/3/(e^2*(d+e*x)*sqrt(d^2-e^2*x^2))],
[1/((d+e*x)*(d^2-e^2*x^2)^(3/2)),x,2,2/3*x/(d^3*sqrt(d^2-e^2*x^2))+(-1/3)/(d*e*(d+e*x)*sqrt(d^2-e^2*x^2))],
[1/(x*(d+e*x)*(d^2-e^2*x^2)^(3/2)),x,6,-arctanh(sqrt(d^2-e^2*x^2)/d)/d^4+1/3*(3*d-2*e*x)/(d^4*sqrt(d^2-e^2*x^2))+1/3/(d^2*(d+e*x)*sqrt(d^2-e^2*x^2))],
[1/(x^2*(d+e*x)*(d^2-e^2*x^2)^(3/2)),x,6,e*arctanh(sqrt(d^2-e^2*x^2)/d)/d^5+1/3*(4*d-3*e*x)/(d^4*x*sqrt(d^2-e^2*x^2))+1/3/(d^2*x*(d+e*x)*sqrt(d^2-e^2*x^2))-8/3*sqrt(d^2-e^2*x^2)/(d^5*x)],
[1/(x^3*(d+e*x)*(d^2-e^2*x^2)^(3/2)),x,7,-5/2*e^2*arctanh(sqrt(d^2-e^2*x^2)/d)/d^6+1/3*(5*d-4*e*x)/(d^4*x^2*sqrt(d^2-e^2*x^2))+1/3/(d^2*x^2*(d+e*x)*sqrt(d^2-e^2*x^2))-5/2*sqrt(d^2-e^2*x^2)/(d^5*x^2)+8/3*e*sqrt(d^2-e^2*x^2)/(d^6*x)],
[x^7/((d+e*x)*(d^2-e^2*x^2)^(5/2)),x,7,1/5*x^6*(d-e*x)/(e^2*(d^2-e^2*x^2)^(5/2))-1/15*x^4*(6*d-7*e*x)/(e^4*(d^2-e^2*x^2)^(3/2))+7/2*d^2*arctan(e*x/sqrt(d^2-e^2*x^2))/e^8+1/15*x^2*(24*d-35*e*x)/(e^6*sqrt(d^2-e^2*x^2))+1/10*(32*d-35*e*x)*sqrt(d^2-e^2*x^2)/e^8],
[x^6/((d+e*x)*(d^2-e^2*x^2)^(5/2)),x,7,1/5*x^5*(d-e*x)/(e^2*(d^2-e^2*x^2)^(5/2))-1/15*x^3*(5*d-6*e*x)/(e^4*(d^2-e^2*x^2)^(3/2))-d*arctan(e*x/sqrt(d^2-e^2*x^2))/e^7+1/5*x*(5*d-8*e*x)/(e^6*sqrt(d^2-e^2*x^2))-16/5*sqrt(d^2-e^2*x^2)/e^7],
[x^5/((d+e*x)*(d^2-e^2*x^2)^(5/2)),x,6,1/5*x^4*(d-e*x)/(e^2*(d^2-e^2*x^2)^(5/2))-1/15*x^2*(4*d-5*e*x)/(e^4*(d^2-e^2*x^2)^(3/2))+arctan(e*x/sqrt(d^2-e^2*x^2))/e^6+1/15*(8*d-15*e*x)/(e^6*sqrt(d^2-e^2*x^2))],
[x^4/((d+e*x)*(d^2-e^2*x^2)^(5/2)),x,5,-1/5*x^4*(d-e*x)/(d*e*(d^2-e^2*x^2)^(5/2))+4/15*d^2/(e^5*(d^2-e^2*x^2)^(3/2))+(-4/5)/(e^5*sqrt(d^2-e^2*x^2))],
[x^3/((d+e*x)*(d^2-e^2*x^2)^(5/2)),x,4,1/5*x^2*(d-e*x)/(e^2*(d^2-e^2*x^2)^(5/2))+1/15*(-2*d+3*e*x)/(e^4*(d^2-e^2*x^2)^(3/2))-1/5*x/(d^2*e^3*sqrt(d^2-e^2*x^2))],
[x^2/((d+e*x)*(d^2-e^2*x^2)^(5/2)),x,3,-1/5*x^2/(d*e*(d+e*x)*(d^2-e^2*x^2)^(3/2))+2/15*(d+e*x)/(d*e^3*(d^2-e^2*x^2)^(3/2))-2/15*x/(d^3*e^2*sqrt(d^2-e^2*x^2))],
[x/((d+e*x)*(d^2-e^2*x^2)^(5/2)),x,3,1/15*x/(d^2*e*(d^2-e^2*x^2)^(3/2))+1/5/(e^2*(d+e*x)*(d^2-e^2*x^2)^(3/2))+2/15*x/(d^4*e*sqrt(d^2-e^2*x^2))],
[1/((d+e*x)*(d^2-e^2*x^2)^(5/2)),x,3,4/15*x/(d^3*(d^2-e^2*x^2)^(3/2))+(-1/5)/(d*e*(d+e*x)*(d^2-e^2*x^2)^(3/2))+8/15*x/(d^5*sqrt(d^2-e^2*x^2))],
[1/(x*(d+e*x)*(d^2-e^2*x^2)^(5/2)),x,7,1/15*(5*d-4*e*x)/(d^4*(d^2-e^2*x^2)^(3/2))+1/5/(d^2*(d+e*x)*(d^2-e^2*x^2)^(3/2))-arctanh(sqrt(d^2-e^2*x^2)/d)/d^6+1/15*(15*d-8*e*x)/(d^6*sqrt(d^2-e^2*x^2))],
[1/(x^2*(d+e*x)*(d^2-e^2*x^2)^(5/2)),x,7,1/15*(6*d-5*e*x)/(d^4*x*(d^2-e^2*x^2)^(3/2))+1/5/(d^2*x*(d+e*x)*(d^2-e^2*x^2)^(3/2))+e*arctanh(sqrt(d^2-e^2*x^2)/d)/d^7+1/5*(8*d-5*e*x)/(d^6*x*sqrt(d^2-e^2*x^2))-16/5*sqrt(d^2-e^2*x^2)/(d^7*x)],
[1/(x^3*(d+e*x)*(d^2-e^2*x^2)^(5/2)),x,8,1/15*(7*d-6*e*x)/(d^4*x^2*(d^2-e^2*x^2)^(3/2))+1/5/(d^2*x^2*(d+e*x)*(d^2-e^2*x^2)^(3/2))-7/2*e^2*arctanh(sqrt(d^2-e^2*x^2)/d)/d^8+1/15*(35*d-24*e*x)/(d^6*x^2*sqrt(d^2-e^2*x^2))-7/2*sqrt(d^2-e^2*x^2)/(d^7*x^2)+16/5*e*sqrt(d^2-e^2*x^2)/(d^8*x)],
[1/(x^4*(d+e*x)*(d^2-e^2*x^2)^(5/2)),x,9,1/15*(8*d-7*e*x)/(d^4*x^3*(d^2-e^2*x^2)^(3/2))+1/5/(d^2*x^3*(d+e*x)*(d^2-e^2*x^2)^(3/2))+7/2*e^3*arctanh(sqrt(d^2-e^2*x^2)/d)/d^9+1/15*(48*d-35*e*x)/(d^6*x^3*sqrt(d^2-e^2*x^2))-64/15*sqrt(d^2-e^2*x^2)/(d^7*x^3)+7/2*e*sqrt(d^2-e^2*x^2)/(d^8*x^2)-128/15*e^2*sqrt(d^2-e^2*x^2)/(d^9*x)],
[x^3/((d+e*x)*(d^2-e^2*x^2)^(7/2)),x,5,1/7*x^2*(d-e*x)/(e^2*(d^2-e^2*x^2)^(7/2))+1/35*(-2*d+3*e*x)/(e^4*(d^2-e^2*x^2)^(5/2))-1/35*x/(d^2*e^3*(d^2-e^2*x^2)^(3/2))-2/35*x/(d^4*e^3*sqrt(d^2-e^2*x^2))],
[x^2/((d+e*x)*(d^2-e^2*x^2)^(7/2)),x,4,-1/7*x^2/(d*e*(d+e*x)*(d^2-e^2*x^2)^(5/2))+2/35*(d+2*e*x)/(d*e^3*(d^2-e^2*x^2)^(5/2))-4/105*x/(d^3*e^2*(d^2-e^2*x^2)^(3/2))-8/105*x/(d^5*e^2*sqrt(d^2-e^2*x^2))],
[x^3/((1+a*x)*sqrt(1-a^2*x^2)),x,4,3/2*arcsin(a*x)/a^4+x^2*(1-a*x)/(a^2*sqrt(1-a^2*x^2))+1/2*(4-3*a*x)*sqrt(1-a^2*x^2)/a^4],
[x^2/((1+a*x)*sqrt(1-a^2*x^2)),x,4,-arcsin(a*x)/a^3-sqrt(1-a^2*x^2)/a^3-sqrt(1-a^2*x^2)/(a^3*(1+a*x))],
[x/((1+a*x)*sqrt(1-a^2*x^2)),x,2,arcsin(a*x)/a^2+sqrt(1-a^2*x^2)/(a^2*(1+a*x))],
[1/((1+a*x)*sqrt(1-a^2*x^2)),x,1,-sqrt(1-a^2*x^2)/(a*(1+a*x))],
[1/(x*(1-a*x)*sqrt(1-a^2*x^2)),x,5,-arctanh(sqrt(1-a^2*x^2))+sqrt(1-a^2*x^2)/(1-a*x)],
[1/(x^2*(1-a*x)*sqrt(1-a^2*x^2)),x,5,-a*arctanh(sqrt(1-a^2*x^2))-2*sqrt(1-a^2*x^2)/x+sqrt(1-a^2*x^2)/(x*(1-a*x))],
[1/(x^3*(1-a*x)*sqrt(1-a^2*x^2)),x,6,-3/2*a^2*arctanh(sqrt(1-a^2*x^2))-3/2*sqrt(1-a^2*x^2)/x^2-2*a*sqrt(1-a^2*x^2)/x+sqrt(1-a^2*x^2)/(x^2*(1-a*x))],

# Integrands of the form x^m / (d+e x)^2 (d^2 - e^2 x^2)^(p/2)

# p>0
[x^5*(d^2-e^2*x^2)^(5/2)/(d+e*x)^2,x,10,-4/21*d^4*x^2*(d^2-e^2*x^2)^(3/2)/e^4+5/24*d^3*x^3*(d^2-e^2*x^2)^(3/2)/e^3-5/21*d^2*x^4*(d^2-e^2*x^2)^(3/2)/e^2+1/4*d*x^5*(d^2-e^2*x^2)^(3/2)/e-1/9*x^6*(d^2-e^2*x^2)^(3/2)-1/2016*d^5*(256*d-315*e*x)*(d^2-e^2*x^2)^(3/2)/e^6-5/64*d^9*arctan(e*x/sqrt(d^2-e^2*x^2))/e^6-5/64*d^7*x*sqrt(d^2-e^2*x^2)/e^5],
[x^4*(d^2-e^2*x^2)^(5/2)/(d+e*x)^2,x,9,8/35*d^3*x^2*(d^2-e^2*x^2)^(3/2)/e^3-13/48*d^2*x^3*(d^2-e^2*x^2)^(3/2)/e^2+2/7*d*x^4*(d^2-e^2*x^2)^(3/2)/e-1/8*x^5*(d^2-e^2*x^2)^(3/2)+1/6720*d^4*(1024*d-1365*e*x)*(d^2-e^2*x^2)^(3/2)/e^5+13/128*d^8*arctan(e*x/sqrt(d^2-e^2*x^2))/e^5+13/128*d^6*x*sqrt(d^2-e^2*x^2)/e^4],
[x^3*(d^2-e^2*x^2)^(5/2)/(d+e*x)^2,x,8,-11/35*d^2*x^2*(d^2-e^2*x^2)^(3/2)/e^2+1/3*d*x^3*(d^2-e^2*x^2)^(3/2)/e-1/7*x^4*(d^2-e^2*x^2)^(3/2)-1/420*d^3*(88*d-105*e*x)*(d^2-e^2*x^2)^(3/2)/e^4-1/8*d^7*arctan(e*x/sqrt(d^2-e^2*x^2))/e^4-1/8*d^5*x*sqrt(d^2-e^2*x^2)/e^3],
[x^2*(d^2-e^2*x^2)^(5/2)/(d+e*x)^2,x,7,2/5*d*x^2*(d^2-e^2*x^2)^(3/2)/e-1/6*x^3*(d^2-e^2*x^2)^(3/2)+1/120*d^2*(32*d-45*e*x)*(d^2-e^2*x^2)^(3/2)/e^3+3/16*d^6*arctan(e*x/sqrt(d^2-e^2*x^2))/e^3+3/16*d^4*x*sqrt(d^2-e^2*x^2)/e^2],
[x*(d^2-e^2*x^2)^(5/2)/(d+e*x)^2,x,6,-1/6*d*x*(d^2-e^2*x^2)^(3/2)/e-2/15*(d^2-e^2*x^2)^(5/2)/e^2-1/3*(d^2-e^2*x^2)^(7/2)/(e^2*(d+e*x)^2)-1/4*d^5*arctan(e*x/sqrt(d^2-e^2*x^2))/e^2-1/4*d^3*x*sqrt(d^2-e^2*x^2)/e],
[(d^2-e^2*x^2)^(5/2)/(d+e*x)^2,x,6,5/12*d*(d^2-e^2*x^2)^(3/2)/e+1/4*(d-e*x)*(d^2-e^2*x^2)^(3/2)/e+5/8*d^4*arctan(e*x/sqrt(d^2-e^2*x^2))/e+5/8*d^2*x*sqrt(d^2-e^2*x^2)],
[(d^2-e^2*x^2)^(5/2)/(x*(d+e*x)^2),x,9,-1/3*(d^2-e^2*x^2)^(3/2)-d^3*arctan(e*x/sqrt(d^2-e^2*x^2))-d^3*arctanh(sqrt(d^2-e^2*x^2)/d)+d*(d-e*x)*sqrt(d^2-e^2*x^2)],
[(d^2-e^2*x^2)^(5/2)/(x^2*(d+e*x)^2),x,9,-(d^2-e^2*x^2)^(3/2)/x-1/2*d^2*e*arctan(e*x/sqrt(d^2-e^2*x^2))+2*d^2*e*arctanh(sqrt(d^2-e^2*x^2)/d)-1/2*e*(4*d+e*x)*sqrt(d^2-e^2*x^2)],
[(d^2-e^2*x^2)^(5/2)/(x^3*(d+e*x)^2),x,9,-1/2*(d^2-e^2*x^2)^(3/2)/x^2+2*d*e^2*arctan(e*x/sqrt(d^2-e^2*x^2))-1/2*d*e^2*arctanh(sqrt(d^2-e^2*x^2)/d)+1/2*e*(4*d+e*x)*sqrt(d^2-e^2*x^2)/x],
[(d^2-e^2*x^2)^(5/2)/(x^4*(d+e*x)^2),x,9,-1/3*(d^2-e^2*x^2)^(3/2)/x^3-e^3*arctan(e*x/sqrt(d^2-e^2*x^2))-e^3*arctanh(sqrt(d^2-e^2*x^2)/d)+e*(d-e*x)*sqrt(d^2-e^2*x^2)/x^2],
[(d^2-e^2*x^2)^(5/2)/(x^5*(d+e*x)^2),x,7,-1/4*(d^2-e^2*x^2)^(3/2)/x^4+2/3*e*(d^2-e^2*x^2)^(3/2)/(d*x^3)+5/8*e^4*arctanh(sqrt(d^2-e^2*x^2)/d)/d-5/8*e^2*sqrt(d^2-e^2*x^2)/x^2],
[(d^2-e^2*x^2)^(5/2)/(x^6*(d+e*x)^2),x,8,-1/5*(d^2-e^2*x^2)^(3/2)/x^5+1/2*e*(d^2-e^2*x^2)^(3/2)/(d*x^4)-7/15*e^2*(d^2-e^2*x^2)^(3/2)/(d^2*x^3)-1/4*e^5*arctanh(sqrt(d^2-e^2*x^2)/d)/d^2+1/4*e^3*sqrt(d^2-e^2*x^2)/(d*x^2)],
[(d^2-e^2*x^2)^(5/2)/(x^7*(d+e*x)^2),x,9,-1/6*(d^2-e^2*x^2)^(3/2)/x^6+2/5*e*(d^2-e^2*x^2)^(3/2)/(d*x^5)-3/8*e^2*(d^2-e^2*x^2)^(3/2)/(d^2*x^4)+4/15*e^3*(d^2-e^2*x^2)^(3/2)/(d^3*x^3)+3/16*e^6*arctanh(sqrt(d^2-e^2*x^2)/d)/d^3-3/16*e^4*sqrt(d^2-e^2*x^2)/(d^2*x^2)],
[(d^2-e^2*x^2)^(5/2)/(x^8*(d+e*x)^2),x,10,-1/7*(d^2-e^2*x^2)^(3/2)/x^7+1/3*e*(d^2-e^2*x^2)^(3/2)/(d*x^6)-11/35*e^2*(d^2-e^2*x^2)^(3/2)/(d^2*x^5)+1/4*e^3*(d^2-e^2*x^2)^(3/2)/(d^3*x^4)-22/105*e^4*(d^2-e^2*x^2)^(3/2)/(d^4*x^3)-1/8*e^7*arctanh(sqrt(d^2-e^2*x^2)/d)/d^4+1/8*e^5*sqrt(d^2-e^2*x^2)/(d^3*x^2)],

# p<0
[x^4/((d+e*x)^2*(d^2-e^2*x^2)^(3/2)),x,7,-1/5*d^3*(d-e*x)^2/(e^5*(d^2-e^2*x^2)^(5/2))+17/15*d^2*(d-e*x)/(e^5*(d^2-e^2*x^2)^(3/2))-arctan(e*x/sqrt(d^2-e^2*x^2))/e^5-2/15*(15*d-13*e*x)/(e^5*sqrt(d^2-e^2*x^2))],
[x^3/((d+e*x)^2*(d^2-e^2*x^2)^(3/2)),x,4,1/5*d^2*(d-e*x)^2/(e^4*(d^2-e^2*x^2)^(5/2))-4/5*d*(d-e*x)/(e^4*(d^2-e^2*x^2)^(3/2))+1/5*(5*d-2*e*x)/(d*e^4*sqrt(d^2-e^2*x^2))],
[x^2/((d+e*x)^2*(d^2-e^2*x^2)^(3/2)),x,4,1/15*x/(d^2*e^2*sqrt(d^2-e^2*x^2))-1/5*d/(e^3*(d+e*x)^2*sqrt(d^2-e^2*x^2))+7/15/(e^3*(d+e*x)*sqrt(d^2-e^2*x^2)),-1/5*d*(d-e*x)^2/(e^3*(d^2-e^2*x^2)^(5/2))+7/15*(d-e*x)/(e^3*(d^2-e^2*x^2)^(3/2))+1/15*x/(d^2*e^2*sqrt(d^2-e^2*x^2))],
[x/((d+e*x)^2*(d^2-e^2*x^2)^(3/2)),x,3,4/15*x/(d^3*e*sqrt(d^2-e^2*x^2))+1/5/(e^2*(d+e*x)^2*sqrt(d^2-e^2*x^2))+(-2/15)/(d*e^2*(d+e*x)*sqrt(d^2-e^2*x^2))],
[1/((d+e*x)^2*(d^2-e^2*x^2)^(3/2)),x,3,2/5*x/(d^4*sqrt(d^2-e^2*x^2))+(-1/5)/(d*e*(d+e*x)^2*sqrt(d^2-e^2*x^2))+(-1/5)/(d^2*e*(d+e*x)*sqrt(d^2-e^2*x^2))],
[1/(x*(d+e*x)^2*(d^2-e^2*x^2)^(3/2)),x,8,2/5*(d-e*x)/(d*(d^2-e^2*x^2)^(5/2))+1/15*(5*d-8*e*x)/(d^3*(d^2-e^2*x^2)^(3/2))-arctanh(sqrt(d^2-e^2*x^2)/d)/d^5+1/15*(15*d-16*e*x)/(d^5*sqrt(d^2-e^2*x^2))],
[1/(x^2*(d+e*x)^2*(d^2-e^2*x^2)^(3/2)),x,8,-2/5*e*(d-e*x)/(d^2*(d^2-e^2*x^2)^(5/2))-1/15*e*(10*d-13*e*x)/(d^4*(d^2-e^2*x^2)^(3/2))+2*e*arctanh(sqrt(d^2-e^2*x^2)/d)/d^6-1/15*e*(30*d-41*e*x)/(d^6*sqrt(d^2-e^2*x^2))-sqrt(d^2-e^2*x^2)/(d^6*x)],
[1/(x^3*(d+e*x)^2*(d^2-e^2*x^2)^(3/2)),x,9,2/5*e^2*(d-e*x)/(d^3*(d^2-e^2*x^2)^(5/2))+1/5*e^2*(5*d-6*e*x)/(d^5*(d^2-e^2*x^2)^(3/2))-9/2*e^2*arctanh(sqrt(d^2-e^2*x^2)/d)/d^7+2/5*e^2*(10*d-11*e*x)/(d^7*sqrt(d^2-e^2*x^2))-1/2*sqrt(d^2-e^2*x^2)/(d^6*x^2)+2*e*sqrt(d^2-e^2*x^2)/(d^7*x)],

# Integrands of the form x^m / (d+e x)^3 (d^2 - e^2 x^2)^(p/2)

# p>0

# p<0
[x^5/((d+e*x)^3*sqrt(d^2-e^2*x^2)),x,8,1/5*d^4*(d-e*x)^3/(e^6*(d^2-e^2*x^2)^(5/2))-23/15*d^3*(d-e*x)^2/(e^6*(d^2-e^2*x^2)^(3/2))+13/2*d^2*arctan(e*x/sqrt(d^2-e^2*x^2))/e^6+127/15*d^2*(d-e*x)/(e^6*sqrt(d^2-e^2*x^2))+3*d*sqrt(d^2-e^2*x^2)/e^6-1/2*x*sqrt(d^2-e^2*x^2)/e^5],
[x^4/((d+e*x)^3*sqrt(d^2-e^2*x^2)),x,7,-1/5*d^3*(d-e*x)^3/(e^5*(d^2-e^2*x^2)^(5/2))+6/5*d^2*(d-e*x)^2/(e^5*(d^2-e^2*x^2)^(3/2))-3*d*arctan(e*x/sqrt(d^2-e^2*x^2))/e^5-24/5*d*(d-e*x)/(e^5*sqrt(d^2-e^2*x^2))-sqrt(d^2-e^2*x^2)/e^5],
[x^3/((d+e*x)^3*sqrt(d^2-e^2*x^2)),x,6,1/5*d^2*(d-e*x)^3/(e^4*(d^2-e^2*x^2)^(5/2))-13/15*d*(d-e*x)^2/(e^4*(d^2-e^2*x^2)^(3/2))+arctan(e*x/sqrt(d^2-e^2*x^2))/e^4+32/15*(d-e*x)/(e^4*sqrt(d^2-e^2*x^2))],
[x^2/((d+e*x)^3*sqrt(d^2-e^2*x^2)),x,4,-1/5*d*sqrt(d^2-e^2*x^2)/(e^3*(d+e*x)^3)+8/15*sqrt(d^2-e^2*x^2)/(e^3*(d+e*x)^2)-7/15*sqrt(d^2-e^2*x^2)/(d*e^3*(d+e*x))],
[x/((d+e*x)^3*sqrt(d^2-e^2*x^2)),x,3,1/5*sqrt(d^2-e^2*x^2)/(e^2*(d+e*x)^3)-1/5*sqrt(d^2-e^2*x^2)/(d*e^2*(d+e*x)^2)-1/5*sqrt(d^2-e^2*x^2)/(d^2*e^2*(d+e*x))],
[1/((d+e*x)^3*sqrt(d^2-e^2*x^2)),x,3,-1/5*sqrt(d^2-e^2*x^2)/(d*e*(d+e*x)^3)-2/15*sqrt(d^2-e^2*x^2)/(d^2*e*(d+e*x)^2)-2/15*sqrt(d^2-e^2*x^2)/(d^3*e*(d+e*x))],
[1/(x*(d+e*x)^3*sqrt(d^2-e^2*x^2)),x,8,4/5*(d-e*x)/(d^2-e^2*x^2)^(5/2)+1/15*(5*d-11*e*x)/(d^2*(d^2-e^2*x^2)^(3/2))-arctanh(sqrt(d^2-e^2*x^2)/d)/d^4+1/15*(15*d-22*e*x)/(d^4*sqrt(d^2-e^2*x^2))],
[1/(x^2*(d+e*x)^3*sqrt(d^2-e^2*x^2)),x,8,-4/5*e*(d-e*x)/(d*(d^2-e^2*x^2)^(5/2))-1/5*e*(5*d-7*e*x)/(d^3*(d^2-e^2*x^2)^(3/2))+3*e*arctanh(sqrt(d^2-e^2*x^2)/d)/d^5-1/5*e*(15*d-19*e*x)/(d^5*sqrt(d^2-e^2*x^2))-sqrt(d^2-e^2*x^2)/(d^5*x)],
[1/(x^3*(d+e*x)^3*sqrt(d^2-e^2*x^2)),x,9,4/5*e^2*(d-e*x)/(d^2*(d^2-e^2*x^2)^(5/2))+1/15*e^2*(25*d-31*e*x)/(d^4*(d^2-e^2*x^2)^(3/2))-13/2*e^2*arctanh(sqrt(d^2-e^2*x^2)/d)/d^6+1/15*e^2*(90*d-107*e*x)/(d^6*sqrt(d^2-e^2*x^2))-1/2*sqrt(d^2-e^2*x^2)/(d^5*x^2)+3*e*sqrt(d^2-e^2*x^2)/(d^6*x)],

# Integrands of the form x^m / (d+e x)^4 (d^2 - e^2 x^2)^(p/2)

# p>0
[x^5*sqrt(d^2-e^2*x^2)/(d+e*x)^4,x,9,1/5*d^4*(d-e*x)^4/(e^6*(d^2-e^2*x^2)^(5/2))-8/5*d^3*(d-e*x)^3/(e^6*(d^2-e^2*x^2)^(3/2))+18*d^3*arctan(e*x/sqrt(d^2-e^2*x^2))/e^6+10*d^2*(d-e*x)^2/(e^6*sqrt(d^2-e^2*x^2))+59/3*d^2*sqrt(d^2-e^2*x^2)/e^6-2*d*x*sqrt(d^2-e^2*x^2)/e^5+1/3*x^2*sqrt(d^2-e^2*x^2)/e^4],
[x^4*sqrt(d^2-e^2*x^2)/(d+e*x)^4,x,7,-1/5*d^3*(d-e*x)^4/(e^5*(d^2-e^2*x^2)^(5/2))+19/15*d^2*(d-e*x)^3/(e^5*(d^2-e^2*x^2)^(3/2))-19/2*d^2*arctan(e*x/sqrt(d^2-e^2*x^2))/e^5-6*d*(d-e*x)^2/(e^5*sqrt(d^2-e^2*x^2))-1/2*(20*d-e*x)*sqrt(d^2-e^2*x^2)/e^5],
[x^3*sqrt(d^2-e^2*x^2)/(d+e*x)^4,x,9,1/5*d^2*(d^2-e^2*x^2)^(3/2)/(e^4*(d+e*x)^4)-14/15*d*(d^2-e^2*x^2)^(3/2)/(e^4*(d+e*x)^3)-(d^2-e^2*x^2)^(3/2)/(e^4*(d+e*x)^2)+4*d*arctan(e*x/sqrt(d^2-e^2*x^2))/e^4+8*d*sqrt(d^2-e^2*x^2)/(e^4*(d+e*x))],
[x^2*sqrt(d^2-e^2*x^2)/(d+e*x)^4,x,8,-1/5*d*(d^2-e^2*x^2)^(3/2)/(e^3*(d+e*x)^4)+3/5*(d^2-e^2*x^2)^(3/2)/(e^3*(d+e*x)^3)-arctan(e*x/sqrt(d^2-e^2*x^2))/e^3-2*sqrt(d^2-e^2*x^2)/(e^3*(d+e*x))],
[x*sqrt(d^2-e^2*x^2)/(d+e*x)^4,x,2,1/5*(d^2-e^2*x^2)^(3/2)/(e^2*(d+e*x)^4)-4/15*(d^2-e^2*x^2)^(3/2)/(d*e^2*(d+e*x)^3)],
[sqrt(d^2-e^2*x^2)/(d+e*x)^4,x,2,-1/5*(d^2-e^2*x^2)^(3/2)/(d*e*(d+e*x)^4)-1/15*(d^2-e^2*x^2)^(3/2)/(d^2*e*(d+e*x)^3)],
[sqrt(d^2-e^2*x^2)/(x*(d+e*x)^4),x,8,8/5*d*(d-e*x)/(d^2-e^2*x^2)^(5/2)-4/5*e*x/(d*(d^2-e^2*x^2)^(3/2))-arctanh(sqrt(d^2-e^2*x^2)/d)/d^3+1/5*(5*d-8*e*x)/(d^3*sqrt(d^2-e^2*x^2))],
[sqrt(d^2-e^2*x^2)/(x^2*(d+e*x)^4),x,8,-8/5*e*(d-e*x)/(d^2-e^2*x^2)^(5/2)-4/15*e*(5*d-8*e*x)/(d^2*(d^2-e^2*x^2)^(3/2))+4*e*arctanh(sqrt(d^2-e^2*x^2)/d)/d^4-1/15*e*(60*d-79*e*x)/(d^4*sqrt(d^2-e^2*x^2))-sqrt(d^2-e^2*x^2)/(d^4*x)],
[sqrt(d^2-e^2*x^2)/(x^3*(d+e*x)^4),x,9,8/5*e^2*(d-e*x)/(d*(d^2-e^2*x^2)^(5/2))+4/15*e^2*(10*d-13*e*x)/(d^3*(d^2-e^2*x^2)^(3/2))-19/2*e^2*arctanh(sqrt(d^2-e^2*x^2)/d)/d^5+1/15*e^2*(135*d-164*e*x)/(d^5*sqrt(d^2-e^2*x^2))-1/2*sqrt(d^2-e^2*x^2)/(d^4*x^2)+4*e*sqrt(d^2-e^2*x^2)/(d^5*x)],
[sqrt(d^2-e^2*x^2)/(x^4*(d+e*x)^4),x,10,-8/5*e^3*(d-e*x)/(d^2*(d^2-e^2*x^2)^(5/2))-4/5*e^3*(5*d-6*e*x)/(d^4*(d^2-e^2*x^2)^(3/2))+18*e^3*arctanh(sqrt(d^2-e^2*x^2)/d)/d^6-1/5*e^3*(80*d-93*e*x)/(d^6*sqrt(d^2-e^2*x^2))-1/3*sqrt(d^2-e^2*x^2)/(d^4*x^3)+2*e*sqrt(d^2-e^2*x^2)/(d^5*x^2)-29/3*e^2*sqrt(d^2-e^2*x^2)/(d^6*x)],
[x^5*(d^2-e^2*x^2)^(5/2)/(d+e*x)^4,x,11,65/4*d^7*arctan(e*x/sqrt(d^2-e^2*x^2))/e^6+d^4*(d-e*x)^4/(e^6*sqrt(d^2-e^2*x^2))+515/21*d^6*sqrt(d^2-e^2*x^2)/e^6-49/4*d^5*x*sqrt(d^2-e^2*x^2)/e^5+121/21*d^4*x^2*sqrt(d^2-e^2*x^2)/e^4-17/6*d^3*x^3*sqrt(d^2-e^2*x^2)/e^3+11/7*d^2*x^4*sqrt(d^2-e^2*x^2)/e^2-2/3*d*x^5*sqrt(d^2-e^2*x^2)/e+1/7*x^6*sqrt(d^2-e^2*x^2)],
[x^4*(d^2-e^2*x^2)^(5/2)/(d+e*x)^4,x,10,-239/16*d^6*arctan(e*x/sqrt(d^2-e^2*x^2))/e^5-d^3*(d-e*x)^4/(e^5*sqrt(d^2-e^2*x^2))-337/15*d^5*sqrt(d^2-e^2*x^2)/e^5+175/16*d^4*x*sqrt(d^2-e^2*x^2)/e^4-71/15*d^3*x^2*sqrt(d^2-e^2*x^2)/e^3+47/24*d^2*x^3*sqrt(d^2-e^2*x^2)/e^2-4/5*d*x^4*sqrt(d^2-e^2*x^2)/e+1/6*x^5*sqrt(d^2-e^2*x^2)],
[x^3*(d^2-e^2*x^2)^(5/2)/(d+e*x)^4,x,9,27/2*d^5*arctan(e*x/sqrt(d^2-e^2*x^2))/e^4+d^2*(d-e*x)^4/(e^4*sqrt(d^2-e^2*x^2))+101/5*d^4*sqrt(d^2-e^2*x^2)/e^4-19/2*d^3*x*sqrt(d^2-e^2*x^2)/e^3+18/5*d^2*x^2*sqrt(d^2-e^2*x^2)/e^2-d*x^3*sqrt(d^2-e^2*x^2)/e+1/5*x^4*sqrt(d^2-e^2*x^2)],
[x^2*(d^2-e^2*x^2)^(5/2)/(d+e*x)^4,x,8,-95/8*d^4*arctan(e*x/sqrt(d^2-e^2*x^2))/e^3-d*(d-e*x)^4/(e^3*sqrt(d^2-e^2*x^2))-95/8*d^3*sqrt(d^2-e^2*x^2)/e^3-95/24*d^2*(d-e*x)*sqrt(d^2-e^2*x^2)/e^3-19/12*d*(d-e*x)^2*sqrt(d^2-e^2*x^2)/e^3-1/4*(d-e*x)^3*sqrt(d^2-e^2*x^2)/e^3],
[x*(d^2-e^2*x^2)^(5/2)/(d+e*x)^4,x,6,20/3*(d^2-e^2*x^2)^(3/2)/e^2+8*(d^2-e^2*x^2)^(5/2)/(e^2*(d+e*x)^2)+(d^2-e^2*x^2)^(7/2)/(e^2*(d+e*x)^4)+10*d^3*arctan(e*x/sqrt(d^2-e^2*x^2))/e^2+10*d*x*sqrt(d^2-e^2*x^2)/e],
[(d^2-e^2*x^2)^(5/2)/(d+e*x)^4,x,5,-5/2*(d^2-e^2*x^2)^(3/2)/(e*(d+e*x))-2*(d^2-e^2*x^2)^(5/2)/(e*(d+e*x)^3)-15/2*d^2*arctan(e*x/sqrt(d^2-e^2*x^2))/e-15/2*d*sqrt(d^2-e^2*x^2)/e],
[(d^2-e^2*x^2)^(5/2)/(x*(d+e*x)^4),x,9,4*d*arctan(e*x/sqrt(d^2-e^2*x^2))-d*arctanh(sqrt(d^2-e^2*x^2)/d)+8*d*(d-e*x)/sqrt(d^2-e^2*x^2)+sqrt(d^2-e^2*x^2)],
[(d^2-e^2*x^2)^(5/2)/(x^2*(d+e*x)^4),x,9,-e*arctan(e*x/sqrt(d^2-e^2*x^2))+4*e*arctanh(sqrt(d^2-e^2*x^2)/d)-8*e*(d-e*x)/sqrt(d^2-e^2*x^2)-sqrt(d^2-e^2*x^2)/x],
[(d^2-e^2*x^2)^(5/2)/(x^3*(d+e*x)^4),x,7,-15/2*e^2*arctanh(sqrt(d^2-e^2*x^2)/d)/d+8*e^2*(d-e*x)/(d*sqrt(d^2-e^2*x^2))-1/2*sqrt(d^2-e^2*x^2)/x^2+4*e*sqrt(d^2-e^2*x^2)/(d*x)],
[(d^2-e^2*x^2)^(5/2)/(x^4*(d+e*x)^4),x,8,10*e^3*arctanh(sqrt(d^2-e^2*x^2)/d)/d^2-8*e^3*(d-e*x)/(d^2*sqrt(d^2-e^2*x^2))-1/3*sqrt(d^2-e^2*x^2)/x^3+2*e*sqrt(d^2-e^2*x^2)/(d*x^2)-23/3*e^2*sqrt(d^2-e^2*x^2)/(d^2*x)],
[(d^2-e^2*x^2)^(5/2)/(x^5*(d+e*x)^4),x,9,-95/8*e^4*arctanh(sqrt(d^2-e^2*x^2)/d)/d^3+8*e^4*(d-e*x)/(d^3*sqrt(d^2-e^2*x^2))-1/4*sqrt(d^2-e^2*x^2)/x^4+4/3*e*sqrt(d^2-e^2*x^2)/(d*x^3)-31/8*e^2*sqrt(d^2-e^2*x^2)/(d^2*x^2)+32/3*e^3*sqrt(d^2-e^2*x^2)/(d^3*x)],
[(d^2-e^2*x^2)^(5/2)/(x^6*(d+e*x)^4),x,10,27/2*e^5*arctanh(sqrt(d^2-e^2*x^2)/d)/d^4-8*e^5*(d-e*x)/(d^4*sqrt(d^2-e^2*x^2))-1/5*sqrt(d^2-e^2*x^2)/x^5+e*sqrt(d^2-e^2*x^2)/(d*x^4)-13/5*e^2*sqrt(d^2-e^2*x^2)/(d^2*x^3)+11/2*e^3*sqrt(d^2-e^2*x^2)/(d^3*x^2)-66/5*e^4*sqrt(d^2-e^2*x^2)/(d^4*x)],
[x^2*sqrt(1-a^2*x^2)/(1-a*x)^4,x,7,1/5*(1-a^2*x^2)^(3/2)/(a^3*(1-a*x)^4)-3/5*(1-a^2*x^2)^(3/2)/(a^3*(1-a*x)^3)-arcsin(a*x)/a^3+2*sqrt(1-a^2*x^2)/(a^3*(1-a*x))],
[x^2*sqrt(1-a^2*x^2)/(1-a*x)^5,x,4,1/7*(1-a^2*x^2)^(3/2)/(a^3*(1-a*x)^5)-12/35*(1-a^2*x^2)^(3/2)/(a^3*(1-a*x)^4)+23/105*(1-a^2*x^2)^(3/2)/(a^3*(1-a*x)^3)],

# p<0
[x^3/((d+e*x)^4*(d^2-e^2*x^2)^(7/2)),x,9,-24/5005*x/(d^3*e^3*(d^2-e^2*x^2)^(5/2))+1/13*d^2/(e^4*(d+e*x)^4*(d^2-e^2*x^2)^(5/2))-30/143*d/(e^4*(d+e*x)^3*(d^2-e^2*x^2)^(5/2))+21/143/(e^4*(d+e*x)^2*(d^2-e^2*x^2)^(5/2))+4/1001/(d*e^4*(d+e*x)*(d^2-e^2*x^2)^(5/2))-32/5005*x/(d^5*e^3*(d^2-e^2*x^2)^(3/2))-64/5005*x/(d^7*e^3*sqrt(d^2-e^2*x^2))],
[x^2/((d+e*x)^4*(d^2-e^2*x^2)^(7/2)),x,8,14/2145*x/(d^4*e^2*(d^2-e^2*x^2)^(5/2))-1/13*d/(e^3*(d+e*x)^4*(d^2-e^2*x^2)^(5/2))+17/143/(e^3*(d+e*x)^3*(d^2-e^2*x^2)^(5/2))+(-7/1287)/(d*e^3*(d+e*x)^2*(d^2-e^2*x^2)^(5/2))+(-7/1287)/(d^2*e^3*(d+e*x)*(d^2-e^2*x^2)^(5/2))+56/6435*x/(d^6*e^2*(d^2-e^2*x^2)^(3/2))+112/6435*x/(d^8*e^2*sqrt(d^2-e^2*x^2))],
[x/((d+e*x)^4*(d^2-e^2*x^2)^(7/2)),x,7,64/2145*x/(d^5*e*(d^2-e^2*x^2)^(5/2))+1/13/(e^2*(d+e*x)^4*(d^2-e^2*x^2)^(5/2))+(-4/143)/(d*e^2*(d+e*x)^3*(d^2-e^2*x^2)^(5/2))+(-32/1287)/(d^2*e^2*(d+e*x)^2*(d^2-e^2*x^2)^(5/2))+(-32/1287)/(d^3*e^2*(d+e*x)*(d^2-e^2*x^2)^(5/2))+256/6435*x/(d^7*e*(d^2-e^2*x^2)^(3/2))+512/6435*x/(d^9*e*sqrt(d^2-e^2*x^2))],
[1/((d+e*x)^4*(d^2-e^2*x^2)^(7/2)),x,7,48/715*x/(d^6*(d^2-e^2*x^2)^(5/2))+(-1/13)/(d*e*(d+e*x)^4*(d^2-e^2*x^2)^(5/2))+(-9/143)/(d^2*e*(d+e*x)^3*(d^2-e^2*x^2)^(5/2))+(-8/143)/(d^3*e*(d+e*x)^2*(d^2-e^2*x^2)^(5/2))+(-8/143)/(d^4*e*(d+e*x)*(d^2-e^2*x^2)^(5/2))+64/715*x/(d^8*(d^2-e^2*x^2)^(3/2))+128/715*x/(d^10*sqrt(d^2-e^2*x^2))],
[1/(x*(d+e*x)^4*(d^2-e^2*x^2)^(7/2)),x,12,8/13*d*(d-e*x)/(d^2-e^2*x^2)^(13/2)-4/13*e*x/(d*(d^2-e^2*x^2)^(11/2))+1/117*(13*d-40*e*x)/(d^3*(d^2-e^2*x^2)^(9/2))+1/819*(117*d-320*e*x)/(d^5*(d^2-e^2*x^2)^(7/2))+1/1365*(273*d-640*e*x)/(d^7*(d^2-e^2*x^2)^(5/2))+1/819*(273*d-512*e*x)/(d^9*(d^2-e^2*x^2)^(3/2))-arctanh(sqrt(d^2-e^2*x^2)/d)/d^11+1/819*(819*d-1024*e*x)/(d^11*sqrt(d^2-e^2*x^2))],
[1/(x^2*(d+e*x)^4*(d^2-e^2*x^2)^(7/2)),x,12,-8/13*e*(d-e*x)/(d^2-e^2*x^2)^(13/2)-4/143*e*(13*d-24*e*x)/(d^2*(d^2-e^2*x^2)^(11/2))-1/1287*e*(572*d-1103*e*x)/(d^4*(d^2-e^2*x^2)^(9/2))-1/9009*e*(5148*d-10111*e*x)/(d^6*(d^2-e^2*x^2)^(7/2))-1/15015*e*(12012*d-23225*e*x)/(d^8*(d^2-e^2*x^2)^(5/2))-1/9009*e*(12012*d-21583*e*x)/(d^10*(d^2-e^2*x^2)^(3/2))+4*e*arctanh(sqrt(d^2-e^2*x^2)/d)/d^12-1/9009*e*(36036*d-52175*e*x)/(d^12*sqrt(d^2-e^2*x^2))-sqrt(d^2-e^2*x^2)/(d^12*x)],

# Integrands of the form x^m (d+e x)^(n/2) (d^2 - e^2 x^2)^(p/2)

# p>0
[sqrt(c-a*c*x)*sqrt(1-a^2*x^2)/x^2,x,4,-c^2*(1-a^2*x^2)^(3/2)/(x*(c-a*c*x)^(3/2))+a*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/sqrt(c-a*c*x))*sqrt(c)-a*c*sqrt(1-a^2*x^2)/sqrt(c-a*c*x)],

# p<0
[sqrt(c-a*c*x)/(x*sqrt(1-a^2*x^2)),x,2,-2*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/sqrt(c-a*c*x))*sqrt(c)],

# Integrands of the form (g x)^(m/2) (d+e x)^(n/2) (d^2 - e^2 x^2)^(p/2)

#  Following pairs of integrands are equal. 
[sqrt(1-a*x)/sqrt(x),x,3,arcsin(sqrt(a)*sqrt(x))/sqrt(a)+sqrt(x)*sqrt(1-a*x)],
[sqrt(1-a^2*x^2)/(sqrt(x)*sqrt(1+a*x)),x,4,arcsin(sqrt(a)*sqrt(x))/sqrt(a)+sqrt(x)*sqrt(1-a*x)],
[sqrt(1+a*x)/sqrt(x),x,3,arcsinh(sqrt(a)*sqrt(x))/sqrt(a)+sqrt(x)*sqrt(1+a*x)],
[sqrt(1-a^2*x^2)/(sqrt(x)*sqrt(1-a*x)),x,4,arcsinh(sqrt(a)*sqrt(x))/sqrt(a)+sqrt(x)*sqrt(1+a*x)],
[sqrt(x)*sqrt(1-a*x),x,4,1/4*arcsin(sqrt(a)*sqrt(x))/a^(3/2)+1/2*x^(3/2)*sqrt(1-a*x)-1/4*sqrt(x)*sqrt(1-a*x)/a],
[sqrt(x)*sqrt(1-a^2*x^2)/sqrt(1+a*x),x,5,1/4*arcsin(sqrt(a)*sqrt(x))/a^(3/2)+1/2*x^(3/2)*sqrt(1-a*x)-1/4*sqrt(x)*sqrt(1-a*x)/a],

# Integrands of the form (g x)^m (d+e x)^n (d^2 - e^2 x^2)^(p/2) when m symbolic

# p>0
[(g*x)^m*(d+e*x)^3*(d^2-e^2*x^2)^(5/2),x,7,-3*d*(g*x)^(1+m)*(d^2-e^2*x^2)^(7/2)/(g*(8+m))-e*(g*x)^(2+m)*(d^2-e^2*x^2)^(7/2)/(g^2*(9+m))+d^7*(11+4*m)*(g*x)^(1+m)*hypergeom([-5/2,1/2*(1+m)],[1/2*(3+m)],e^2*x^2/d^2)*sqrt(d^2-e^2*x^2)/(g*(1+m)*(8+m)*sqrt(1-e^2*x^2/d^2))+d^6*e*(29+4*m)*(g*x)^(2+m)*hypergeom([-5/2,1/2*(2+m)],[1/2*(4+m)],e^2*x^2/d^2)*sqrt(d^2-e^2*x^2)/(g^2*(2+m)*(9+m)*sqrt(1-e^2*x^2/d^2))],
[(g*x)^m*(d+e*x)^2*(d^2-e^2*x^2)^(5/2),x,6,-(g*x)^(1+m)*(d^2-e^2*x^2)^(7/2)/(g*(8+m))+d^6*(9+2*m)*(g*x)^(1+m)*hypergeom([-5/2,1/2*(1+m)],[1/2*(3+m)],e^2*x^2/d^2)*sqrt(d^2-e^2*x^2)/(g*(1+m)*(8+m)*sqrt(1-e^2*x^2/d^2))+2*d^5*e*(g*x)^(2+m)*hypergeom([-5/2,1/2*(2+m)],[1/2*(4+m)],e^2*x^2/d^2)*sqrt(d^2-e^2*x^2)/(g^2*(2+m)*sqrt(1-e^2*x^2/d^2))],
[(g*x)^m*(d+e*x)*(d^2-e^2*x^2)^(5/2),x,5,d^5*(g*x)^(1+m)*hypergeom([-5/2,1/2*(1+m)],[1/2*(3+m)],e^2*x^2/d^2)*sqrt(d^2-e^2*x^2)/(g*(1+m)*sqrt(1-e^2*x^2/d^2))+d^4*e*(g*x)^(2+m)*hypergeom([-5/2,1/2*(2+m)],[1/2*(4+m)],e^2*x^2/d^2)*sqrt(d^2-e^2*x^2)/(g^2*(2+m)*sqrt(1-e^2*x^2/d^2))],
[(g*x)^m*(d^2-e^2*x^2)^(5/2),x,2,d^4*(g*x)^(1+m)*hypergeom([-5/2,1/2*(1+m)],[1/2*(3+m)],e^2*x^2/d^2)*sqrt(d^2-e^2*x^2)/(g*(1+m)*sqrt(1-e^2*x^2/d^2))],
[(g*x)^m*(d^2-e^2*x^2)^(5/2)/(d+e*x),x,8,d^3*(g*x)^(1+m)*hypergeom([-3/2,1/2*(1+m)],[1/2*(3+m)],e^2*x^2/d^2)*sqrt(d^2-e^2*x^2)/(g*(1+m)*sqrt(1-e^2*x^2/d^2))-d^2*e*(g*x)^(2+m)*hypergeom([-3/2,1/2*(2+m)],[1/2*(4+m)],e^2*x^2/d^2)*sqrt(d^2-e^2*x^2)/(g^2*(2+m)*sqrt(1-e^2*x^2/d^2))],
[(g*x)^m*(d^2-e^2*x^2)^(5/2)/(d+e*x)^2,x,7,-(g*x)^(1+m)*(d^2-e^2*x^2)^(3/2)/(g*(4+m))+d^2*(5+2*m)*(g*x)^(1+m)*hypergeom([-1/2,1/2*(1+m)],[1/2*(3+m)],e^2*x^2/d^2)*sqrt(d^2-e^2*x^2)/(g*(1+m)*(4+m)*sqrt(1-e^2*x^2/d^2))-2*d*e*(g*x)^(2+m)*hypergeom([-1/2,1/2*(2+m)],[1/2*(4+m)],e^2*x^2/d^2)*sqrt(d^2-e^2*x^2)/(g^2*(2+m)*sqrt(1-e^2*x^2/d^2))],
[(g*x)^m*(d^2-e^2*x^2)^(5/2)/(d+e*x)^3,x,8,-3*d*(g*x)^(1+m)*sqrt(d^2-e^2*x^2)/(g*(2+m))+e*(g*x)^(2+m)*sqrt(d^2-e^2*x^2)/(g^2*(3+m))+d^3*(5+4*m)*(g*x)^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],e^2*x^2/d^2)*sqrt(1-e^2*x^2/d^2)/(g*(1+m)*(2+m)*sqrt(d^2-e^2*x^2))-d^2*e*(11+4*m)*(g*x)^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],e^2*x^2/d^2)*sqrt(1-e^2*x^2/d^2)/(g^2*(2+m)*(3+m)*sqrt(d^2-e^2*x^2))],

# p<0
[(g*x)^m*(d+e*x)^3/(d^2-e^2*x^2)^(7/2),x,6,4/5*(g*x)^(1+m)*(d+e*x)/(g*(d^2-e^2*x^2)^(5/2))+1/5*(1-4*m)*(g*x)^(1+m)*hypergeom([5/2,1/2*(1+m)],[1/2*(3+m)],e^2*x^2/d^2)*sqrt(1-e^2*x^2/d^2)/(d^3*g*(1+m)*sqrt(d^2-e^2*x^2))+1/5*e*(7-4*m)*(g*x)^(2+m)*hypergeom([5/2,1/2*(2+m)],[1/2*(4+m)],e^2*x^2/d^2)*sqrt(1-e^2*x^2/d^2)/(d^4*g^2*(2+m)*sqrt(d^2-e^2*x^2))],
[(g*x)^m*(d+e*x)^2/(d^2-e^2*x^2)^(7/2),x,6,2/5*(g*x)^(1+m)*(d+e*x)/(d*g*(d^2-e^2*x^2)^(5/2))+1/5*(3-2*m)*(g*x)^(1+m)*hypergeom([5/2,1/2*(1+m)],[1/2*(3+m)],e^2*x^2/d^2)*sqrt(1-e^2*x^2/d^2)/(d^4*g*(1+m)*sqrt(d^2-e^2*x^2))+2/5*e*(3-m)*(g*x)^(2+m)*hypergeom([5/2,1/2*(2+m)],[1/2*(4+m)],e^2*x^2/d^2)*sqrt(1-e^2*x^2/d^2)/(d^5*g^2*(2+m)*sqrt(d^2-e^2*x^2))],
[(g*x)^m*(d+e*x)/(d^2-e^2*x^2)^(7/2),x,5,(g*x)^(1+m)*hypergeom([1,1/2*(-4+m)],[1/2*(3+m)],e^2*x^2/d^2)/(d*g*(1+m)*(d^2-e^2*x^2)^(5/2))+e*(g*x)^(2+m)*hypergeom([1,1/2*(-3+m)],[1/2*(4+m)],e^2*x^2/d^2)/(d^2*g^2*(2+m)*(d^2-e^2*x^2)^(5/2)),(g*x)^(1+m)*hypergeom([7/2,1/2*(1+m)],[1/2*(3+m)],e^2*x^2/d^2)*sqrt(1-e^2*x^2/d^2)/(d^5*g*(1+m)*sqrt(d^2-e^2*x^2))+e*(g*x)^(2+m)*hypergeom([7/2,1/2*(2+m)],[1/2*(4+m)],e^2*x^2/d^2)*sqrt(1-e^2*x^2/d^2)/(d^6*g^2*(2+m)*sqrt(d^2-e^2*x^2))],
[(g*x)^m/(d^2-e^2*x^2)^(7/2),x,2,(g*x)^(1+m)*hypergeom([7/2,1/2*(1+m)],[1/2*(3+m)],e^2*x^2/d^2)*sqrt(1-e^2*x^2/d^2)/(d^6*g*(1+m)*sqrt(d^2-e^2*x^2))],
[(g*x)^m/((d+e*x)*(d^2-e^2*x^2)^(7/2)),x,8,(g*x)^(1+m)*hypergeom([9/2,1/2*(1+m)],[1/2*(3+m)],e^2*x^2/d^2)*sqrt(1-e^2*x^2/d^2)/(d^7*g*(1+m)*sqrt(d^2-e^2*x^2))-e*(g*x)^(2+m)*hypergeom([9/2,1/2*(2+m)],[1/2*(4+m)],e^2*x^2/d^2)*sqrt(1-e^2*x^2/d^2)/(d^8*g^2*(2+m)*sqrt(d^2-e^2*x^2))],
[(g*x)^m/((d+e*x)^2*(d^2-e^2*x^2)^(7/2)),x,7,2/9*(g*x)^(1+m)*(d-e*x)/(d*g*(d^2-e^2*x^2)^(9/2))+1/9*(7-2*m)*(g*x)^(1+m)*hypergeom([9/2,1/2*(1+m)],[1/2*(3+m)],e^2*x^2/d^2)*sqrt(1-e^2*x^2/d^2)/(d^8*g*(1+m)*sqrt(d^2-e^2*x^2))-2/9*e*(7-m)*(g*x)^(2+m)*hypergeom([9/2,1/2*(2+m)],[1/2*(4+m)],e^2*x^2/d^2)*sqrt(1-e^2*x^2/d^2)/(d^9*g^2*(2+m)*sqrt(d^2-e^2*x^2))],
[(g*x)^m/((d+e*x)^3*(d^2-e^2*x^2)^(7/2)),x,7,4/11*(g*x)^(1+m)*(d-e*x)/(g*(d^2-e^2*x^2)^(11/2))+1/11*(7-4*m)*(g*x)^(1+m)*hypergeom([11/2,1/2*(1+m)],[1/2*(3+m)],e^2*x^2/d^2)*sqrt(1-e^2*x^2/d^2)/(d^9*g*(1+m)*sqrt(d^2-e^2*x^2))-1/11*e*(25-4*m)*(g*x)^(2+m)*hypergeom([11/2,1/2*(2+m)],[1/2*(4+m)],e^2*x^2/d^2)*sqrt(1-e^2*x^2/d^2)/(d^10*g^2*(2+m)*sqrt(d^2-e^2*x^2))],

# Integrands of the form x^m (d+e x)^n (d^2 - e^2 x^2)^p when p symbolic

# n>0
[x^5*(d+e*x)*(d^2-e^2*x^2)^p,x,6,-1/2*d^5*(d^2-e^2*x^2)^(1+p)/(e^6*(1+p))+d^3*(d^2-e^2*x^2)^(2+p)/(e^6*(2+p))-1/2*d*(d^2-e^2*x^2)^(3+p)/(e^6*(3+p))+1/7*e*x^7*(d^2-e^2*x^2)^p*hypergeom([7/2,-p],[9/2],e^2*x^2/d^2)/(1-e^2*x^2/d^2)^p],
[x^4*(d+e*x)*(d^2-e^2*x^2)^p,x,6,-1/2*d^4*(d^2-e^2*x^2)^(1+p)/(e^5*(1+p))+d^2*(d^2-e^2*x^2)^(2+p)/(e^5*(2+p))-1/2*(d^2-e^2*x^2)^(3+p)/(e^5*(3+p))+1/5*d*x^5*(d^2-e^2*x^2)^p*hypergeom([5/2,-p],[7/2],e^2*x^2/d^2)/(1-e^2*x^2/d^2)^p],
[x^3*(d+e*x)*(d^2-e^2*x^2)^p,x,6,-1/2*d^3*(d^2-e^2*x^2)^(1+p)/(e^4*(1+p))+1/2*d*(d^2-e^2*x^2)^(2+p)/(e^4*(2+p))+1/5*e*x^5*(d^2-e^2*x^2)^p*hypergeom([5/2,-p],[7/2],e^2*x^2/d^2)/(1-e^2*x^2/d^2)^p],
[x^2*(d+e*x)*(d^2-e^2*x^2)^p,x,6,-1/2*d^2*(d^2-e^2*x^2)^(1+p)/(e^3*(1+p))+1/2*(d^2-e^2*x^2)^(2+p)/(e^3*(2+p))+1/3*d*x^3*(d^2-e^2*x^2)^p*hypergeom([3/2,-p],[5/2],e^2*x^2/d^2)/(1-e^2*x^2/d^2)^p],
[x*(d+e*x)*(d^2-e^2*x^2)^p,x,4,-1/2*d*(d^2-e^2*x^2)^(1+p)/(e^2*(1+p))+1/3*e*x^3*(d^2-e^2*x^2)^p*hypergeom([3/2,-p],[5/2],e^2*x^2/d^2)/(1-e^2*x^2/d^2)^p],
[(d+e*x)*(d^2-e^2*x^2)^p,x,3,-1/2*(d^2-e^2*x^2)^(1+p)/(e*(1+p))+d*x*(d^2-e^2*x^2)^p*hypergeom([1/2,-p],[3/2],e^2*x^2/d^2)/(1-e^2*x^2/d^2)^p],
[(d+e*x)*(d^2-e^2*x^2)^p/x,x,5,e*x*(d^2-e^2*x^2)^p*hypergeom([1/2,-p],[3/2],e^2*x^2/d^2)/(1-e^2*x^2/d^2)^p-1/2*(d^2-e^2*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1-e^2*x^2/d^2)/(d*(1+p))],
[(d+e*x)*(d^2-e^2*x^2)^p/x^2,x,5,-d*(d^2-e^2*x^2)^p*hypergeom([-1/2,-p],[1/2],e^2*x^2/d^2)/(x*(1-e^2*x^2/d^2)^p)-1/2*e*(d^2-e^2*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1-e^2*x^2/d^2)/(d^2*(1+p))],
[(d+e*x)*(d^2-e^2*x^2)^p/x^3,x,5,-e*(d^2-e^2*x^2)^p*hypergeom([-1/2,-p],[1/2],e^2*x^2/d^2)/(x*(1-e^2*x^2/d^2)^p)-1/2*e^2*(d^2-e^2*x^2)^(1+p)*hypergeom([2,1+p],[2+p],1-e^2*x^2/d^2)/(d^3*(1+p))],
[x^5*(d+e*x)^2*(d^2-e^2*x^2)^p,x,7,-d^6*(d^2-e^2*x^2)^(1+p)/(e^6*(1+p))+5/2*d^4*(d^2-e^2*x^2)^(2+p)/(e^6*(2+p))-2*d^2*(d^2-e^2*x^2)^(3+p)/(e^6*(3+p))+1/2*(d^2-e^2*x^2)^(4+p)/(e^6*(4+p))+2/7*d*e*x^7*(d^2-e^2*x^2)^p*hypergeom([7/2,-p],[9/2],e^2*x^2/d^2)/(1-e^2*x^2/d^2)^p],
[x^4*(d+e*x)^2*(d^2-e^2*x^2)^p,x,8,-d^5*(d^2-e^2*x^2)^(1+p)/(e^5*(1+p))-x^5*(d^2-e^2*x^2)^(1+p)/(7+2*p)+2*d^3*(d^2-e^2*x^2)^(2+p)/(e^5*(2+p))-d*(d^2-e^2*x^2)^(3+p)/(e^5*(3+p))+2/5*d^2*(6+p)*x^5*(d^2-e^2*x^2)^p*hypergeom([5/2,-p],[7/2],e^2*x^2/d^2)/((7+2*p)*(1-e^2*x^2/d^2)^p)],
[x^3*(d+e*x)^2*(d^2-e^2*x^2)^p,x,7,-d^4*(d^2-e^2*x^2)^(1+p)/(e^4*(1+p))+3/2*d^2*(d^2-e^2*x^2)^(2+p)/(e^4*(2+p))-1/2*(d^2-e^2*x^2)^(3+p)/(e^4*(3+p))+2/5*d*e*x^5*(d^2-e^2*x^2)^p*hypergeom([5/2,-p],[7/2],e^2*x^2/d^2)/(1-e^2*x^2/d^2)^p],
[x^2*(d+e*x)^2*(d^2-e^2*x^2)^p,x,8,-d^3*(d^2-e^2*x^2)^(1+p)/(e^3*(1+p))-x^3*(d^2-e^2*x^2)^(1+p)/(5+2*p)+d*(d^2-e^2*x^2)^(2+p)/(e^3*(2+p))+2/3*d^2*(4+p)*x^3*(d^2-e^2*x^2)^p*hypergeom([3/2,-p],[5/2],e^2*x^2/d^2)/((5+2*p)*(1-e^2*x^2/d^2)^p)],
[x*(d+e*x)^2*(d^2-e^2*x^2)^p,x,7,-d^2*(d^2-e^2*x^2)^(1+p)/(e^2*(1+p))+1/2*(d^2-e^2*x^2)^(2+p)/(e^2*(2+p))+2/3*d*e*x^3*(d^2-e^2*x^2)^p*hypergeom([3/2,-p],[5/2],e^2*x^2/d^2)/(1-e^2*x^2/d^2)^p],
[(d+e*x)^2*(d^2-e^2*x^2)^p,x,2,-2^(2+p)*d*(1+e*x/d)^(-1-p)*(d^2-e^2*x^2)^(1+p)*hypergeom([-2-p,1+p],[2+p],1/2*(d-e*x)/d)/(e*(1+p))],
[(d+e*x)^2*(d^2-e^2*x^2)^p/x,x,7,-1/2*(d^2-e^2*x^2)^(1+p)/(1+p)+2*d*e*x*(d^2-e^2*x^2)^p*hypergeom([1/2,-p],[3/2],e^2*x^2/d^2)/(1-e^2*x^2/d^2)^p-1/2*(d^2-e^2*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1-e^2*x^2/d^2)/(1+p)],
[(d+e*x)^2*(d^2-e^2*x^2)^p/x^2,x,6,-(d^2-e^2*x^2)^(1+p)/x-2*e^2*p*x*(d^2-e^2*x^2)^p*hypergeom([1/2,-p],[3/2],e^2*x^2/d^2)/(1-e^2*x^2/d^2)^p-e*(d^2-e^2*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1-e^2*x^2/d^2)/(d*(1+p))],
[(d+e*x)^2*(d^2-e^2*x^2)^p/x^3,x,6,-1/2*(d^2-e^2*x^2)^(1+p)/x^2-2*d*e*(d^2-e^2*x^2)^p*hypergeom([-1/2,-p],[1/2],e^2*x^2/d^2)/(x*(1-e^2*x^2/d^2)^p)-1/2*e^2*(1-p)*(d^2-e^2*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1-e^2*x^2/d^2)/(d^2*(1+p))],
[x^5*(d+e*x)^3*(d^2-e^2*x^2)^p,x,7,-2*d^7*(d^2-e^2*x^2)^(1+p)/(e^6*(1+p))-e*x^7*(d^2-e^2*x^2)^(1+p)/(9+2*p)+11/2*d^5*(d^2-e^2*x^2)^(2+p)/(e^6*(2+p))-5*d^3*(d^2-e^2*x^2)^(3+p)/(e^6*(3+p))+3/2*d*(d^2-e^2*x^2)^(4+p)/(e^6*(4+p))+2/7*d^2*e*(17+3*p)*x^7*(d^2-e^2*x^2)^p*hypergeom([7/2,-p],[9/2],e^2*x^2/d^2)/((9+2*p)*(1-e^2*x^2/d^2)^p)],
[x^4*(d+e*x)^3*(d^2-e^2*x^2)^p,x,7,-2*d^6*(d^2-e^2*x^2)^(1+p)/(e^5*(1+p))-3*d*x^5*(d^2-e^2*x^2)^(1+p)/(7+2*p)+9/2*d^4*(d^2-e^2*x^2)^(2+p)/(e^5*(2+p))-3*d^2*(d^2-e^2*x^2)^(3+p)/(e^5*(3+p))+1/2*(d^2-e^2*x^2)^(4+p)/(e^5*(4+p))+2/5*d^3*(11+p)*x^5*(d^2-e^2*x^2)^p*hypergeom([5/2,-p],[7/2],e^2*x^2/d^2)/((7+2*p)*(1-e^2*x^2/d^2)^p)],
[x^3*(d+e*x)^3*(d^2-e^2*x^2)^p,x,7,-2*d^5*(d^2-e^2*x^2)^(1+p)/(e^4*(1+p))-e*x^5*(d^2-e^2*x^2)^(1+p)/(7+2*p)+7/2*d^3*(d^2-e^2*x^2)^(2+p)/(e^4*(2+p))-3/2*d*(d^2-e^2*x^2)^(3+p)/(e^4*(3+p))+2/5*d^2*e*(13+3*p)*x^5*(d^2-e^2*x^2)^p*hypergeom([5/2,-p],[7/2],e^2*x^2/d^2)/((7+2*p)*(1-e^2*x^2/d^2)^p)],
[x^2*(d+e*x)^3*(d^2-e^2*x^2)^p,x,7,-2*d^4*(d^2-e^2*x^2)^(1+p)/(e^3*(1+p))-3*d*x^3*(d^2-e^2*x^2)^(1+p)/(5+2*p)+5/2*d^2*(d^2-e^2*x^2)^(2+p)/(e^3*(2+p))-1/2*(d^2-e^2*x^2)^(3+p)/(e^3*(3+p))+2/3*d^3*(7+p)*x^3*(d^2-e^2*x^2)^p*hypergeom([3/2,-p],[5/2],e^2*x^2/d^2)/((5+2*p)*(1-e^2*x^2/d^2)^p)],
[x*(d+e*x)^3*(d^2-e^2*x^2)^p,x,3,-(d+e*x)^3*(d^2-e^2*x^2)^(1+p)/(e^2*(5+2*p))-3*2^(3+p)*d^3*(1+e*x/d)^(-1-p)*(d^2-e^2*x^2)^(1+p)*hypergeom([-3-p,1+p],[2+p],1/2*(d-e*x)/d)/(e^2*(5+7*p+2*p^2))],
[(d+e*x)^3*(d^2-e^2*x^2)^p,x,2,-2^(3+p)*d^2*(1+e*x/d)^(-1-p)*(d^2-e^2*x^2)^(1+p)*hypergeom([-3-p,1+p],[2+p],1/2*(d-e*x)/d)/(e*(1+p))],
[(d+e*x)^3*(d^2-e^2*x^2)^p/x,x,7,-3/2*d*(d^2-e^2*x^2)^(1+p)/(1+p)-e*x*(d^2-e^2*x^2)^(1+p)/(3+2*p)+2*d^2*e*(5+3*p)*x*(d^2-e^2*x^2)^p*hypergeom([1/2,-p],[3/2],e^2*x^2/d^2)/((3+2*p)*(1-e^2*x^2/d^2)^p)-1/2*d*(d^2-e^2*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1-e^2*x^2/d^2)/(1+p)],
[(d+e*x)^3*(d^2-e^2*x^2)^p/x^2,x,8,-1/2*e*(d^2-e^2*x^2)^(1+p)/(1+p)-d*(d^2-e^2*x^2)^(1+p)/x+2*d*e^2*(1-p)*x*(d^2-e^2*x^2)^p*hypergeom([1/2,-p],[3/2],e^2*x^2/d^2)/(1-e^2*x^2/d^2)^p-3/2*e*(d^2-e^2*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1-e^2*x^2/d^2)/(1+p)],
[(d+e*x)^3*(d^2-e^2*x^2)^p/x^3,x,7,-1/2*d*(d^2-e^2*x^2)^(1+p)/x^2-3*e*(d^2-e^2*x^2)^(1+p)/x-2*e^3*(1+3*p)*x*(d^2-e^2*x^2)^p*hypergeom([1/2,-p],[3/2],e^2*x^2/d^2)/(1-e^2*x^2/d^2)^p-1/2*e^2*(3-p)*(d^2-e^2*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1-e^2*x^2/d^2)/(d*(1+p))],

# n<0
[x^4*(d^2-e^2*x^2)^p/(d+e*x),x,7,1/2*d^4*(d^2-e^2*x^2)^p/(e^5*p)-d^2*(d^2-e^2*x^2)^(1+p)/(e^5*(1+p))+1/2*(d^2-e^2*x^2)^(2+p)/(e^5*(2+p))+1/5*x^5*(d^2-e^2*x^2)^p*hypergeom([5/2,1-p],[7/2],e^2*x^2/d^2)/(d*(1-e^2*x^2/d^2)^p)],
[x^3*(d^2-e^2*x^2)^p/(d+e*x),x,7,-1/2*d^3*(d^2-e^2*x^2)^p/(e^4*p)+1/2*d*(d^2-e^2*x^2)^(1+p)/(e^4*(1+p))-1/5*e*x^5*(d^2-e^2*x^2)^p*hypergeom([5/2,1-p],[7/2],e^2*x^2/d^2)/(d^2*(1-e^2*x^2/d^2)^p)],
[x^2*(d^2-e^2*x^2)^p/(d+e*x),x,7,1/2*d^2*(d^2-e^2*x^2)^p/(e^3*p)-1/2*(d^2-e^2*x^2)^(1+p)/(e^3*(1+p))+1/3*x^3*(d^2-e^2*x^2)^p*hypergeom([3/2,1-p],[5/2],e^2*x^2/d^2)/(d*(1-e^2*x^2/d^2)^p)],
[x*(d^2-e^2*x^2)^p/(d+e*x),x,5,-1/2*d*(d^2-e^2*x^2)^p/(e^2*p)-1/3*e*x^3*(d^2-e^2*x^2)^p*hypergeom([3/2,1-p],[5/2],e^2*x^2/d^2)/(d^2*(1-e^2*x^2/d^2)^p)],
[(d^2-e^2*x^2)^p/(d+e*x),x,2,-2^(-1+p)*(1+e*x/d)^(-1-p)*(d^2-e^2*x^2)^(1+p)*hypergeom([1-p,1+p],[2+p],1/2*(d-e*x)/d)/(d^2*e*(1+p))],
[(d^2-e^2*x^2)^p/(x*(d+e*x)),x,6,-e*x*(d^2-e^2*x^2)^p*hypergeom([1/2,1-p],[3/2],e^2*x^2/d^2)/(d^2*(1-e^2*x^2/d^2)^p)-1/2*(d^2-e^2*x^2)^p*hypergeom([1,p],[1+p],1-e^2*x^2/d^2)/(d*p)],
[(d^2-e^2*x^2)^p/(x^2*(d+e*x)),x,6,-(d^2-e^2*x^2)^p*hypergeom([-1/2,1-p],[1/2],e^2*x^2/d^2)/(d*x*(1-e^2*x^2/d^2)^p)+1/2*e*(d^2-e^2*x^2)^p*hypergeom([1,p],[1+p],1-e^2*x^2/d^2)/(d^2*p)],
[(d^2-e^2*x^2)^p/(x^3*(d+e*x)),x,6,e*(d^2-e^2*x^2)^p*hypergeom([-1/2,1-p],[1/2],e^2*x^2/d^2)/(d^2*x*(1-e^2*x^2/d^2)^p)-1/2*e^2*(d^2-e^2*x^2)^p*hypergeom([2,p],[1+p],1-e^2*x^2/d^2)/(d^3*p)],
[x^5*(d^2-e^2*x^2)^p/(d+e*x)^2,x,8,d^6*(d^2-e^2*x^2)^(-1+p)/(e^6*(1-p))+5/2*d^4*(d^2-e^2*x^2)^p/(e^6*p)-2*d^2*(d^2-e^2*x^2)^(1+p)/(e^6*(1+p))+1/2*(d^2-e^2*x^2)^(2+p)/(e^6*(2+p))-2/7*e*x^7*(d^2-e^2*x^2)^p*hypergeom([7/2,2-p],[9/2],e^2*x^2/d^2)/(d^3*(1-e^2*x^2/d^2)^p)],
[x^4*(d^2-e^2*x^2)^p/(d+e*x)^2,x,9,-d^5*(d^2-e^2*x^2)^(-1+p)/(e^5*(1-p))-x^5*(d^2-e^2*x^2)^(-1+p)/(3+2*p)-2*d^3*(d^2-e^2*x^2)^p/(e^5*p)+d*(d^2-e^2*x^2)^(1+p)/(e^5*(1+p))+2/5*(4+p)*x^5*(d^2-e^2*x^2)^p*hypergeom([5/2,2-p],[7/2],e^2*x^2/d^2)/(d^2*(3+2*p)*(1-e^2*x^2/d^2)^p)],
[x^3*(d^2-e^2*x^2)^p/(d+e*x)^2,x,8,d^4*(d^2-e^2*x^2)^(-1+p)/(e^4*(1-p))+3/2*d^2*(d^2-e^2*x^2)^p/(e^4*p)-1/2*(d^2-e^2*x^2)^(1+p)/(e^4*(1+p))-2/5*e*x^5*(d^2-e^2*x^2)^p*hypergeom([5/2,2-p],[7/2],e^2*x^2/d^2)/(d^3*(1-e^2*x^2/d^2)^p)],
[x^2*(d^2-e^2*x^2)^p/(d+e*x)^2,x,9,-d^3*(d^2-e^2*x^2)^(-1+p)/(e^3*(1-p))-x^3*(d^2-e^2*x^2)^(-1+p)/(1+2*p)-d*(d^2-e^2*x^2)^p/(e^3*p)+2/3*(2+p)*x^3*(d^2-e^2*x^2)^p*hypergeom([3/2,2-p],[5/2],e^2*x^2/d^2)/(d^2*(1+2*p)*(1-e^2*x^2/d^2)^p)],
[x*(d^2-e^2*x^2)^p/(d+e*x)^2,x,3,1/2*(d^2-e^2*x^2)^(1+p)/(e^2*(1-p)*(d+e*x)^2)-2^(-1+p)*(1+e*x/d)^(-1-p)*(d^2-e^2*x^2)^(1+p)*hypergeom([1-p,1+p],[2+p],1/2*(d-e*x)/d)/(d^2*e^2*(1-p^2))],
[(d^2-e^2*x^2)^p/(d+e*x)^2,x,2,-2^(-2+p)*(1+e*x/d)^(-1-p)*(d^2-e^2*x^2)^(1+p)*hypergeom([2-p,1+p],[2+p],1/2*(d-e*x)/d)/(d^3*e*(1+p))],
[(d^2-e^2*x^2)^p/(x*(d+e*x)^2),x,8,(d^2-e^2*x^2)^(-1+p)/(1-p)-2*e*x*(d^2-e^2*x^2)^p*hypergeom([1/2,2-p],[3/2],e^2*x^2/d^2)/(d^3*(1-e^2*x^2/d^2)^p)-1/2*(d^2-e^2*x^2)^p*hypergeom([1,p],[1+p],1-e^2*x^2/d^2)/(d^2*p)],
[(d^2-e^2*x^2)^p/(x^2*(d+e*x)^2),x,7,-(d^2-e^2*x^2)^(-1+p)/x+2*e^2*(2-p)*x*(d^2-e^2*x^2)^p*hypergeom([1/2,2-p],[3/2],e^2*x^2/d^2)/(d^4*(1-e^2*x^2/d^2)^p)-e*(d^2-e^2*x^2)^(-1+p)*hypergeom([1,-1+p],[p],1-e^2*x^2/d^2)/(d*(1-p))],
[(d^2-e^2*x^2)^p/(x^3*(d+e*x)^2),x,7,-1/2*(d^2-e^2*x^2)^(-1+p)/x^2+2*e*(d^2-e^2*x^2)^p*hypergeom([-1/2,2-p],[1/2],e^2*x^2/d^2)/(d^3*x*(1-e^2*x^2/d^2)^p)+1/2*e^2*(3-p)*(d^2-e^2*x^2)^(-1+p)*hypergeom([1,-1+p],[p],1-e^2*x^2/d^2)/(d^2*(1-p))],
[(d^2-e^2*x^2)^p/(x^4*(d+e*x)^2),x,7,-1/3*(d^2-e^2*x^2)^(-1+p)/x^3-2/3*e^2*(4-p)*(d^2-e^2*x^2)^p*hypergeom([-1/2,2-p],[1/2],e^2*x^2/d^2)/(d^4*x*(1-e^2*x^2/d^2)^p)-e^3*(d^2-e^2*x^2)^(-1+p)*hypergeom([2,-1+p],[p],1-e^2*x^2/d^2)/(d^3*(1-p))],
[(d^2-e^2*x^2)^p/(x^5*(d+e*x)^2),x,7,-1/4*(d^2-e^2*x^2)^(-1+p)/x^4+2/3*e*(d^2-e^2*x^2)^p*hypergeom([-3/2,2-p],[-1/2],e^2*x^2/d^2)/(d^3*x^3*(1-e^2*x^2/d^2)^p)+1/4*e^4*(5-p)*(d^2-e^2*x^2)^(-1+p)*hypergeom([2,-1+p],[p],1-e^2*x^2/d^2)/(d^4*(1-p))],
[x^4*(d^2-e^2*x^2)^p/(d+e*x)^3,x,8,-2*d^6*(d^2-e^2*x^2)^(-2+p)/(e^5*(2-p))-3*d*x^5*(d^2-e^2*x^2)^(-2+p)/(1+2*p)+9/2*d^4*(d^2-e^2*x^2)^(-1+p)/(e^5*(1-p))+3*d^2*(d^2-e^2*x^2)^p/(e^5*p)-1/2*(d^2-e^2*x^2)^(1+p)/(e^5*(1+p))+2/5*(8+p)*x^5*(d^2-e^2*x^2)^p*hypergeom([5/2,3-p],[7/2],e^2*x^2/d^2)/(d^3*(1+2*p)*(1-e^2*x^2/d^2)^p)],
[x^3*(d^2-e^2*x^2)^p/(d+e*x)^3,x,8,2*d^5*(d^2-e^2*x^2)^(-2+p)/(e^4*(2-p))+e*x^5*(d^2-e^2*x^2)^(-2+p)/(1+2*p)-7/2*d^3*(d^2-e^2*x^2)^(-1+p)/(e^4*(1-p))-3/2*d*(d^2-e^2*x^2)^p/(e^4*p)-2/5*e*(4+3*p)*x^5*(d^2-e^2*x^2)^p*hypergeom([5/2,3-p],[7/2],e^2*x^2/d^2)/(d^4*(1+2*p)*(1-e^2*x^2/d^2)^p)],
[x^2*(d^2-e^2*x^2)^p/(d+e*x)^3,x,4,-1/2*d*(d^2-e^2*x^2)^(1+p)/(e^3*(2-p)*(d+e*x)^3)-1/2*(d^2-e^2*x^2)^(1+p)/(e^3*p*(d+e*x)^2)+2^(-3+p)*(4+p)*(1+e*x/d)^(-1-p)*(d^2-e^2*x^2)^(1+p)*hypergeom([2-p,1+p],[2+p],1/2*(d-e*x)/d)/(d^2*e^3*p*(2+p-p^2))],
[x*(d^2-e^2*x^2)^p/(d+e*x)^3,x,3,1/2*(d^2-e^2*x^2)^(1+p)/(e^2*(2-p)*(d+e*x)^3)-3*2^(-3+p)*(1+e*x/d)^(-1-p)*(d^2-e^2*x^2)^(1+p)*hypergeom([2-p,1+p],[2+p],1/2*(d-e*x)/d)/(d^3*e^2*(2+p-p^2))],
[(d^2-e^2*x^2)^p/(d+e*x)^3,x,2,-2^(-3+p)*(1+e*x/d)^(-1-p)*(d^2-e^2*x^2)^(1+p)*hypergeom([3-p,1+p],[2+p],1/2*(d-e*x)/d)/(d^4*e*(1+p))],
[(d^2-e^2*x^2)^p/(x*(d+e*x)^3),x,8,2*d*(d^2-e^2*x^2)^(-2+p)/(2-p)-e*x*(d^2-e^2*x^2)^(-2+p)/(3-2*p)-2*e*(4-3*p)*x*(d^2-e^2*x^2)^p*hypergeom([1/2,3-p],[3/2],e^2*x^2/d^2)/(d^4*(3-2*p)*(1-e^2*x^2/d^2)^p)+1/2*(d^2-e^2*x^2)^(-1+p)*hypergeom([1,-1+p],[p],1-e^2*x^2/d^2)/(d*(1-p))],
[(d^2-e^2*x^2)^p/(x^2*(d+e*x)^3),x,9,-2*e*(d^2-e^2*x^2)^(-2+p)/(2-p)-d*(d^2-e^2*x^2)^(-2+p)/x+2*e^2*(4-p)*x*(d^2-e^2*x^2)^p*hypergeom([1/2,3-p],[3/2],e^2*x^2/d^2)/(d^5*(1-e^2*x^2/d^2)^p)-3/2*e*(d^2-e^2*x^2)^(-1+p)*hypergeom([1,-1+p],[p],1-e^2*x^2/d^2)/(d^2*(1-p))],
[(d^2-e^2*x^2)^p/(x^3*(d+e*x)^3),x,8,-1/2*d*(d^2-e^2*x^2)^(-2+p)/x^2+3*e*(d^2-e^2*x^2)^(-2+p)/x-2*e^3*(8-3*p)*x*(d^2-e^2*x^2)^p*hypergeom([1/2,3-p],[3/2],e^2*x^2/d^2)/(d^6*(1-e^2*x^2/d^2)^p)+1/2*e^2*(6-p)*(d^2-e^2*x^2)^(-2+p)*hypergeom([1,-2+p],[-1+p],1-e^2*x^2/d^2)/(d*(2-p))],
[(d^2-e^2*x^2)^p/(x^4*(d+e*x)^3),x,8,-1/3*d*(d^2-e^2*x^2)^(-2+p)/x^3+3/2*e*(d^2-e^2*x^2)^(-2+p)/x^2-2/3*e^2*(8-p)*(d^2-e^2*x^2)^p*hypergeom([-1/2,3-p],[1/2],e^2*x^2/d^2)/(d^5*x*(1-e^2*x^2/d^2)^p)-1/2*e^3*(10-3*p)*(d^2-e^2*x^2)^(-2+p)*hypergeom([1,-2+p],[-1+p],1-e^2*x^2/d^2)/(d^2*(2-p))],
[(d^2-e^2*x^2)^p/(x^5*(d+e*x)^3),x,8,-1/4*d*(d^2-e^2*x^2)^(-2+p)/x^4+e*(d^2-e^2*x^2)^(-2+p)/x^3+2*e^3*(4-p)*(d^2-e^2*x^2)^p*hypergeom([-1/2,3-p],[1/2],e^2*x^2/d^2)/(d^6*x*(1-e^2*x^2/d^2)^p)+1/4*e^4*(10-p)*(d^2-e^2*x^2)^(-2+p)*hypergeom([2,-2+p],[-1+p],1-e^2*x^2/d^2)/(d^3*(2-p))],
[x^4*(d^2-e^2*x^2)^p/(d+e*x)^4,x,9,-4*d^7*(d^2-e^2*x^2)^(-3+p)/(e^5*(3-p))+d^2*(13+12*p)*x^5*(d^2-e^2*x^2)^(-3+p)/(1-4*p^2)-e^2*x^7*(d^2-e^2*x^2)^(-3+p)/(1+2*p)+10*d^5*(d^2-e^2*x^2)^(-2+p)/(e^5*(2-p))-8*d^3*(d^2-e^2*x^2)^(-1+p)/(e^5*(1-p))-2*d*(d^2-e^2*x^2)^p/(e^5*p)-4/5*(16+15*p+p^2)*x^5*(d^2-e^2*x^2)^p*hypergeom([5/2,4-p],[7/2],e^2*x^2/d^2)/(d^4*(1-4*p^2)*(1-e^2*x^2/d^2)^p)],
[x^3*(d^2-e^2*x^2)^p/(d+e*x)^4,x,5,1/2*d^2*(d^2-e^2*x^2)^(1+p)/(e^4*(3-p)*(d+e*x)^4)-d*(1+2*p)*(d^2-e^2*x^2)^(1+p)/(e^4*(1-2*p)*p*(d+e*x)^3)-1/2*(d^2-e^2*x^2)^(1+p)/(e^4*p*(d+e*x)^2)+3*2^(-2+p)*(2+p)*(1+e*x/d)^(-1-p)*(d^2-e^2*x^2)^(1+p)*hypergeom([3-p,1+p],[2+p],1/2*(d-e*x)/d)/(d^2*e^4*p*(3-4*p-5*p^2+2*p^3))],
[x^2*(d^2-e^2*x^2)^p/(d+e*x)^4,x,4,-1/2*d*(d^2-e^2*x^2)^(1+p)/(e^3*(3-p)*(d+e*x)^4)+(d^2-e^2*x^2)^(1+p)/(e^3*(1-2*p)*(d+e*x)^3)-2^(-3+p)*(7+p)*(1+e*x/d)^(-1-p)*(d^2-e^2*x^2)^(1+p)*hypergeom([3-p,1+p],[2+p],1/2*(d-e*x)/d)/(d^3*e^3*(3-4*p-5*p^2+2*p^3))],
[x*(d^2-e^2*x^2)^p/(d+e*x)^4,x,3,1/2*(d^2-e^2*x^2)^(1+p)/(e^2*(3-p)*(d+e*x)^4)-2^(-2+p)*(1+e*x/d)^(-1-p)*(d^2-e^2*x^2)^(1+p)*hypergeom([3-p,1+p],[2+p],1/2*(d-e*x)/d)/(d^4*e^2*(3+2*p-p^2))],
[(d^2-e^2*x^2)^p/(d+e*x)^4,x,2,-2^(-4+p)*(1+e*x/d)^(-1-p)*(d^2-e^2*x^2)^(1+p)*hypergeom([4-p,1+p],[2+p],1/2*(d-e*x)/d)/(d^5*e*(1+p))],
[(d^2-e^2*x^2)^p/(x*(d+e*x)^4),x,9,4*d^2*(d^2-e^2*x^2)^(-3+p)/(3-p)-4*d*e*x*(d^2-e^2*x^2)^(-3+p)/(5-2*p)-1/2*(d^2-e^2*x^2)^(-2+p)/(2-p)-8*e*(2-p)*x*(d^2-e^2*x^2)^p*hypergeom([1/2,4-p],[3/2],e^2*x^2/d^2)/(d^5*(5-2*p)*(1-e^2*x^2/d^2)^p)+1/2*(d^2-e^2*x^2)^(-2+p)*hypergeom([1,-2+p],[-1+p],1-e^2*x^2/d^2)/(2-p)],
[(d^2-e^2*x^2)^p/(x^2*(d+e*x)^4),x,9,-4*d*e*(d^2-e^2*x^2)^(-3+p)/(3-p)-d^2*(d^2-e^2*x^2)^(-3+p)/x+e^2*x*(d^2-e^2*x^2)^(-3+p)/(5-2*p)+4*e^2*(16-9*p+p^2)*x*(d^2-e^2*x^2)^p*hypergeom([1/2,4-p],[3/2],e^2*x^2/d^2)/(d^6*(5-2*p)*(1-e^2*x^2/d^2)^p)-2*e*(d^2-e^2*x^2)^(-2+p)*hypergeom([1,-2+p],[-1+p],1-e^2*x^2/d^2)/(d*(2-p))],
[(d^2-e^2*x^2)^p/(x^3*(d+e*x)^4),x,10,1/2*e^2*(11-p)*(d^2-e^2*x^2)^(-3+p)/(3-p)-1/2*d^2*(d^2-e^2*x^2)^(-3+p)/x^2+4*d*e*(d^2-e^2*x^2)^(-3+p)/x-8*e^3*(4-p)*x*(d^2-e^2*x^2)^p*hypergeom([1/2,4-p],[3/2],e^2*x^2/d^2)/(d^7*(1-e^2*x^2/d^2)^p)+1/2*e^2*(10-p)*(d^2-e^2*x^2)^(-2+p)*hypergeom([1,-2+p],[-1+p],1-e^2*x^2/d^2)/(d^2*(2-p))],
[(d^2-e^2*x^2)^p/(x^4*(d+e*x)^4),x,9,-1/3*d^2*(d^2-e^2*x^2)^(-3+p)/x^3+2*d*e*(d^2-e^2*x^2)^(-3+p)/x^2-1/3*e^2*(27-2*p)*(d^2-e^2*x^2)^(-3+p)/x+4/3*e^4*(48-17*p+p^2)*x*(d^2-e^2*x^2)^p*hypergeom([1/2,4-p],[3/2],e^2*x^2/d^2)/(d^8*(1-e^2*x^2/d^2)^p)-2*e^3*(5-p)*(d^2-e^2*x^2)^(-3+p)*hypergeom([1,-3+p],[-2+p],1-e^2*x^2/d^2)/(d*(3-p))],
[(d^2-e^2*x^2)^p/(x^5*(d+e*x)^4),x,9,-1/4*d^2*(d^2-e^2*x^2)^(-3+p)/x^4+4/3*d*e*(d^2-e^2*x^2)^(-3+p)/x^3-1/4*e^2*(17-p)*(d^2-e^2*x^2)^(-3+p)/x^2+8/3*e^3*(6-p)*(d^2-e^2*x^2)^p*hypergeom([-1/2,4-p],[1/2],e^2*x^2/d^2)/(d^7*x*(1-e^2*x^2/d^2)^p)+1/4*e^4*(70-21*p+p^2)*(d^2-e^2*x^2)^(-3+p)*hypergeom([1,-3+p],[-2+p],1-e^2*x^2/d^2)/(d^2*(3-p))],

# Integrands of the form (g x)^m (d+e x)^n (d^2 - e^2 x^2)^p when m and p symbolic
[(g*x)^m*(d+e*x)^3*(d^2-e^2*x^2)^p,x,7,-3*d*(g*x)^(1+m)*(d^2-e^2*x^2)^(1+p)/(g*(3+m+2*p))-e*(g*x)^(2+m)*(d^2-e^2*x^2)^(1+p)/(g^2*(4+m+2*p))+2*d^3*(3+2*m+p)*(g*x)^(1+m)*(d^2-e^2*x^2)^p*hypergeom([1/2*(1+m),-p],[1/2*(3+m)],e^2*x^2/d^2)/(g*(1+m)*(3+m+2*p)*(1-e^2*x^2/d^2)^p)+2*d^2*e*(7+2*m+3*p)*(g*x)^(2+m)*(d^2-e^2*x^2)^p*hypergeom([1/2*(2+m),-p],[1/2*(4+m)],e^2*x^2/d^2)/(g^2*(2+m)*(4+m+2*p)*(1-e^2*x^2/d^2)^p)],
[(g*x)^m*(d+e*x)^2*(d^2-e^2*x^2)^p,x,6,-(g*x)^(1+m)*(d^2-e^2*x^2)^(1+p)/(g*(3+m+2*p))+2*d^2*(2+m+p)*(g*x)^(1+m)*(d^2-e^2*x^2)^p*hypergeom([1/2*(1+m),-p],[1/2*(3+m)],e^2*x^2/d^2)/(g*(1+m)*(3+m+2*p)*(1-e^2*x^2/d^2)^p)+2*d*e*(g*x)^(2+m)*(d^2-e^2*x^2)^p*hypergeom([1/2*(2+m),-p],[1/2*(4+m)],e^2*x^2/d^2)/(g^2*(2+m)*(1-e^2*x^2/d^2)^p)],
[(g*x)^m*(d+e*x)*(d^2-e^2*x^2)^p,x,5,d*(g*x)^(1+m)*(d^2-e^2*x^2)^p*hypergeom([1/2*(1+m),-p],[1/2*(3+m)],e^2*x^2/d^2)/(g*(1+m)*(1-e^2*x^2/d^2)^p)+e*(g*x)^(2+m)*(d^2-e^2*x^2)^p*hypergeom([1/2*(2+m),-p],[1/2*(4+m)],e^2*x^2/d^2)/(g^2*(2+m)*(1-e^2*x^2/d^2)^p)],
[(g*x)^m*(d^2-e^2*x^2)^p,x,2,(g*x)^(1+m)*(d^2-e^2*x^2)^p*hypergeom([1/2*(1+m),-p],[1/2*(3+m)],e^2*x^2/d^2)/(g*(1+m)*(1-e^2*x^2/d^2)^p)],
[(g*x)^m*(d^2-e^2*x^2)^p/(d+e*x),x,8,(g*x)^(1+m)*(d^2-e^2*x^2)^p*hypergeom([1/2*(1+m),1-p],[1/2*(3+m)],e^2*x^2/d^2)/(d*g*(1+m)*(1-e^2*x^2/d^2)^p)-e*(g*x)^(2+m)*(d^2-e^2*x^2)^p*hypergeom([1/2*(2+m),1-p],[1/2*(4+m)],e^2*x^2/d^2)/(d^2*g^2*(2+m)*(1-e^2*x^2/d^2)^p)],
[(g*x)^m*(d^2-e^2*x^2)^p/(d+e*x)^2,x,7,(g*x)^(1+m)*(d^2-e^2*x^2)^(-1+p)/(g*(1-m-2*p))-2*(m+p)*(g*x)^(1+m)*(d^2-e^2*x^2)^p*hypergeom([1/2*(1+m),2-p],[1/2*(3+m)],e^2*x^2/d^2)/(d^2*g*(1+m)*(1-m-2*p)*(1-e^2*x^2/d^2)^p)-2*e*(g*x)^(2+m)*(d^2-e^2*x^2)^p*hypergeom([1/2*(2+m),2-p],[1/2*(4+m)],e^2*x^2/d^2)/(d^3*g^2*(2+m)*(1-e^2*x^2/d^2)^p)],
[(g*x)^m*(d^2-e^2*x^2)^p/(d+e*x)^3,x,8,3*d*(g*x)^(1+m)*(d^2-e^2*x^2)^(-2+p)/(g*(3-m-2*p))-e*(g*x)^(2+m)*(d^2-e^2*x^2)^(-2+p)/(g^2*(2-m-2*p))-2*(2*m+p)*(g*x)^(1+m)*(d^2-e^2*x^2)^p*hypergeom([1/2*(1+m),3-p],[1/2*(3+m)],e^2*x^2/d^2)/(d^3*g*(1+m)*(3-m-2*p)*(1-e^2*x^2/d^2)^p)-2*e*(2-2*m-3*p)*(g*x)^(2+m)*(d^2-e^2*x^2)^p*hypergeom([1/2*(2+m),3-p],[1/2*(4+m)],e^2*x^2/d^2)/(d^4*g^2*(2+m)*(2-m-2*p)*(1-e^2*x^2/d^2)^p)],
[(g*x)^m*(1-a^2*x^2)^p/(1+a*x),x,6,(g*x)^(1+m)*hypergeom([1/2*(1+m),1-p],[1/2*(3+m)],a^2*x^2)/(g*(1+m))-a*(g*x)^(2+m)*hypergeom([1/2*(2+m),1-p],[1/2*(4+m)],a^2*x^2)/(g^2*(2+m))],

# Integrands of the form (g x)^m (d+e x)^n (d^2 - e^2 x^2)^p when n and p symbolic
[(g*x)^m*(d+e*x)^n*(d^2-e^2*x^2)^p,x,4,(g*x)^(1+m)*(d+e*x)^n*(1+e*x/d)^(-n-p)*(d^2-e^2*x^2)^p*AppellF1(1+m,-p,-n-p,2+m,e*x/d,-e*x/d)/(g*(1+m)*(1-e*x/d)^p)],

# Integrands of the form (g x)^m (d+e x)^n (a+b x+c x^2)^p when b=0

# Integrands of the form x^m (d+e x)^(n/2) (a+c x^2)^p
[x*sqrt(1+x)/(1+x^2),x,11,2*sqrt(1+x)+1/2*log(1+x+sqrt(2)-sqrt(1+x)*sqrt(2*(1+sqrt(2))))*sqrt(1/2*(1+sqrt(2)))-1/2*log(1+x+sqrt(2)+sqrt(1+x)*sqrt(2*(1+sqrt(2))))*sqrt(1/2*(1+sqrt(2)))+arctan((-2*sqrt(1+x)+sqrt(2*(1+sqrt(2))))/sqrt(2*(-1+sqrt(2))))/sqrt(2*(1+sqrt(2)))-arctan((2*sqrt(1+x)+sqrt(2*(1+sqrt(2))))/sqrt(2*(-1+sqrt(2))))/sqrt(2*(1+sqrt(2)))],

# Integrands of the form x^m / (d+e x)^1 (a+c x^2)^(p/2)

# p>0
[x^4*sqrt(a+c*x^2)/(d+e*x),x,9,1/60*(47*c*d^2-8*a*e^2)*(a+c*x^2)^(3/2)/(c^2*e^3)-13/20*d*(d+e*x)*(a+c*x^2)^(3/2)/(c*e^3)+1/5*(d+e*x)^2*(a+c*x^2)^(3/2)/(c*e^3)-1/8*d*(8*c^2*d^4+4*a*c*d^2*e^2-a^2*e^4)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/(c^(3/2)*e^6)-d^4*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))*sqrt(c*d^2+a*e^2)/e^6+1/8*d*(8*c*d^3-e*(4*c*d^2-a*e^2)*x)*sqrt(a+c*x^2)/(c*e^5)],
[x^3*sqrt(a+c*x^2)/(d+e*x),x,8,-7/12*d*(a+c*x^2)^(3/2)/(c*e^2)+1/4*(d+e*x)*(a+c*x^2)^(3/2)/(c*e^2)+1/8*(8*c^2*d^4+4*a*c*d^2*e^2-a^2*e^4)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/(c^(3/2)*e^5)+d^3*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))*sqrt(c*d^2+a*e^2)/e^5-1/8*(8*c*d^3-e*(4*c*d^2-a*e^2)*x)*sqrt(a+c*x^2)/(c*e^4)],
[x^2*sqrt(a+c*x^2)/(d+e*x),x,8,1/3*(a+c*x^2)^(3/2)/(c*e)-1/2*d*(2*c*d^2+a*e^2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/(e^4*sqrt(c))-d^2*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))*sqrt(c*d^2+a*e^2)/e^4+1/2*d*(2*d-e*x)*sqrt(a+c*x^2)/e^3],
[x*sqrt(a+c*x^2)/(d+e*x),x,6,1/2*(2*c*d^2+a*e^2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/(e^3*sqrt(c))+d*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))*sqrt(c*d^2+a*e^2)/e^3-1/2*(2*d-e*x)*sqrt(a+c*x^2)/e^2],
[sqrt(a+c*x^2)/(d+e*x),x,6,-d*arctanh(x*sqrt(c)/sqrt(a+c*x^2))*sqrt(c)/e^2-arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))*sqrt(c*d^2+a*e^2)/e^2+sqrt(a+c*x^2)/e],
[sqrt(a+c*x^2)/(x*(d+e*x)),x,9,-arctanh(sqrt(a+c*x^2)/sqrt(a))*sqrt(a)/d+arctanh(x*sqrt(c)/sqrt(a+c*x^2))*sqrt(c)/e+arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))*sqrt(c*d^2+a*e^2)/(d*e)],
[sqrt(a+c*x^2)/(x^2*(d+e*x)),x,15,e*arctanh(sqrt(a+c*x^2)/sqrt(a))*sqrt(a)/d^2-arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))*sqrt(c*d^2+a*e^2)/d^2-sqrt(a+c*x^2)/(d*x)],
[sqrt(a+c*x^2)/(x^3*(d+e*x)),x,19,-1/2*c*arctanh(sqrt(a+c*x^2)/sqrt(a))/(d*sqrt(a))-e^2*arctanh(sqrt(a+c*x^2)/sqrt(a))*sqrt(a)/d^3+e*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))*sqrt(c*d^2+a*e^2)/d^3-1/2*sqrt(a+c*x^2)/(d*x^2)+e*sqrt(a+c*x^2)/(d^2*x)],
[sqrt(a+c*x^2)/(x^4*(d+e*x)),x,20,-1/3*(a+c*x^2)^(3/2)/(a*d*x^3)+1/2*c*e*arctanh(sqrt(a+c*x^2)/sqrt(a))/(d^2*sqrt(a))+e^3*arctanh(sqrt(a+c*x^2)/sqrt(a))*sqrt(a)/d^4-e^2*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))*sqrt(c*d^2+a*e^2)/d^4+1/2*e*sqrt(a+c*x^2)/(d^2*x^2)-e^2*sqrt(a+c*x^2)/(d^3*x)],
[sqrt(a+c*x^2)/(x^5*(d+e*x)),x,25,1/3*e*(a+c*x^2)^(3/2)/(a*d^2*x^3)+1/8*c^2*arctanh(sqrt(a+c*x^2)/sqrt(a))/(a^(3/2)*d)-1/2*c*e^2*arctanh(sqrt(a+c*x^2)/sqrt(a))/(d^3*sqrt(a))-e^4*arctanh(sqrt(a+c*x^2)/sqrt(a))*sqrt(a)/d^5+e^3*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))*sqrt(c*d^2+a*e^2)/d^5-1/4*sqrt(a+c*x^2)/(d*x^4)-1/8*c*sqrt(a+c*x^2)/(a*d*x^2)-1/2*e^2*sqrt(a+c*x^2)/(d^3*x^2)+e^3*sqrt(a+c*x^2)/(d^4*x)],

# p<0
[x^4/((d+e*x)*sqrt(a+c*x^2)),x,8,-1/2*d*(2*c*d^2-a*e^2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/(c^(3/2)*e^4)-d^4*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(e^4*sqrt(c*d^2+a*e^2))+1/6*(11*c*d^2-4*a*e^2)*sqrt(a+c*x^2)/(c^2*e^3)-7/6*d*(d+e*x)*sqrt(a+c*x^2)/(c*e^3)+1/3*(d+e*x)^2*sqrt(a+c*x^2)/(c*e^3)],
[x^3/((d+e*x)*sqrt(a+c*x^2)),x,7,1/2*(2*c*d^2-a*e^2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/(c^(3/2)*e^3)+d^3*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(e^3*sqrt(c*d^2+a*e^2))-3/2*d*sqrt(a+c*x^2)/(c*e^2)+1/2*(d+e*x)*sqrt(a+c*x^2)/(c*e^2)],
[x^2/((d+e*x)*sqrt(a+c*x^2)),x,7,-d*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/(e^2*sqrt(c))-d^2*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(e^2*sqrt(c*d^2+a*e^2))+sqrt(a+c*x^2)/(c*e)],
[x/((d+e*x)*sqrt(a+c*x^2)),x,5,arctanh(x*sqrt(c)/sqrt(a+c*x^2))/(e*sqrt(c))+d*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(e*sqrt(c*d^2+a*e^2))],
[1/((d+e*x)*sqrt(a+c*x^2)),x,2,-arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/sqrt(c*d^2+a*e^2)],
[1/(x*(d+e*x)*sqrt(a+c*x^2)),x,7,-arctanh(sqrt(a+c*x^2)/sqrt(a))/(d*sqrt(a))+e*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(d*sqrt(c*d^2+a*e^2))],
[1/(x^2*(d+e*x)*sqrt(a+c*x^2)),x,8,e*arctanh(sqrt(a+c*x^2)/sqrt(a))/(d^2*sqrt(a))-e^2*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(d^2*sqrt(c*d^2+a*e^2))-sqrt(a+c*x^2)/(a*d*x)],
[1/(x^3*(d+e*x)*sqrt(a+c*x^2)),x,12,1/2*c*arctanh(sqrt(a+c*x^2)/sqrt(a))/(a^(3/2)*d)-e^2*arctanh(sqrt(a+c*x^2)/sqrt(a))/(d^3*sqrt(a))+e^3*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(d^3*sqrt(c*d^2+a*e^2))-1/2*sqrt(a+c*x^2)/(a*d*x^2)+e*sqrt(a+c*x^2)/(a*d^2*x)],
[x^4/((d+e*x)*(a+c*x^2)^(3/2)),x,7,-d*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/(c^(3/2)*e^2)-d^4*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(e^2*(c*d^2+a*e^2)^(3/2))+a*(a*e+c*d*x)/(c^2*(c*d^2+a*e^2)*sqrt(a+c*x^2))+sqrt(a+c*x^2)/(c^2*e)],
[x^3/((d+e*x)*(a+c*x^2)^(3/2)),x,6,arctanh(x*sqrt(c)/sqrt(a+c*x^2))/(c^(3/2)*e)+d^3*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(e*(c*d^2+a*e^2)^(3/2))+a*(d-e*x)/(c*(c*d^2+a*e^2)*sqrt(a+c*x^2))],
[x^2/((d+e*x)*(a+c*x^2)^(3/2)),x,4,-d^2*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(3/2)+(-a*e-c*d*x)/(c*(c*d^2+a*e^2)*sqrt(a+c*x^2))],
[x/((d+e*x)*(a+c*x^2)^(3/2)),x,4,d*e*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(3/2)+(-d+e*x)/((c*d^2+a*e^2)*sqrt(a+c*x^2))],
[1/((d+e*x)*(a+c*x^2)^(3/2)),x,4,-e^2*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(3/2)+(a*e+c*d*x)/(a*(c*d^2+a*e^2)*sqrt(a+c*x^2))],
[1/(x*(d+e*x)*(a+c*x^2)^(3/2)),x,10,e^3*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(d*(c*d^2+a*e^2)^(3/2))-arctanh(sqrt(a+c*x^2)/sqrt(a))/(a^(3/2)*d)+1/(a*d*sqrt(a+c*x^2))-e*(a*e+c*d*x)/(a*d*(c*d^2+a*e^2)*sqrt(a+c*x^2))],
[1/(x^2*(d+e*x)*(a+c*x^2)^(3/2)),x,12,-e^4*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(d^2*(c*d^2+a*e^2)^(3/2))+e*arctanh(sqrt(a+c*x^2)/sqrt(a))/(a^(3/2)*d^2)-e/(a*d^2*sqrt(a+c*x^2))+(-1)/(a*d*x*sqrt(a+c*x^2))-2*c*x/(a^2*d*sqrt(a+c*x^2))+e^2*(a*e+c*d*x)/(a*d^2*(c*d^2+a*e^2)*sqrt(a+c*x^2))],
[1/(x^3*(d+e*x)*(a+c*x^2)^(3/2)),x,17,e^5*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(d^3*(c*d^2+a*e^2)^(3/2))+3/2*c*arctanh(sqrt(a+c*x^2)/sqrt(a))/(a^(5/2)*d)-e^2*arctanh(sqrt(a+c*x^2)/sqrt(a))/(a^(3/2)*d^3)-3/2*c/(a^2*d*sqrt(a+c*x^2))+e^2/(a*d^3*sqrt(a+c*x^2))+(-1/2)/(a*d*x^2*sqrt(a+c*x^2))+e/(a*d^2*x*sqrt(a+c*x^2))+2*c*e*x/(a^2*d^2*sqrt(a+c*x^2))-e^3*(a*e+c*d*x)/(a*d^3*(c*d^2+a*e^2)*sqrt(a+c*x^2))],

# Integrands of the form x^m / (d+e x)^2 (a+c x^2)^(p/2)

# p>0

# p<0
[x^5/((d+e*x)^2*sqrt(a+c*x^2)),x,9,-d*(4*c*d^2-a*e^2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/(c^(3/2)*e^5)-d^4*(4*c*d^2+5*a*e^2)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(e^5*(c*d^2+a*e^2)^(3/2))+1/3*(13*c*d^2-2*a*e^2)*sqrt(a+c*x^2)/(c^2*e^4)+d^5*sqrt(a+c*x^2)/(e^4*(c*d^2+a*e^2)*(d+e*x))-5/3*d*(d+e*x)*sqrt(a+c*x^2)/(c*e^4)+1/3*(d+e*x)^2*sqrt(a+c*x^2)/(c*e^4)],
[x^4/((d+e*x)^2*sqrt(a+c*x^2)),x,8,1/2*(6*c*d^2-a*e^2)*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/(c^(3/2)*e^4)+d^3*(3*c*d^2+4*a*e^2)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(e^4*(c*d^2+a*e^2)^(3/2))-5/2*d*sqrt(a+c*x^2)/(c*e^3)-d^4*sqrt(a+c*x^2)/(e^3*(c*d^2+a*e^2)*(d+e*x))+1/2*(d+e*x)*sqrt(a+c*x^2)/(c*e^3)],
[x^3/((d+e*x)^2*sqrt(a+c*x^2)),x,7,-d^2*(2*c*d^2+3*a*e^2)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(e^3*(c*d^2+a*e^2)^(3/2))-2*d*arctanh(x*sqrt(c)/sqrt(a+c*x^2))/(e^3*sqrt(c))+sqrt(a+c*x^2)/(c*e^2)+d^3*sqrt(a+c*x^2)/(e^2*(c*d^2+a*e^2)*(d+e*x))],
[x^2/((d+e*x)^2*sqrt(a+c*x^2)),x,6,d*(c*d^2+2*a*e^2)*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(e^2*(c*d^2+a*e^2)^(3/2))+arctanh(x*sqrt(c)/sqrt(a+c*x^2))/(e^2*sqrt(c))-d^2*sqrt(a+c*x^2)/(e*(c*d^2+a*e^2)*(d+e*x))],
[x/((d+e*x)^2*sqrt(a+c*x^2)),x,3,-a*e*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(3/2)+d*sqrt(a+c*x^2)/((c*d^2+a*e^2)*(d+e*x))],
[1/((d+e*x)^2*sqrt(a+c*x^2)),x,3,-c*d*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(3/2)-e*sqrt(a+c*x^2)/((c*d^2+a*e^2)*(d+e*x))],
[1/(x*(d+e*x)^2*sqrt(a+c*x^2)),x,10,c*e*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(c*d^2+a*e^2)^(3/2)-arctanh(sqrt(a+c*x^2)/sqrt(a))/(d^2*sqrt(a))+e*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(d^2*sqrt(c*d^2+a*e^2))+e^2*sqrt(a+c*x^2)/(d*(c*d^2+a*e^2)*(d+e*x))],
[1/(x^2*(d+e*x)^2*sqrt(a+c*x^2)),x,11,-c*e^2*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(d*(c*d^2+a*e^2)^(3/2))+2*e*arctanh(sqrt(a+c*x^2)/sqrt(a))/(d^3*sqrt(a))-2*e^2*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(d^3*sqrt(c*d^2+a*e^2))-sqrt(a+c*x^2)/(a*d^2*x)-e^3*sqrt(a+c*x^2)/(d^2*(c*d^2+a*e^2)*(d+e*x))],
[1/(x^3*(d+e*x)^2*sqrt(a+c*x^2)),x,15,c*e^3*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(d^2*(c*d^2+a*e^2)^(3/2))+1/2*c*arctanh(sqrt(a+c*x^2)/sqrt(a))/(a^(3/2)*d^2)-3*e^2*arctanh(sqrt(a+c*x^2)/sqrt(a))/(d^4*sqrt(a))+3*e^3*arctanh((a*e-c*d*x)/(sqrt(c*d^2+a*e^2)*sqrt(a+c*x^2)))/(d^4*sqrt(c*d^2+a*e^2))-1/2*sqrt(a+c*x^2)/(a*d^2*x^2)+2*e*sqrt(a+c*x^2)/(a*d^3*x)+e^4*sqrt(a+c*x^2)/(d^3*(c*d^2+a*e^2)*(d+e*x))],

# Integrands of the form x^m (d+e x)^n (a+c x^2)^p when n symbolic

# p>0
[x^2*(a+b*x)^n*(c+d*x^2),x,2,a^2*(b^2*c+a^2*d)*(a+b*x)^(1+n)/(b^5*(1+n))-2*a*(b^2*c+2*a^2*d)*(a+b*x)^(2+n)/(b^5*(2+n))+(b^2*c+6*a^2*d)*(a+b*x)^(3+n)/(b^5*(3+n))-4*a*d*(a+b*x)^(4+n)/(b^5*(4+n))+d*(a+b*x)^(5+n)/(b^5*(5+n))],
[x*(a+b*x)^n*(c+d*x^2),x,2,-a*(b^2*c+a^2*d)*(a+b*x)^(1+n)/(b^4*(1+n))+(b^2*c+3*a^2*d)*(a+b*x)^(2+n)/(b^4*(2+n))-3*a*d*(a+b*x)^(3+n)/(b^4*(3+n))+d*(a+b*x)^(4+n)/(b^4*(4+n))],
[(a+b*x)^n*(c+d*x^2),x,2,(b^2*c+a^2*d)*(a+b*x)^(1+n)/(b^3*(1+n))-2*a*d*(a+b*x)^(2+n)/(b^3*(2+n))+d*(a+b*x)^(3+n)/(b^3*(3+n))],
[(a+b*x)^n*(c+d*x^2)/x,x,3,-a*d*(a+b*x)^(1+n)/(b^2*(1+n))+d*(a+b*x)^(2+n)/(b^2*(2+n))-c*(a+b*x)^(1+n)*hypergeom([1,1+n],[2+n],1+b*x/a)/(a*(1+n))],
[x^2*(a+b*x)^n*(c+d*x^2)^2,x,2,a^2*(b^2*c+a^2*d)^2*(a+b*x)^(1+n)/(b^7*(1+n))-2*a*(b^2*c+a^2*d)*(b^2*c+3*a^2*d)*(a+b*x)^(2+n)/(b^7*(2+n))+(b^4*c^2+12*a^2*b^2*c*d+15*a^4*d^2)*(a+b*x)^(3+n)/(b^7*(3+n))-4*a*d*(2*b^2*c+5*a^2*d)*(a+b*x)^(4+n)/(b^7*(4+n))+d*(2*b^2*c+15*a^2*d)*(a+b*x)^(5+n)/(b^7*(5+n))-6*a*d^2*(a+b*x)^(6+n)/(b^7*(6+n))+d^2*(a+b*x)^(7+n)/(b^7*(7+n))],
[x*(a+b*x)^n*(c+d*x^2)^2,x,2,-a*(b^2*c+a^2*d)^2*(a+b*x)^(1+n)/(b^6*(1+n))+(b^2*c+a^2*d)*(b^2*c+5*a^2*d)*(a+b*x)^(2+n)/(b^6*(2+n))-2*a*d*(3*b^2*c+5*a^2*d)*(a+b*x)^(3+n)/(b^6*(3+n))+2*d*(b^2*c+5*a^2*d)*(a+b*x)^(4+n)/(b^6*(4+n))-5*a*d^2*(a+b*x)^(5+n)/(b^6*(5+n))+d^2*(a+b*x)^(6+n)/(b^6*(6+n))],
[(a+b*x)^n*(c+d*x^2)^2,x,2,(b^2*c+a^2*d)^2*(a+b*x)^(1+n)/(b^5*(1+n))-4*a*d*(b^2*c+a^2*d)*(a+b*x)^(2+n)/(b^5*(2+n))+2*d*(b^2*c+3*a^2*d)*(a+b*x)^(3+n)/(b^5*(3+n))-4*a*d^2*(a+b*x)^(4+n)/(b^5*(4+n))+d^2*(a+b*x)^(5+n)/(b^5*(5+n))],
[(a+b*x)^n*(c+d*x^2)^2/x,x,4,-a*d*(2*b^2*c+a^2*d)*(a+b*x)^(1+n)/(b^4*(1+n))+d*(2*b^2*c+3*a^2*d)*(a+b*x)^(2+n)/(b^4*(2+n))-3*a*d^2*(a+b*x)^(3+n)/(b^4*(3+n))+d^2*(a+b*x)^(4+n)/(b^4*(4+n))-c^2*(a+b*x)^(1+n)*hypergeom([1,1+n],[2+n],1+b*x/a)/(a*(1+n))],
[x^2*(a+b*x)^n*(c+d*x^2)^3,x,2,a^2*(b^2*c+a^2*d)^3*(a+b*x)^(1+n)/(b^9*(1+n))-2*a*(b^2*c+a^2*d)^2*(b^2*c+4*a^2*d)*(a+b*x)^(2+n)/(b^9*(2+n))+(b^2*c+a^2*d)*(b^4*c^2+17*a^2*b^2*c*d+28*a^4*d^2)*(a+b*x)^(3+n)/(b^9*(3+n))-4*a*d*(3*b^4*c^2+15*a^2*b^2*c*d+14*a^4*d^2)*(a+b*x)^(4+n)/(b^9*(4+n))+d*(3*b^4*c^2+45*a^2*b^2*c*d+70*a^4*d^2)*(a+b*x)^(5+n)/(b^9*(5+n))-2*a*d^2*(9*b^2*c+28*a^2*d)*(a+b*x)^(6+n)/(b^9*(6+n))+d^2*(3*b^2*c+28*a^2*d)*(a+b*x)^(7+n)/(b^9*(7+n))-8*a*d^3*(a+b*x)^(8+n)/(b^9*(8+n))+d^3*(a+b*x)^(9+n)/(b^9*(9+n))],
[x*(a+b*x)^n*(c+d*x^2)^3,x,2,-a*(b^2*c+a^2*d)^3*(a+b*x)^(1+n)/(b^8*(1+n))+(b^2*c+a^2*d)^2*(b^2*c+7*a^2*d)*(a+b*x)^(2+n)/(b^8*(2+n))-3*a*d*(b^2*c+a^2*d)*(3*b^2*c+7*a^2*d)*(a+b*x)^(3+n)/(b^8*(3+n))+d*(3*b^4*c^2+30*a^2*b^2*c*d+35*a^4*d^2)*(a+b*x)^(4+n)/(b^8*(4+n))-5*a*d^2*(3*b^2*c+7*a^2*d)*(a+b*x)^(5+n)/(b^8*(5+n))+3*d^2*(b^2*c+7*a^2*d)*(a+b*x)^(6+n)/(b^8*(6+n))-7*a*d^3*(a+b*x)^(7+n)/(b^8*(7+n))+d^3*(a+b*x)^(8+n)/(b^8*(8+n))],
[(a+b*x)^n*(c+d*x^2)^3,x,2,(b^2*c+a^2*d)^3*(a+b*x)^(1+n)/(b^7*(1+n))-6*a*d*(b^2*c+a^2*d)^2*(a+b*x)^(2+n)/(b^7*(2+n))+3*d*(b^2*c+a^2*d)*(b^2*c+5*a^2*d)*(a+b*x)^(3+n)/(b^7*(3+n))-4*a*d^2*(3*b^2*c+5*a^2*d)*(a+b*x)^(4+n)/(b^7*(4+n))+3*d^2*(b^2*c+5*a^2*d)*(a+b*x)^(5+n)/(b^7*(5+n))-6*a*d^3*(a+b*x)^(6+n)/(b^7*(6+n))+d^3*(a+b*x)^(7+n)/(b^7*(7+n))],
[(a+b*x)^n*(c+d*x^2)^3/x,x,4,-a*d*(3*b^4*c^2+3*a^2*b^2*c*d+a^4*d^2)*(a+b*x)^(1+n)/(b^6*(1+n))+d*(3*b^4*c^2+9*a^2*b^2*c*d+5*a^4*d^2)*(a+b*x)^(2+n)/(b^6*(2+n))-a*d^2*(9*b^2*c+10*a^2*d)*(a+b*x)^(3+n)/(b^6*(3+n))+d^2*(3*b^2*c+10*a^2*d)*(a+b*x)^(4+n)/(b^6*(4+n))-5*a*d^3*(a+b*x)^(5+n)/(b^6*(5+n))+d^3*(a+b*x)^(6+n)/(b^6*(6+n))-c^3*(a+b*x)^(1+n)*hypergeom([1,1+n],[2+n],1+b*x/a)/(a*(1+n))],

# p<0
[x^4*(d+e*x)^n/(a+c*x^2),x,6,(c*d^2-a*e^2)*(d+e*x)^(1+n)/(c^2*e^3*(1+n))-2*d*(d+e*x)^(2+n)/(c*e^3*(2+n))+(d+e*x)^(3+n)/(c*e^3*(3+n))+1/2*(-a)^(3/2)*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(-e*sqrt(-a)+d*sqrt(c)))/(c^2*(1+n)*(-e*sqrt(-a)+d*sqrt(c)))-1/2*(-a)^(3/2)*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(c^2*(1+n)*(e*sqrt(-a)+d*sqrt(c)))],
[x^3*(d+e*x)^n/(a+c*x^2),x,6,-d*(d+e*x)^(1+n)/(c*e^2*(1+n))+(d+e*x)^(2+n)/(c*e^2*(2+n))+1/2*a*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(-e*sqrt(-a)+d*sqrt(c)))/(c^(3/2)*(1+n)*(-e*sqrt(-a)+d*sqrt(c)))+1/2*a*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/(c^(3/2)*(1+n)*(e*sqrt(-a)+d*sqrt(c)))],
[x^2*(d+e*x)^n/(a+c*x^2),x,6,(d+e*x)^(1+n)/(c*e*(1+n))+1/2*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(-e*sqrt(-a)+d*sqrt(c)))*sqrt(-a)/(c*(1+n)*(-e*sqrt(-a)+d*sqrt(c)))-1/2*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))*sqrt(-a)/(c*(1+n)*(e*sqrt(-a)+d*sqrt(c)))],
[x*(d+e*x)^n/(a+c*x^2),x,4,-1/2*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(-e*sqrt(-a)+d*sqrt(c)))/((1+n)*sqrt(c)*(-e*sqrt(-a)+d*sqrt(c)))-1/2*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/((1+n)*sqrt(c)*(e*sqrt(-a)+d*sqrt(c)))],
[(d+e*x)^n/(a+c*x^2),x,4,1/2*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(-e*sqrt(-a)+d*sqrt(c)))/((1+n)*sqrt(-a)*(-e*sqrt(-a)+d*sqrt(c)))-1/2*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/((1+n)*sqrt(-a)*(e*sqrt(-a)+d*sqrt(c)))],
[(d+e*x)^n/(x*(a+c*x^2)),x,7,-(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],1+e*x/d)/(a*d*(1+n))+1/2*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(-e*sqrt(-a)+d*sqrt(c)))*sqrt(c)/(a*(1+n)*(-e*sqrt(-a)+d*sqrt(c)))+1/2*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))*sqrt(c)/(a*(1+n)*(e*sqrt(-a)+d*sqrt(c)))],
[(d+e*x)^n/(x^2*(a+c*x^2)),x,7,e*(d+e*x)^(1+n)*hypergeom([2,1+n],[2+n],1+e*x/d)/(a*d^2*(1+n))+1/2*c*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(-e*sqrt(-a)+d*sqrt(c)))/((-a)^(3/2)*(1+n)*(-e*sqrt(-a)+d*sqrt(c)))-1/2*c*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/((-a)^(3/2)*(1+n)*(e*sqrt(-a)+d*sqrt(c)))],
[x^4*(d+e*x)^n/(a+c*x^2)^2,x,5,(d+e*x)^(1+n)/(c^2*e*(1+n))+1/2*a*(a*e+c*d*x)*(d+e*x)^(1+n)/(c^2*(c*d^2+a*e^2)*(a+c*x^2))-1/4*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))*(3*c*d^2*sqrt(-a)+a*e^2*(3+n)*sqrt(-a)-a*d*e*n*sqrt(c))/(c^2*(c*d^2+a*e^2)*(1+n)*(e*sqrt(-a)+d*sqrt(c)))+1/4*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(-e*sqrt(-a)+d*sqrt(c)))*(3*c*d^2*sqrt(-a)+a*e^2*(3+n)*sqrt(-a)+a*d*e*n*sqrt(c))/(c^2*(c*d^2+a*e^2)*(1+n)*(-e*sqrt(-a)+d*sqrt(c)))],
[x^3*(d+e*x)^n/(a+c*x^2)^2,x,5,1/2*a*(d-e*x)*(d+e*x)^(1+n)/(c*(c*d^2+a*e^2)*(a+c*x^2))+1/4*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(-e*sqrt(-a)+d*sqrt(c)))*(d*e*n*sqrt(-a)+(-2*c*d^2-a*e^2*(2+n))/sqrt(c))/(c*(c*d^2+a*e^2)*(1+n)*(-e*sqrt(-a)+d*sqrt(c)))-1/4*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))*(2*c*d^2+a*e^2*(2+n)+d*e*n*sqrt(-a)*sqrt(c))/(c^(3/2)*(c*d^2+a*e^2)*(1+n)*(e*sqrt(-a)+d*sqrt(c)))],
[x^2*(d+e*x)^n/(a+c*x^2)^2,x,5,-1/2*(a*e+c*d*x)*(d+e*x)^(1+n)/(c*(c*d^2+a*e^2)*(a+c*x^2))+1/4*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(-e*sqrt(-a)+d*sqrt(c)))*(c*d^2+a*e^2*(1+n)-d*e*n*sqrt(-a)*sqrt(c))/(c*(c*d^2+a*e^2)*(1+n)*sqrt(-a)*(-e*sqrt(-a)+d*sqrt(c)))-1/4*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))*(c*d^2+a*e^2*(1+n)+d*e*n*sqrt(-a)*sqrt(c))/(c*(c*d^2+a*e^2)*(1+n)*sqrt(-a)*(e*sqrt(-a)+d*sqrt(c)))],
[x*(d+e*x)^n/(a+c*x^2)^2,x,5,-1/2*(d-e*x)*(d+e*x)^(1+n)/((c*d^2+a*e^2)*(a+c*x^2))+1/4*e*n*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(-e*sqrt(-a)+d*sqrt(c)))*(e*sqrt(-a)+d*sqrt(c))/((c*d^2+a*e^2)*(1+n)*sqrt(-a)*sqrt(c)*(-e*sqrt(-a)+d*sqrt(c)))+1/4*e*n*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))*(a*e+d*sqrt(-a)*sqrt(c))/(a*(c*d^2+a*e^2)*(1+n)*sqrt(c)*(e*sqrt(-a)+d*sqrt(c)))],
[(d+e*x)^n/(a+c*x^2)^2,x,5,1/2*(a*e+c*d*x)*(d+e*x)^(1+n)/(a*(c*d^2+a*e^2)*(a+c*x^2))+1/4*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))*(c*d^2+a*e^2*(1-n)-d*e*n*sqrt(-a)*sqrt(c))/((-a)^(3/2)*(c*d^2+a*e^2)*(1+n)*(e*sqrt(-a)+d*sqrt(c)))-1/4*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(-e*sqrt(-a)+d*sqrt(c)))*(c*d^2+a*e^2*(1-n)+d*e*n*sqrt(-a)*sqrt(c))/((-a)^(3/2)*(c*d^2+a*e^2)*(1+n)*(-e*sqrt(-a)+d*sqrt(c)))],
[(d+e*x)^n/(x*(a+c*x^2)^2),x,12,1/2*c*(d-e*x)*(d+e*x)^(1+n)/(a*(c*d^2+a*e^2)*(a+c*x^2))-(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],1+e*x/d)/(a^2*d*(1+n))+1/2*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(-e*sqrt(-a)+d*sqrt(c)))*sqrt(c)/(a^2*(1+n)*(-e*sqrt(-a)+d*sqrt(c)))+1/2*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))*sqrt(c)/(a^2*(1+n)*(e*sqrt(-a)+d*sqrt(c)))+1/4*e*n*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(-e*sqrt(-a)+d*sqrt(c)))*sqrt(c)*(e*sqrt(-a)+d*sqrt(c))/((-a)^(3/2)*(c*d^2+a*e^2)*(1+n)*(-e*sqrt(-a)+d*sqrt(c)))-1/4*e*n*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))*sqrt(c)*(a*e+d*sqrt(-a)*sqrt(c))/(a^2*(c*d^2+a*e^2)*(1+n)*(e*sqrt(-a)+d*sqrt(c)))],
[(d+e*x)^n/(x^2*(a+c*x^2)^2),x,12,-1/2*c*(a*e+c*d*x)*(d+e*x)^(1+n)/(a^2*(c*d^2+a*e^2)*(a+c*x^2))+e*(d+e*x)^(1+n)*hypergeom([2,1+n],[2+n],1+e*x/d)/(a^2*d^2*(1+n))-1/2*c*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(-e*sqrt(-a)+d*sqrt(c)))/((-a)^(5/2)*(1+n)*(-e*sqrt(-a)+d*sqrt(c)))+1/2*c*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))/((-a)^(5/2)*(1+n)*(e*sqrt(-a)+d*sqrt(c)))+1/4*c*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(e*sqrt(-a)+d*sqrt(c)))*(c*d^2+a*e^2*(1-n)-d*e*n*sqrt(-a)*sqrt(c))/((-a)^(5/2)*(c*d^2+a*e^2)*(1+n)*(e*sqrt(-a)+d*sqrt(c)))-1/4*c*(d+e*x)^(1+n)*hypergeom([1,1+n],[2+n],(d+e*x)*sqrt(c)/(-e*sqrt(-a)+d*sqrt(c)))*(c*d^2+a*e^2*(1-n)+d*e*n*sqrt(-a)*sqrt(c))/((-a)^(5/2)*(c*d^2+a*e^2)*(1+n)*(-e*sqrt(-a)+d*sqrt(c)))],

# Integrands of the form (g x)^m (d+e x)^n (a+c x^2)^p when m and n symbolic
[(g*x)^m*(d+e*x)^n*(a+c*x^2)^2,x,6,-c*d*(2+m)*(c*d^2*(12+7*m+m^2)+2*a*e^2*(20+m^2+9*n+n^2+m*(9+2*n)))*(g*x)^(1+m)*(d+e*x)^(1+n)/(e^4*g*(2+m+n)*(3+m+n)*(4+m+n)*(5+m+n))+c*(c*d^2*(12+7*m+m^2)+2*a*e^2*(20+m^2+9*n+n^2+m*(9+2*n)))*(g*x)^(2+m)*(d+e*x)^(1+n)/(e^3*g^2*(3+m+n)*(4+m+n)*(5+m+n))-c^2*d*(4+m)*(g*x)^(3+m)*(d+e*x)^(1+n)/(e^2*g^3*(4+m+n)*(5+m+n))+c^2*(g*x)^(4+m)*(d+e*x)^(1+n)/(e*g^4*(5+m+n))+(a^2*e^4*(2+m+n)*(3+m+n)*(4+m+n)*(5+m+n)+c*d^2*(1+m)*(2+m)*(c*d^2*(12+7*m+m^2)+2*a*e^2*(20+m^2+9*n+n^2+m*(9+2*n))))*(g*x)^(1+m)*(d+e*x)^n*hypergeom([1+m,-n],[2+m],-e*x/d)/(e^4*g*(1+m)*(2+m+n)*(3+m+n)*(4+m+n)*(5+m+n)*(1+e*x/d)^n)],
[(g*x)^m*(d+e*x)^n*(a+c*x^2),x,4,-c*d*(2+m)*(g*x)^(1+m)*(d+e*x)^(1+n)/(e^2*g*(2+m+n)*(3+m+n))+c*(g*x)^(2+m)*(d+e*x)^(1+n)/(e*g^2*(3+m+n))+(c*d^2*(1+m)*(2+m)+a*e^2*(2+m+n)*(3+m+n))*(g*x)^(1+m)*(d+e*x)^n*hypergeom([1+m,-n],[2+m],-e*x/d)/(e^2*g*(1+m)*(2+m+n)*(3+m+n)*(1+e*x/d)^n)],
[(g*x)^m*(d+e*x)^n/(a+c*x^2),x,6,1/2*(g*x)^(1+m)*(d+e*x)^n*AppellF1(1+m,-n,1,2+m,-e*x/d,-x*sqrt(c)/sqrt(-a))/(a*g*(1+m)*(1+e*x/d)^n)+1/2*(g*x)^(1+m)*(d+e*x)^n*AppellF1(1+m,-n,1,2+m,-e*x/d,x*sqrt(c)/sqrt(-a))/(a*g*(1+m)*(1+e*x/d)^n)],
[(g*x)^m*(d+e*x)^n/(a+c*x^2)^2,x,12,1/4*(g*x)^(1+m)*(d+e*x)^n*AppellF1(1+m,-n,1,2+m,-e*x/d,-x*sqrt(c)/sqrt(-a))/(a^2*g*(1+m)*(1+e*x/d)^n)+1/4*(g*x)^(1+m)*(d+e*x)^n*AppellF1(1+m,-n,1,2+m,-e*x/d,x*sqrt(c)/sqrt(-a))/(a^2*g*(1+m)*(1+e*x/d)^n)+1/4*(g*x)^(1+m)*(d+e*x)^n*AppellF1(1+m,-n,2,2+m,-e*x/d,-x*sqrt(c)/sqrt(-a))/(a^2*g*(1+m)*(1+e*x/d)^n)+1/4*(g*x)^(1+m)*(d+e*x)^n*AppellF1(1+m,-n,2,2+m,-e*x/d,x*sqrt(c)/sqrt(-a))/(a^2*g*(1+m)*(1+e*x/d)^n)],

# Integrands of the form x^m (d+e x)^n (a+c x^2)^p when p symbolic

# n>0
[x^5*(d+e*x)*(a+b*x^2)^p,x,6,1/2*a^2*d*(a+b*x^2)^(1+p)/(b^3*(1+p))-a*d*(a+b*x^2)^(2+p)/(b^3*(2+p))+1/2*d*(a+b*x^2)^(3+p)/(b^3*(3+p))+1/7*e*x^7*(a+b*x^2)^p*hypergeom([7/2,-p],[9/2],-b*x^2/a)/(1+b*x^2/a)^p],
[x^4*(d+e*x)*(a+b*x^2)^p,x,6,1/2*a^2*e*(a+b*x^2)^(1+p)/(b^3*(1+p))-a*e*(a+b*x^2)^(2+p)/(b^3*(2+p))+1/2*e*(a+b*x^2)^(3+p)/(b^3*(3+p))+1/5*d*x^5*(a+b*x^2)^p*hypergeom([5/2,-p],[7/2],-b*x^2/a)/(1+b*x^2/a)^p],
[x^3*(d+e*x)*(a+b*x^2)^p,x,6,-1/2*a*d*(a+b*x^2)^(1+p)/(b^2*(1+p))+1/2*d*(a+b*x^2)^(2+p)/(b^2*(2+p))+1/5*e*x^5*(a+b*x^2)^p*hypergeom([5/2,-p],[7/2],-b*x^2/a)/(1+b*x^2/a)^p],
[x^2*(d+e*x)*(a+b*x^2)^p,x,6,-1/2*a*e*(a+b*x^2)^(1+p)/(b^2*(1+p))+1/2*e*(a+b*x^2)^(2+p)/(b^2*(2+p))+1/3*d*x^3*(a+b*x^2)^p*hypergeom([3/2,-p],[5/2],-b*x^2/a)/(1+b*x^2/a)^p],
[x*(d+e*x)*(a+b*x^2)^p,x,4,1/2*d*(a+b*x^2)^(1+p)/(b*(1+p))+1/3*e*x^3*(a+b*x^2)^p*hypergeom([3/2,-p],[5/2],-b*x^2/a)/(1+b*x^2/a)^p],
[(d+e*x)*(a+b*x^2)^p,x,3,1/2*e*(a+b*x^2)^(1+p)/(b*(1+p))+d*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(1+b*x^2/a)^p],
[(d+e*x)*(a+b*x^2)^p/x,x,5,e*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(1+b*x^2/a)^p-1/2*d*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1+b*x^2/a)/(a*(1+p))],
[(d+e*x)*(a+b*x^2)^p/x^2,x,5,-d*(a+b*x^2)^p*hypergeom([-1/2,-p],[1/2],-b*x^2/a)/(x*(1+b*x^2/a)^p)-1/2*e*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1+b*x^2/a)/(a*(1+p))],
[(d+e*x)*(a+b*x^2)^p/x^3,x,5,-e*(a+b*x^2)^p*hypergeom([-1/2,-p],[1/2],-b*x^2/a)/(x*(1+b*x^2/a)^p)+1/2*b*d*(a+b*x^2)^(1+p)*hypergeom([2,1+p],[2+p],1+b*x^2/a)/(a^2*(1+p))],
[x^5*(d+e*x)^2*(a+b*x^2)^p,x,7,1/2*a^2*(b*d^2-a*e^2)*(a+b*x^2)^(1+p)/(b^4*(1+p))-1/2*a*(2*b*d^2-3*a*e^2)*(a+b*x^2)^(2+p)/(b^4*(2+p))+1/2*(b*d^2-3*a*e^2)*(a+b*x^2)^(3+p)/(b^4*(3+p))+1/2*e^2*(a+b*x^2)^(4+p)/(b^4*(4+p))+2/7*d*e*x^7*(a+b*x^2)^p*hypergeom([7/2,-p],[9/2],-b*x^2/a)/(1+b*x^2/a)^p],
[x^4*(d+e*x)^2*(a+b*x^2)^p,x,8,a^2*d*e*(a+b*x^2)^(1+p)/(b^3*(1+p))+e^2*x^5*(a+b*x^2)^(1+p)/(b*(7+2*p))-2*a*d*e*(a+b*x^2)^(2+p)/(b^3*(2+p))+d*e*(a+b*x^2)^(3+p)/(b^3*(3+p))-1/5*(5*a*e^2-b*d^2*(7+2*p))*x^5*(a+b*x^2)^p*hypergeom([5/2,-p],[7/2],-b*x^2/a)/(b*(7+2*p)*(1+b*x^2/a)^p)],
[x^3*(d+e*x)^2*(a+b*x^2)^p,x,7,-1/2*a*(b*d^2-a*e^2)*(a+b*x^2)^(1+p)/(b^3*(1+p))+1/2*(b*d^2-2*a*e^2)*(a+b*x^2)^(2+p)/(b^3*(2+p))+1/2*e^2*(a+b*x^2)^(3+p)/(b^3*(3+p))+2/5*d*e*x^5*(a+b*x^2)^p*hypergeom([5/2,-p],[7/2],-b*x^2/a)/(1+b*x^2/a)^p],
[x^2*(d+e*x)^2*(a+b*x^2)^p,x,8,-a*d*e*(a+b*x^2)^(1+p)/(b^2*(1+p))+e^2*x^3*(a+b*x^2)^(1+p)/(b*(5+2*p))+d*e*(a+b*x^2)^(2+p)/(b^2*(2+p))-1/3*(3*a*e^2-b*d^2*(5+2*p))*x^3*(a+b*x^2)^p*hypergeom([3/2,-p],[5/2],-b*x^2/a)/(b*(5+2*p)*(1+b*x^2/a)^p)],
[x*(d+e*x)^2*(a+b*x^2)^p,x,7,1/2*(b*d^2-a*e^2)*(a+b*x^2)^(1+p)/(b^2*(1+p))+1/2*e^2*(a+b*x^2)^(2+p)/(b^2*(2+p))+2/3*d*e*x^3*(a+b*x^2)^p*hypergeom([3/2,-p],[5/2],-b*x^2/a)/(1+b*x^2/a)^p],
[(d+e*x)^2*(a+b*x^2)^p,x,4,d*e*(2+p)*(a+b*x^2)^(1+p)/(b*(3+5*p+2*p^2))+e*(d+e*x)*(a+b*x^2)^(1+p)/(b*(3+2*p))-(a*e^2-b*d^2*(3+2*p))*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(b*(3+2*p)*(1+b*x^2/a)^p)],
[(d+e*x)^2*(a+b*x^2)^p/x,x,7,1/2*e^2*(a+b*x^2)^(1+p)/(b*(1+p))+2*d*e*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(1+b*x^2/a)^p-1/2*d^2*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1+b*x^2/a)/(a*(1+p))],
[(d+e*x)^2*(a+b*x^2)^p/x^2,x,6,-d^2*(a+b*x^2)^(1+p)/(a*x)+(a*e^2+b*d^2*(1+2*p))*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(a*(1+b*x^2/a)^p)-d*e*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1+b*x^2/a)/(a*(1+p))],
[(d+e*x)^2*(a+b*x^2)^p/x^3,x,6,-1/2*d^2*(a+b*x^2)^(1+p)/(a*x^2)-2*d*e*(a+b*x^2)^p*hypergeom([-1/2,-p],[1/2],-b*x^2/a)/(x*(1+b*x^2/a)^p)-1/2*(a*e^2+b*d^2*p)*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1+b*x^2/a)/(a^2*(1+p))],
[x^5*(d+e*x)^3*(a+b*x^2)^p,x,7,1/2*a^2*d*(b*d^2-3*a*e^2)*(a+b*x^2)^(1+p)/(b^4*(1+p))+e^3*x^7*(a+b*x^2)^(1+p)/(b*(9+2*p))-1/2*a*d*(2*b*d^2-9*a*e^2)*(a+b*x^2)^(2+p)/(b^4*(2+p))+1/2*d*(b*d^2-9*a*e^2)*(a+b*x^2)^(3+p)/(b^4*(3+p))+3/2*d*e^2*(a+b*x^2)^(4+p)/(b^4*(4+p))-1/7*e*(7*a*e^2-3*b*d^2*(9+2*p))*x^7*(a+b*x^2)^p*hypergeom([7/2,-p],[9/2],-b*x^2/a)/(b*(9+2*p)*(1+b*x^2/a)^p)],
[x^4*(d+e*x)^3*(a+b*x^2)^p,x,7,1/2*a^2*e*(3*b*d^2-a*e^2)*(a+b*x^2)^(1+p)/(b^4*(1+p))+3*d*e^2*x^5*(a+b*x^2)^(1+p)/(b*(7+2*p))-3/2*a*e*(2*b*d^2-a*e^2)*(a+b*x^2)^(2+p)/(b^4*(2+p))+3/2*e*(b*d^2-a*e^2)*(a+b*x^2)^(3+p)/(b^4*(3+p))+1/2*e^3*(a+b*x^2)^(4+p)/(b^4*(4+p))-1/5*d*(15*a*e^2-b*d^2*(7+2*p))*x^5*(a+b*x^2)^p*hypergeom([5/2,-p],[7/2],-b*x^2/a)/(b*(7+2*p)*(1+b*x^2/a)^p)],
[x^3*(d+e*x)^3*(a+b*x^2)^p,x,7,-1/2*a*d*(b*d^2-3*a*e^2)*(a+b*x^2)^(1+p)/(b^3*(1+p))+e^3*x^5*(a+b*x^2)^(1+p)/(b*(7+2*p))+1/2*d*(b*d^2-6*a*e^2)*(a+b*x^2)^(2+p)/(b^3*(2+p))+3/2*d*e^2*(a+b*x^2)^(3+p)/(b^3*(3+p))-1/5*e*(5*a*e^2-3*b*d^2*(7+2*p))*x^5*(a+b*x^2)^p*hypergeom([5/2,-p],[7/2],-b*x^2/a)/(b*(7+2*p)*(1+b*x^2/a)^p)],
[x^2*(d+e*x)^3*(a+b*x^2)^p,x,7,-1/2*a*e*(3*b*d^2-a*e^2)*(a+b*x^2)^(1+p)/(b^3*(1+p))+3*d*e^2*x^3*(a+b*x^2)^(1+p)/(b*(5+2*p))+1/2*e*(3*b*d^2-2*a*e^2)*(a+b*x^2)^(2+p)/(b^3*(2+p))+1/2*e^3*(a+b*x^2)^(3+p)/(b^3*(3+p))-1/3*d*(9*a*e^2-b*d^2*(5+2*p))*x^3*(a+b*x^2)^p*hypergeom([3/2,-p],[5/2],-b*x^2/a)/(b*(5+2*p)*(1+b*x^2/a)^p)],
[x*(d+e*x)^3*(a+b*x^2)^p,x,7,1/2*d*(b*d^2-3*a*e^2)*(a+b*x^2)^(1+p)/(b^2*(1+p))+e^3*x^3*(a+b*x^2)^(1+p)/(b*(5+2*p))+3/2*d*e^2*(a+b*x^2)^(2+p)/(b^2*(2+p))-e*(a*e^2-b*d^2*(5+2*p))*x^3*(a+b*x^2)^p*hypergeom([3/2,-p],[5/2],-b*x^2/a)/(b*(5+2*p)*(1+b*x^2/a)^p)],
[(d+e*x)^3*(a+b*x^2)^p,x,4,-1/2*e*(a*e^2-3*b*d^2*(2+p))*(a+b*x^2)^(1+p)/(b^2*(1+p)*(2+p))+3*d*e^2*x*(a+b*x^2)^(1+p)/(b*(3+2*p))+1/2*e^3*x^2*(a+b*x^2)^(1+p)/(b*(2+p))-d*(3*a*e^2-b*d^2*(3+2*p))*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(b*(3+2*p)*(1+b*x^2/a)^p),1/2*e*(d+e*x)^2*(a+b*x^2)^(1+p)/(b*(2+p))-1/2*e*((3+2*p)*(a*e^2-b*d^2*(5+2*p))-2*b*d*e*(1+p)*(3+p)*x)*(a+b*x^2)^(1+p)/(b^2*(2+p)*(3+5*p+2*p^2))-d*(3*a*e^2-b*d^2*(3+2*p))*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(b*(3+2*p)*(1+b*x^2/a)^p)],
[(d+e*x)^3*(a+b*x^2)^p/x,x,7,3/2*d*e^2*(a+b*x^2)^(1+p)/(b*(1+p))+e^3*x*(a+b*x^2)^(1+p)/(b*(3+2*p))-e*(a*e^2-3*b*d^2*(3+2*p))*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(b*(3+2*p)*(1+b*x^2/a)^p)-1/2*d^3*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1+b*x^2/a)/(a*(1+p))],
[(d+e*x)^3*(a+b*x^2)^p/x^2,x,8,1/2*e^3*(a+b*x^2)^(1+p)/(b*(1+p))-d^3*(a+b*x^2)^(1+p)/(a*x)+d*(3*a*e^2+b*d^2*(1+2*p))*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(a*(1+b*x^2/a)^p)-3/2*d^2*e*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1+b*x^2/a)/(a*(1+p))],
[(d+e*x)^3*(a+b*x^2)^p/x^3,x,7,-1/2*d^3*(a+b*x^2)^(1+p)/(a*x^2)-3*d^2*e*(a+b*x^2)^(1+p)/(a*x)+e*(a*e^2+3*b*d^2*(1+2*p))*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(a*(1+b*x^2/a)^p)-1/2*d*(3*a*e^2+b*d^2*p)*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1+b*x^2/a)/(a^2*(1+p))],

# n<0
[x^4*(a+b*x^2)^p/(d+e*x),x,7,1/2*(b*d^2-a*e^2)*(a+b*x^2)^(1+p)/(b^2*e^3*(1+p))+1/2*(a+b*x^2)^(2+p)/(b^2*e*(2+p))+1/5*x^5*(a+b*x^2)^p*AppellF1(5/2,-p,1,7/2,-b*x^2/a,e^2*x^2/d^2)/(d*(1+b*x^2/a)^p)-1/2*d^4*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(e^3*(b*d^2+a*e^2)*(1+p))],
[x^3*(a+b*x^2)^p/(d+e*x),x,6,-1/2*d*(a+b*x^2)^(1+p)/(b*e^2*(1+p))-1/5*e*x^5*(a+b*x^2)^p*AppellF1(5/2,-p,1,7/2,-b*x^2/a,e^2*x^2/d^2)/(d^2*(1+b*x^2/a)^p)+1/2*d^3*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(e^2*(b*d^2+a*e^2)*(1+p))],
[x^2*(a+b*x^2)^p/(d+e*x),x,6,1/2*(a+b*x^2)^(1+p)/(b*e*(1+p))+1/3*x^3*(a+b*x^2)^p*AppellF1(3/2,-p,1,5/2,-b*x^2/a,e^2*x^2/d^2)/(d*(1+b*x^2/a)^p)-1/2*d^2*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(e*(b*d^2+a*e^2)*(1+p))],
[x*(a+b*x^2)^p/(d+e*x),x,9,-x*(a+b*x^2)^p*AppellF1(1/2,-p,1,3/2,-b*x^2/a,e^2*x^2/d^2)/(e*(1+b*x^2/a)^p)+x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(e*(1+b*x^2/a)^p)+1/2*d*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/((b*d^2+a*e^2)*(1+p))],
[(a+b*x^2)^p/(d+e*x),x,6,x*(a+b*x^2)^p*AppellF1(1/2,-p,1,3/2,-b*x^2/a,e^2*x^2/d^2)/(d*(1+b*x^2/a)^p)-1/2*e*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/((b*d^2+a*e^2)*(1+p))],
[(a+b*x^2)^p/(x*(d+e*x)),x,7,-e*x*(a+b*x^2)^p*AppellF1(1/2,-p,1,3/2,-b*x^2/a,e^2*x^2/d^2)/(d^2*(1+b*x^2/a)^p)+1/2*e^2*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(d*(b*d^2+a*e^2)*(1+p))-1/2*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1+b*x^2/a)/(a*d*(1+p))],
[(a+b*x^2)^p/(x^2*(d+e*x)),x,7,-(a+b*x^2)^p*AppellF1(-1/2,-p,1,1/2,-b*x^2/a,e^2*x^2/d^2)/(d*x*(1+b*x^2/a)^p)-1/2*e^3*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(d^2*(b*d^2+a*e^2)*(1+p))+1/2*e*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1+b*x^2/a)/(a*d^2*(1+p))],
[(a+b*x^2)^p/(x^3*(d+e*x)),x,8,-1/2*(a+b*x^2)^(1+p)/(a*d*x^2)+e*(a+b*x^2)^p*AppellF1(-1/2,-p,1,1/2,-b*x^2/a,e^2*x^2/d^2)/(d^2*x*(1+b*x^2/a)^p)+1/2*e^4*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(d^3*(b*d^2+a*e^2)*(1+p))-1/2*(a*e^2+b*d^2*p)*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1+b*x^2/a)/(a^2*d^3*(1+p))],
[x^4*(a+b*x^2)^p/(d+e*x)^2,x,12,-d*(4+3*p)*(a+b*x^2)^(1+p)/(b*e^3*(1+p)*(3+2*p))-d^4*(a+b*x^2)^(1+p)/(e^3*(b*d^2+a*e^2)*(d+e*x))+(d+e*x)*(a+b*x^2)^(1+p)/(b*e^3*(3+2*p))-2*d^2*(2*a*e^2+b*d^2*(2+p))*x*(a+b*x^2)^p*AppellF1(1/2,-p,1,3/2,-b*x^2/a,e^2*x^2/d^2)/(e^4*(b*d^2+a*e^2)*(1+b*x^2/a)^p)-(a^2*e^4-2*a*b*d^2*e^2*(4+3*p)-2*b^2*d^4*(6+7*p+2*p^2))*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(b*e^4*(b*d^2+a*e^2)*(3+2*p)*(1+b*x^2/a)^p)+d^3*(2*a*e^2+b*d^2*(2+p))*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(e^3*(b*d^2+a*e^2)^2*(1+p))],
[x^3*(a+b*x^2)^p/(d+e*x)^2,x,11,1/2*(a+b*x^2)^(1+p)/(b*e^2*(1+p))+d^3*(a+b*x^2)^(1+p)/(e^2*(b*d^2+a*e^2)*(d+e*x))+d*(3*a*e^2+b*d^2*(3+2*p))*x*(a+b*x^2)^p*AppellF1(1/2,-p,1,3/2,-b*x^2/a,e^2*x^2/d^2)/(e^3*(b*d^2+a*e^2)*(1+b*x^2/a)^p)-d*(2*a*e^2+b*d^2*(3+2*p))*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(e^3*(b*d^2+a*e^2)*(1+b*x^2/a)^p)-1/2*d^2*(3*a*e^2+b*d^2*(3+2*p))*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(e^2*(b*d^2+a*e^2)^2*(1+p))],
[x^2*(a+b*x^2)^p/(d+e*x)^2,x,10,-d^2*(a+b*x^2)^(1+p)/(e*(b*d^2+a*e^2)*(d+e*x))-2*(a*e^2+b*d^2*(1+p))*x*(a+b*x^2)^p*AppellF1(1/2,-p,1,3/2,-b*x^2/a,e^2*x^2/d^2)/(e^2*(b*d^2+a*e^2)*(1+b*x^2/a)^p)+(a*e^2+2*b*d^2*(1+p))*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(e^2*(b*d^2+a*e^2)*(1+b*x^2/a)^p)+d*(a*e^2+b*d^2*(1+p))*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(e*(b*d^2+a*e^2)^2*(1+p))],
[x*(a+b*x^2)^p/(d+e*x)^2,x,10,d*(a+b*x^2)^(1+p)/((b*d^2+a*e^2)*(d+e*x))+(a*e^2+b*d^2*(1+2*p))*x*(a+b*x^2)^p*AppellF1(1/2,-p,1,3/2,-b*x^2/a,e^2*x^2/d^2)/(d*e*(b*d^2+a*e^2)*(1+b*x^2/a)^p)-b*d*(1+2*p)*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(e*(b*d^2+a*e^2)*(1+b*x^2/a)^p)-1/2*(a*e^2+b*d^2*(1+2*p))*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/((b*d^2+a*e^2)^2*(1+p))],
[(a+b*x^2)^p/(d+e*x)^2,x,8,e^2*x*(a+b*x^2)^(1+p)/((b*d^2+a*e^2)*(d^2-e^2*x^2))-2*b*p*x*(a+b*x^2)^p*AppellF1(1/2,-p,1,3/2,-b*x^2/a,e^2*x^2/d^2)/((b*d^2+a*e^2)*(1+b*x^2/a)^p)+b*(1+2*p)*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/((b*d^2+a*e^2)*(1+b*x^2/a)^p)-b*d*e*(a+b*x^2)^(1+p)*hypergeom([2,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/((b*d^2+a*e^2)^2*(1+p)),x*(a+b*x^2)^p*AppellF1(1/2,-p,2,3/2,-b*x^2/a,e^2*x^2/d^2)/(d^2*(1+b*x^2/a)^p)+1/3*e^2*x^3*(a+b*x^2)^p*AppellF1(3/2,-p,2,5/2,-b*x^2/a,e^2*x^2/d^2)/(d^4*(1+b*x^2/a)^p)-b*d*e*(a+b*x^2)^(1+p)*hypergeom([2,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/((b*d^2+a*e^2)^2*(1+p))],
[(a+b*x^2)^p/(x*(d+e*x)^2),x,18,-e*x*(a+b*x^2)^p*AppellF1(1/2,-p,1,3/2,-b*x^2/a,e^2*x^2/d^2)/(d^3*(1+b*x^2/a)^p)-e*x*(a+b*x^2)^p*AppellF1(1/2,-p,2,3/2,-b*x^2/a,e^2*x^2/d^2)/(d^3*(1+b*x^2/a)^p)-1/3*e^3*x^3*(a+b*x^2)^p*AppellF1(3/2,-p,2,5/2,-b*x^2/a,e^2*x^2/d^2)/(d^5*(1+b*x^2/a)^p)+1/2*e^2*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(d^2*(b*d^2+a*e^2)*(1+p))-1/2*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1+b*x^2/a)/(a*d^2*(1+p))+b*e^2*(a+b*x^2)^(1+p)*hypergeom([2,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/((b*d^2+a*e^2)^2*(1+p))],
[(a+b*x^2)^p/(x^2*(d+e*x)^2),x,20,2*e^2*x*(a+b*x^2)^p*AppellF1(1/2,-p,1,3/2,-b*x^2/a,e^2*x^2/d^2)/(d^4*(1+b*x^2/a)^p)+e^2*x*(a+b*x^2)^p*AppellF1(1/2,-p,2,3/2,-b*x^2/a,e^2*x^2/d^2)/(d^4*(1+b*x^2/a)^p)+1/3*e^4*x^3*(a+b*x^2)^p*AppellF1(3/2,-p,2,5/2,-b*x^2/a,e^2*x^2/d^2)/(d^6*(1+b*x^2/a)^p)-(a+b*x^2)^p*hypergeom([-1/2,-p],[1/2],-b*x^2/a)/(d^2*x*(1+b*x^2/a)^p)-e^3*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(d^3*(b*d^2+a*e^2)*(1+p))+e*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1+b*x^2/a)/(a*d^3*(1+p))-b*e^3*(a+b*x^2)^(1+p)*hypergeom([2,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(d*(b*d^2+a*e^2)^2*(1+p))],
[x^4*(a+b*x^2)^p/(d+e*x)^3,x,12,1/2*(a+b*x^2)^(1+p)/(b*e^3*(1+p))-1/2*d^4*(a+b*x^2)^(1+p)/(e^3*(b*d^2+a*e^2)*(d+e*x)^2)+d^3*(4*a*e^2+b*d^2*(3+p))*(a+b*x^2)^(1+p)/(e^3*(b*d^2+a*e^2)^2*(d+e*x))+d*(6*a^2*e^4+3*a*b*d^2*e^2*(4+3*p)+b^2*d^4*(6+7*p+2*p^2))*x*(a+b*x^2)^p*AppellF1(1/2,-p,1,3/2,-b*x^2/a,e^2*x^2/d^2)/(e^4*(b*d^2+a*e^2)^2*(1+b*x^2/a)^p)-d*(3*a^2*e^4+2*a*b*d^2*e^2*(5+4*p)+b^2*d^4*(6+7*p+2*p^2))*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(e^4*(b*d^2+a*e^2)^2*(1+b*x^2/a)^p)-1/2*d^2*(6*a^2*e^4+3*a*b*d^2*e^2*(4+3*p)+b^2*d^4*(6+7*p+2*p^2))*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(e^3*(b*d^2+a*e^2)^3*(1+p))],
[x^3*(a+b*x^2)^p/(d+e*x)^3,x,11,1/2*d^3*(a+b*x^2)^(1+p)/(e^2*(b*d^2+a*e^2)*(d+e*x)^2)-d^2*(3*a*e^2+b*d^2*(2+p))*(a+b*x^2)^(1+p)/(e^2*(b*d^2+a*e^2)^2*(d+e*x))-(3*a^2*e^4+a*b*d^2*e^2*(6+7*p)+b^2*d^4*(3+5*p+2*p^2))*x*(a+b*x^2)^p*AppellF1(1/2,-p,1,3/2,-b*x^2/a,e^2*x^2/d^2)/(e^3*(b*d^2+a*e^2)^2*(1+b*x^2/a)^p)+(a^2*e^4+a*b*d^2*e^2*(5+6*p)+b^2*d^4*(3+5*p+2*p^2))*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(e^3*(b*d^2+a*e^2)^2*(1+b*x^2/a)^p)+1/2*d*(3*a^2*e^4+a*b*d^2*e^2*(6+7*p)+b^2*d^4*(3+5*p+2*p^2))*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(e^2*(b*d^2+a*e^2)^3*(1+p))],
[x^2*(a+b*x^2)^p/(d+e*x)^3,x,11,-1/2*d^2*(a+b*x^2)^(1+p)/(e*(b*d^2+a*e^2)*(d+e*x)^2)+d*(2*a*e^2+b*d^2*(1+p))*(a+b*x^2)^(1+p)/(e*(b*d^2+a*e^2)^2*(d+e*x))+(a^2*e^4+a*b*d^2*e^2*(2+5*p)+b^2*d^4*(1+3*p+2*p^2))*x*(a+b*x^2)^p*AppellF1(1/2,-p,1,3/2,-b*x^2/a,e^2*x^2/d^2)/(d*e^2*(b*d^2+a*e^2)^2*(1+b*x^2/a)^p)-b*d*(1+2*p)*(2*a*e^2+b*d^2*(1+p))*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(e^2*(b*d^2+a*e^2)^2*(1+b*x^2/a)^p)-1/2*(a^2*e^4+a*b*d^2*e^2*(2+5*p)+b^2*d^4*(1+3*p+2*p^2))*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(e*(b*d^2+a*e^2)^3*(1+p))],
[x*(a+b*x^2)^p/(d+e*x)^3,x,11,1/2*d*(a+b*x^2)^(1+p)/((b*d^2+a*e^2)*(d+e*x)^2)-(a*e^2+b*d^2*p)*(a+b*x^2)^(1+p)/((b*d^2+a*e^2)^2*(d+e*x))-b*p*(3*a*e^2+b*d^2*(1+2*p))*x*(a+b*x^2)^p*AppellF1(1/2,-p,1,3/2,-b*x^2/a,e^2*x^2/d^2)/(e*(b*d^2+a*e^2)^2*(1+b*x^2/a)^p)+b*(1+2*p)*(a*e^2+b*d^2*p)*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(e*(b*d^2+a*e^2)^2*(1+b*x^2/a)^p)+1/2*b*d*p*(3*a*e^2+b*d^2*(1+2*p))*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/((b*d^2+a*e^2)^3*(1+p))],
[(a+b*x^2)^p/(d+e*x)^3,x,11,-1/4*d^2*e*(a+b*x^2)^(1+p)/((b*d^2+a*e^2)*(d^2-e^2*x^2)^2)+x*(a+b*x^2)^p*AppellF1(1/2,-p,3,3/2,-b*x^2/a,e^2*x^2/d^2)/(d^3*(1+b*x^2/a)^p)+e^2*x^3*(a+b*x^2)^p*AppellF1(3/2,-p,3,5/2,-b*x^2/a,e^2*x^2/d^2)/(d^5*(1+b*x^2/a)^p)+1/4*b*e*(2*a*e^2+b*d^2*(1+p))*(a+b*x^2)^(1+p)*hypergeom([2,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/((b*d^2+a*e^2)^3*(1+p))-3/2*b^2*d^2*e*(a+b*x^2)^(1+p)*hypergeom([3,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/((b*d^2+a*e^2)^3*(1+p))],
[(a+b*x^2)^p/(x*(d+e*x)^3),x,29,1/4*d*e^2*(a+b*x^2)^(1+p)/((b*d^2+a*e^2)*(d^2-e^2*x^2)^2)-e*x*(a+b*x^2)^p*AppellF1(1/2,-p,1,3/2,-b*x^2/a,e^2*x^2/d^2)/(d^4*(1+b*x^2/a)^p)-e*x*(a+b*x^2)^p*AppellF1(1/2,-p,2,3/2,-b*x^2/a,e^2*x^2/d^2)/(d^4*(1+b*x^2/a)^p)-e*x*(a+b*x^2)^p*AppellF1(1/2,-p,3,3/2,-b*x^2/a,e^2*x^2/d^2)/(d^4*(1+b*x^2/a)^p)-1/3*e^3*x^3*(a+b*x^2)^p*AppellF1(3/2,-p,2,5/2,-b*x^2/a,e^2*x^2/d^2)/(d^6*(1+b*x^2/a)^p)-e^3*x^3*(a+b*x^2)^p*AppellF1(3/2,-p,3,5/2,-b*x^2/a,e^2*x^2/d^2)/(d^6*(1+b*x^2/a)^p)+1/2*e^2*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(d^3*(b*d^2+a*e^2)*(1+p))-1/2*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1+b*x^2/a)/(a*d^3*(1+p))+b*e^2*(a+b*x^2)^(1+p)*hypergeom([2,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(d*(b*d^2+a*e^2)^2*(1+p))-1/4*b*e^2*(2*a*e^2+b*d^2*(1+p))*(a+b*x^2)^(1+p)*hypergeom([2,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(d*(b*d^2+a*e^2)^3*(1+p))+3/2*b^2*d*e^2*(a+b*x^2)^(1+p)*hypergeom([3,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/((b*d^2+a*e^2)^3*(1+p))],
[(a+b*x^2)^p/(x^2*(d+e*x)^3),x,31,-1/4*e^3*(a+b*x^2)^(1+p)/((b*d^2+a*e^2)*(d^2-e^2*x^2)^2)+3*e^2*x*(a+b*x^2)^p*AppellF1(1/2,-p,1,3/2,-b*x^2/a,e^2*x^2/d^2)/(d^5*(1+b*x^2/a)^p)+2*e^2*x*(a+b*x^2)^p*AppellF1(1/2,-p,2,3/2,-b*x^2/a,e^2*x^2/d^2)/(d^5*(1+b*x^2/a)^p)+e^2*x*(a+b*x^2)^p*AppellF1(1/2,-p,3,3/2,-b*x^2/a,e^2*x^2/d^2)/(d^5*(1+b*x^2/a)^p)+2/3*e^4*x^3*(a+b*x^2)^p*AppellF1(3/2,-p,2,5/2,-b*x^2/a,e^2*x^2/d^2)/(d^7*(1+b*x^2/a)^p)+e^4*x^3*(a+b*x^2)^p*AppellF1(3/2,-p,3,5/2,-b*x^2/a,e^2*x^2/d^2)/(d^7*(1+b*x^2/a)^p)-(a+b*x^2)^p*hypergeom([-1/2,-p],[1/2],-b*x^2/a)/(d^3*x*(1+b*x^2/a)^p)-3/2*e^3*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(d^4*(b*d^2+a*e^2)*(1+p))+3/2*e*(a+b*x^2)^(1+p)*hypergeom([1,1+p],[2+p],1+b*x^2/a)/(a*d^4*(1+p))-2*b*e^3*(a+b*x^2)^(1+p)*hypergeom([2,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(d^2*(b*d^2+a*e^2)^2*(1+p))+1/4*b*e^3*(2*a*e^2+b*d^2*(1+p))*(a+b*x^2)^(1+p)*hypergeom([2,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/(d^2*(b*d^2+a*e^2)^3*(1+p))-3/2*b^2*e^3*(a+b*x^2)^(1+p)*hypergeom([3,1+p],[2+p],e^2*(a+b*x^2)/(b*d^2+a*e^2))/((b*d^2+a*e^2)^3*(1+p))],

# Integrands of the form (g x)^m (d+e x)^n (a+c x^2)^p when m and p symbolic
[(g*x)^m*(d+e*x)^3*(a+c*x^2)^p,x,7,3*d*e^2*(g*x)^(1+m)*(a+c*x^2)^(1+p)/(c*g*(3+m+2*p))+e^3*(g*x)^(2+m)*(a+c*x^2)^(1+p)/(c*g^2*(4+m+2*p))-d*(3*a*e^2*(1+m)-c*d^2*(3+m+2*p))*(g*x)^(1+m)*(a+c*x^2)^p*hypergeom([1/2*(1+m),-p],[1/2*(3+m)],-c*x^2/a)/(c*g*(1+m)*(3+m+2*p)*(1+c*x^2/a)^p)-e*(a*e^2*(2+m)-3*c*d^2*(4+m+2*p))*(g*x)^(2+m)*(a+c*x^2)^p*hypergeom([1/2*(2+m),-p],[1/2*(4+m)],-c*x^2/a)/(c*g^2*(2+m)*(4+m+2*p)*(1+c*x^2/a)^p)],
[(g*x)^m*(d+e*x)^2*(a+c*x^2)^p,x,6,e^2*(g*x)^(1+m)*(a+c*x^2)^(1+p)/(c*g*(3+m+2*p))-(a*e^2*(1+m)-c*d^2*(3+m+2*p))*(g*x)^(1+m)*(a+c*x^2)^p*hypergeom([1/2*(1+m),-p],[1/2*(3+m)],-c*x^2/a)/(c*g*(1+m)*(3+m+2*p)*(1+c*x^2/a)^p)+2*d*e*(g*x)^(2+m)*(a+c*x^2)^p*hypergeom([1/2*(2+m),-p],[1/2*(4+m)],-c*x^2/a)/(g^2*(2+m)*(1+c*x^2/a)^p)],
[(g*x)^m*(d+e*x)*(a+c*x^2)^p,x,5,d*(g*x)^(1+m)*(a+c*x^2)^p*hypergeom([1/2*(1+m),-p],[1/2*(3+m)],-c*x^2/a)/(g*(1+m)*(1+c*x^2/a)^p)+e*(g*x)^(2+m)*(a+c*x^2)^p*hypergeom([1/2*(2+m),-p],[1/2*(4+m)],-c*x^2/a)/(g^2*(2+m)*(1+c*x^2/a)^p)],
[(g*x)^m*(a+c*x^2)^p,x,2,(g*x)^(1+m)*(a+c*x^2)^p*hypergeom([1/2*(1+m),-p],[1/2*(3+m)],-c*x^2/a)/(g*(1+m)*(1+c*x^2/a)^p)],
[(g*x)^m*(a+c*x^2)^p/(d+e*x),x,5,x*(g*x)^m*(a+c*x^2)^p*AppellF1(1/2*(1+m),-p,1,1/2*(3+m),-c*x^2/a,e^2*x^2/d^2)/(d*(1+m)*(1+c*x^2/a)^p)-e*x^2*(g*x)^m*(a+c*x^2)^p*AppellF1(1/2*(2+m),-p,1,1/2*(4+m),-c*x^2/a,e^2*x^2/d^2)/(d^2*(2+m)*(1+c*x^2/a)^p)],
[(g*x)^m*(a+c*x^2)^p/(d+e*x)^2,x,8,x*(g*x)^m*(a+c*x^2)^p*AppellF1(1/2*(1+m),-p,2,1/2*(3+m),-c*x^2/a,e^2*x^2/d^2)/(d^2*(1+m)*(1+c*x^2/a)^p)-2*e*x^2*(g*x)^m*(a+c*x^2)^p*AppellF1(1/2*(2+m),-p,2,1/2*(4+m),-c*x^2/a,e^2*x^2/d^2)/(d^3*(2+m)*(1+c*x^2/a)^p)+e^2*x^3*(g*x)^m*(a+c*x^2)^p*AppellF1(1/2*(3+m),-p,2,1/2*(5+m),-c*x^2/a,e^2*x^2/d^2)/(d^4*(3+m)*(1+c*x^2/a)^p)],
[(g*x)^m*(a+c*x^2)^p/(d+e*x)^3,x,10,x*(g*x)^m*(a+c*x^2)^p*AppellF1(1/2*(1+m),-p,3,1/2*(3+m),-c*x^2/a,e^2*x^2/d^2)/(d^3*(1+m)*(1+c*x^2/a)^p)-3*e*x^2*(g*x)^m*(a+c*x^2)^p*AppellF1(1/2*(2+m),-p,3,1/2*(4+m),-c*x^2/a,e^2*x^2/d^2)/(d^4*(2+m)*(1+c*x^2/a)^p)+3*e^2*x^3*(g*x)^m*(a+c*x^2)^p*AppellF1(1/2*(3+m),-p,3,1/2*(5+m),-c*x^2/a,e^2*x^2/d^2)/(d^5*(3+m)*(1+c*x^2/a)^p)-e^3*x^4*(g*x)^m*(a+c*x^2)^p*AppellF1(1/2*(4+m),-p,3,1/2*(6+m),-c*x^2/a,e^2*x^2/d^2)/(d^6*(4+m)*(1+c*x^2/a)^p)],

# Integrands of the form x^m (d+e x)^n (a+b x+c x^2)^p when b^2-4 a c=0

# Integrands of the form x^m (d+e x)^n (a+b x+c x^2)^p when c d^2-b d e+a e^2=0

# Integrands of the form x^2 / (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^(p/2)

# p>0
[x^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d+e*x),x,6,1/128*(c*d^2-a*e^2)*(35*c^3*d^6+15*a*c^2*d^4*e^2+9*a^2*c*d^2*e^4+5*a^3*e^6)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(7/2)*d^(7/2)*e^(9/2))+1/24*(a/(c*d)-7*d/e^2)*x^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)+1/4*x^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/e-1/192*(105*c^3*d^6-25*a*c^2*d^4*e^2-17*a^2*c*d^2*e^4-15*a^3*e^6-2*c*d*e*(35*c^2*d^4-6*a*c*d^2*e^2-5*a^2*e^4)*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*e^4)],
[x^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d+e*x),x,5,-1/16*(c*d^2-a*e^2)*(5*c^2*d^4+2*a*c*d^2*e^2+a^2*e^4)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(5/2)*d^(5/2)*e^(7/2))+1/3*x^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/e+1/24*((5*c*d^2-3*a*e^2)*(3*c*d^2+a*e^2)-2*c*d*e*(5*c*d^2-a*e^2)*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*e^3)],
[x*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d+e*x),x,4,1/2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c*d*e*(d+e*x))+1/8*(c*d^2-a*e^2)*(3*c*d^2+a*e^2)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(3/2)*d^(3/2)*e^(5/2))-1/4*(a/(c*d)+3*d/e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)],
[sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d+e*x),x,3,-1/2*(c*d^2-a*e^2)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(e^(3/2)*sqrt(c)*sqrt(d))+sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/e],
[sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(x*(d+e*x)),x,6,arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))*sqrt(c)*sqrt(d)/sqrt(e)-arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))*sqrt(a)*sqrt(e)/sqrt(d)],
[sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(x^2*(d+e*x)),x,4,-1/2*(c*d^2-a*e^2)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(d^(3/2)*sqrt(a)*sqrt(e))-sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d*x)],
[sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(x^3*(d+e*x)),x,5,1/8*(c*d^2-a*e^2)*(c*d^2+3*a*e^2)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(a^(3/2)*d^(5/2)*e^(3/2))-1/2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d*x^2)-1/4*(c/(a*e)-3*e/d^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/x],
[sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(x^4*(d+e*x)),x,6,-1/16*(c*d^2-a*e^2)*(c^2*d^4+2*a*c*d^2*e^2+5*a^2*e^4)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(a^(5/2)*d^(7/2)*e^(5/2))-1/3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d*x^3)-1/12*(c/(a*e)-5*e/d^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/x^2+1/24*(3*c*d^2-5*a*e^2)*(c*d^2+3*a*e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(a^2*d^3*e^2*x)],
[sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(x^5*(d+e*x)),x,7,1/128*(c*d^2-a*e^2)*(5*c^3*d^6+9*a*c^2*d^4*e^2+15*a^2*c*d^2*e^4+35*a^3*e^6)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(a^(7/2)*d^(9/2)*e^(7/2))-1/4*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d*x^4)-1/24*(c/(a*e)-7*e/d^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/x^3+1/96*(5*c^2*d^4+6*a*c*d^2*e^2-35*a^2*e^4)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(a^2*d^3*e^2*x^2)-1/192*(15*c^3*d^6+17*a*c^2*d^4*e^2+25*a^2*c*d^2*e^4-105*a^3*e^6)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(a^3*d^4*e^3*x)],
[x^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x),x,7,1/20*(a/(c*d)-3*d/e^2)*x^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)+1/6*x^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/e-1/960*(105*c^3*d^6-21*a*c^2*d^4*e^2-33*a^2*c*d^2*e^4-35*a^3*e^6-6*c*d*e*(21*c^2*d^4-6*a*c*d^2*e^2-7*a^2*e^4)*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^3*d^3*e^4)-1/1024*(c*d^2-a*e^2)^3*(21*c^3*d^6+21*a*c^2*d^4*e^2+15*a^2*c*d^2*e^4+7*a^3*e^6)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(9/2)*d^(9/2)*e^(11/2))+1/512*(21*c^4*d^8-6*a^2*c^2*d^4*e^4-8*a^3*c*d^2*e^6-7*a^4*e^8)*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^4*d^4*e^5)],
[x^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x),x,6,1/5*x^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/e+1/240*(35*c^2*d^4-12*a*c*d^2*e^2-15*a^2*e^4-6*c*d*e*(7*c*d^2-3*a*e^2)*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^2*d^2*e^3)+1/256*(c*d^2-a*e^2)^3*(7*c^2*d^4+6*a*c*d^2*e^2+3*a^2*e^4)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(7/2)*d^(7/2)*e^(9/2))-1/128*(c*d^2-a*e^2)*(7*c^2*d^4+6*a*c*d^2*e^2+3*a^2*e^4)*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*e^4)],
[x*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x),x,5,-1/24*(3*a/(c*d)+5*d/e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)+1/4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c*d*e*(d+e*x))-1/128*(c*d^2-a*e^2)^3*(5*c*d^2+3*a*e^2)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(5/2)*d^(5/2)*e^(7/2))+1/64*(c*d^2-a*e^2)*(5*c*d^2+3*a*e^2)*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*e^3)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x),x,4,1/3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/e+1/16*(c*d^2-a*e^2)^3*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(3/2)*d^(3/2)*e^(5/2))+1/8*(a/(c*d)-d/e^2)*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(x*(d+e*x)),x,7,-1/8*(c^2*d^4-6*a*c*d^2*e^2-3*a^2*e^4)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(e^(3/2)*sqrt(c)*sqrt(d))-a^(3/2)*e^(3/2)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))*sqrt(d)+1/4*(c*d^2+5*a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/e],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(x^2*(d+e*x)),x,7,1/2*(c*d^2+3*a*e^2)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))*sqrt(c)*sqrt(d)/sqrt(e)-1/2*(3*c*d^2+a*e^2)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))*sqrt(a)*sqrt(e)/sqrt(d)-(a*e-c*d*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/x],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(x^3*(d+e*x)),x,7,-1/8*(3*c^2*d^4+6*a*c*d^2*e^2-a^2*e^4)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(d^(3/2)*sqrt(a)*sqrt(e))+c^(3/2)*d^(3/2)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))*sqrt(e)-1/4*(2*a*d*e+(5*c*d^2+a*e^2)*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d*x^2)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(x^4*(d+e*x)),x,5,-1/3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d*x^3)+1/16*(c*d^2-a*e^2)^3*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(a^(3/2)*d^(5/2)*e^(3/2))-1/8*(c/(a*e)-e/d^2)*(2*a*d*e+(c*d^2+a*e^2)*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/x^2],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(x^5*(d+e*x)),x,6,-1/4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d*x^4)-1/24*(3*c/(a*e)-5*e/d^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/x^3-1/128*(c*d^2-a*e^2)^3*(3*c*d^2+5*a*e^2)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(a^(5/2)*d^(7/2)*e^(5/2))+1/64*(c*d^2-a*e^2)*(3*c*d^2+5*a*e^2)*(2*a*d*e+(c*d^2+a*e^2)*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(a^2*d^3*e^2*x^2)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(x^6*(d+e*x)),x,7,-1/5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d*x^5)-1/40*(3*c/(a*e)-7*e/d^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/x^4+1/240*(15*c^2*d^4+12*a*c*d^2*e^2-35*a^2*e^4)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(a^2*d^3*e^2*x^3)+1/256*(c*d^2-a*e^2)^3*(3*c^2*d^4+6*a*c*d^2*e^2+7*a^2*e^4)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(a^(7/2)*d^(9/2)*e^(7/2))-1/128*(c*d^2-a*e^2)*(3*c^2*d^4+6*a*c*d^2*e^2+7*a^2*e^4)*(2*a*d*e+(c*d^2+a*e^2)*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(a^3*d^4*e^3*x^2)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(x^7*(d+e*x)),x,8,-1/6*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d*x^6)-1/20*(c/(a*e)-3*e/d^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/x^5+1/160*(7*c^2*d^4+6*a*c*d^2*e^2-21*a^2*e^4)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(a^2*d^3*e^2*x^4)-1/960*(35*c^3*d^6+33*a*c^2*d^4*e^2+21*a^2*c*d^2*e^4-105*a^3*e^6)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(a^3*d^4*e^3*x^3)-1/1024*(c*d^2-a*e^2)^3*(7*c^3*d^6+15*a*c^2*d^4*e^2+21*a^2*c*d^2*e^4+21*a^3*e^6)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(a^(9/2)*d^(11/2)*e^(9/2))+1/512*(7*c^4*d^8+8*a*c^3*d^6*e^2+6*a^2*c^2*d^4*e^4-21*a^4*e^8)*(2*a*d*e+(c*d^2+a*e^2)*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(a^4*d^5*e^4*x^2)],
[x^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x),x,8,1/2048*(c*d^2-a*e^2)*(33*c^3*d^6+45*a*c^2*d^4*e^2+35*a^2*c*d^2*e^4+15*a^3*e^6)*(c*d^2+a*e^2+2*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^4*d^4*e^5)+1/112*(5*a/(c*d)-11*d/e^2)*x^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)+1/8*x^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/e-1/4480*(231*c^3*d^6-15*a*c^2*d^4*e^2-95*a^2*c*d^2*e^4-105*a^3*e^6-10*c*d*e*(33*c^2*d^4-10*a*c*d^2*e^2-15*a^2*e^4)*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^3*d^3*e^4)+3/32768*(c*d^2-a*e^2)^5*(33*c^3*d^6+45*a*c^2*d^4*e^2+35*a^2*c*d^2*e^4+15*a^3*e^6)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(11/2)*d^(11/2)*e^(13/2))-3/16384*(c*d^2-a*e^2)^3*(33*c^3*d^6+45*a*c^2*d^4*e^2+35*a^2*c*d^2*e^4+15*a^3*e^6)*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^5*d^5*e^6)],
[x^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x),x,7,-1/384*(c*d^2-a*e^2)*(9*c^2*d^4+10*a*c*d^2*e^2+5*a^2*e^4)*(c*d^2+a*e^2+2*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^3*d^3*e^4)+1/7*x^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/e+1/840*(63*c^2*d^4-20*a*c*d^2*e^2-35*a^2*e^4-10*c*d*e*(9*c*d^2-5*a*e^2)*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^2*d^2*e^3)-1/2048*(c*d^2-a*e^2)^5*(9*c^2*d^4+10*a*c*d^2*e^2+5*a^2*e^4)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(9/2)*d^(9/2)*e^(11/2))+1/1024*(c*d^2-a*e^2)^3*(9*c^2*d^4+10*a*c*d^2*e^2+5*a^2*e^4)*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^4*d^4*e^5)],
[x*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x),x,6,1/192*(c*d^2-a*e^2)*(7*c*d^2+5*a*e^2)*(c*d^2+a*e^2+2*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^2*d^2*e^3)-1/60*(5*a/(c*d)+7*d/e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)+1/6*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c*d*e*(d+e*x))+1/1024*(c*d^2-a*e^2)^5*(7*c*d^2+5*a*e^2)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(7/2)*d^(7/2)*e^(9/2))-1/512*(c*d^2-a*e^2)^3*(7*c*d^2+5*a*e^2)*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*e^4)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x),x,5,1/16*(a/(c*d)-d/e^2)*(c*d^2+a*e^2+2*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)+1/5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/e-3/256*(c*d^2-a*e^2)^5*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(5/2)*d^(5/2)*e^(7/2))+3/128*(c*d^2-a*e^2)^3*(c*d^2+a*e^2+2*c*d*e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*e^3)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(x*(d+e*x)),x,8,1/24*(3*c*d^2+11*a*e^2+6*c*d*e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/e-a^(5/2)*d^(3/2)*e^(5/2)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))+1/128*(3*c^4*d^8-20*a*c^3*d^6*e^2+90*a^2*c^2*d^4*e^4+60*a^3*c*d^2*e^6-5*a^4*e^8)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(3/2)*d^(3/2)*e^(5/2))-1/64*(3*c^3*d^6-11*a*c^2*d^4*e^2-83*a^2*c*d^2*e^4-5*a^3*e^6+2*c*d*e*(c*d^2-5*a*e^2)*(3*c*d^2+a*e^2)*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*e^2)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(x^2*(d+e*x)),x,8,-1/3*(3*a*e-c*d*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/x-1/16*(c^3*d^6-15*a*c^2*d^4*e^2-45*a^2*c*d^2*e^4-5*a^3*e^6)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(e^(3/2)*sqrt(c)*sqrt(d))-1/2*a^(3/2)*e^(3/2)*(5*c*d^2+3*a*e^2)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))*sqrt(d)+1/8*(c^2*d^4+28*a*c*d^2*e^2+19*a^2*e^4+2*c*d*e*(c*d^2+7*a*e^2)*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/e],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(x^3*(d+e*x)),x,8,-1/2*(a*e-c*d*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/x^2+3/8*(c^2*d^4+10*a*c*d^2*e^2+5*a^2*e^4)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))*sqrt(c)*sqrt(d)/sqrt(e)-3/8*(5*c^2*d^4+10*a*c*d^2*e^2+a^2*e^4)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))*sqrt(a)*sqrt(e)/sqrt(d)-3/4*(a*e*(3*c*d^2+a*e^2)-c*d*(c*d^2+3*a*e^2)*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/x],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(x^4*(d+e*x)),x,8,-1/12*(4*a*d*e+3*(3*c*d^2+a*e^2)*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d*x^3)-1/16*(5*c^3*d^6+45*a*c^2*d^4*e^2+15*a^2*c*d^2*e^4-a^3*e^6)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(d^(3/2)*sqrt(a)*sqrt(e))+1/2*c^(3/2)*d^(3/2)*(3*c*d^2+5*a*e^2)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))*sqrt(e)-1/8*(5*c^2*d^4+12*a*c*d^2*e^2-a^2*e^4-2*c*d*e*(7*c*d^2+a*e^2)*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(d*x)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(x^5*(d+e*x)),x,8,-1/24*(6*a*d*e+(11*c*d^2+3*a*e^2)*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d*x^4)+1/128*(5*c^4*d^8-60*a*c^3*d^6*e^2-90*a^2*c^2*d^4*e^4+20*a^3*c*d^2*e^6-3*a^4*e^8)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(a^(3/2)*d^(5/2)*e^(3/2))+c^(5/2)*d^(5/2)*e^(3/2)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))-1/64*(2*a*d*e*(5*c*d^2-a*e^2)*(c*d^2+3*a*e^2)+(5*c^3*d^6+83*a*c^2*d^4*e^2+11*a^2*c*d^2*e^4-3*a^3*e^6)*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(a*d^2*e*x^2)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(x^6*(d+e*x)),x,6,-1/16*(c/(a*e)-e/d^2)*(2*a*d*e+(c*d^2+a*e^2)*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/x^4-1/5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d*x^5)-3/256*(c*d^2-a*e^2)^5*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(a^(5/2)*d^(7/2)*e^(5/2))+3/128*(c*d^2-a*e^2)^3*(2*a*d*e+(c*d^2+a*e^2)*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(a^2*d^3*e^2*x^2)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(x^7*(d+e*x)),x,7,1/192*(c*d^2-a*e^2)*(5*c*d^2+7*a*e^2)*(2*a*d*e+(c*d^2+a*e^2)*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(a^2*d^3*e^2*x^4)-1/6*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d*x^6)-1/60*(5*c/(a*e)-7*e/d^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/x^5+1/1024*(c*d^2-a*e^2)^5*(5*c*d^2+7*a*e^2)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(a^(7/2)*d^(9/2)*e^(7/2))-1/512*(c*d^2-a*e^2)^3*(5*c*d^2+7*a*e^2)*(2*a*d*e+(c*d^2+a*e^2)*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(a^3*d^4*e^3*x^2)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(x^8*(d+e*x)),x,8,-1/384*(c*d^2-a*e^2)*(5*c^2*d^4+10*a*c*d^2*e^2+9*a^2*e^4)*(2*a*d*e+(c*d^2+a*e^2)*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(a^3*d^4*e^3*x^4)-1/7*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d*x^7)-1/84*(5*c/(a*e)-9*e/d^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/x^6+1/840*(35*c^2*d^4+20*a*c*d^2*e^2-63*a^2*e^4)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(a^2*d^3*e^2*x^5)-1/2048*(c*d^2-a*e^2)^5*(5*c^2*d^4+10*a*c*d^2*e^2+9*a^2*e^4)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(a^(9/2)*d^(11/2)*e^(9/2))+1/1024*(c*d^2-a*e^2)^3*(5*c^2*d^4+10*a*c*d^2*e^2+9*a^2*e^4)*(2*a*d*e+(c*d^2+a*e^2)*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(a^4*d^5*e^4*x^2)],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(x^9*(d+e*x)),x,9,1/2048*(c*d^2-a*e^2)*(15*c^3*d^6+35*a*c^2*d^4*e^2+45*a^2*c*d^2*e^4+33*a^3*e^6)*(2*a*d*e+(c*d^2+a*e^2)*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(a^4*d^5*e^4*x^4)-1/8*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d*x^8)-1/112*(5*c/(a*e)-11*e/d^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/x^7+1/448*(15*c^2*d^4+10*a*c*d^2*e^2-33*a^2*e^4)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(a^2*d^3*e^2*x^6)-1/4480*(105*c^3*d^6+95*a*c^2*d^4*e^2+15*a^2*c*d^2*e^4-231*a^3*e^6)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(a^3*d^4*e^3*x^5)+3/32768*(c*d^2-a*e^2)^5*(15*c^3*d^6+35*a*c^2*d^4*e^2+45*a^2*c*d^2*e^4+33*a^3*e^6)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(a^(11/2)*d^(13/2)*e^(11/2))-3/16384*(c*d^2-a*e^2)^3*(15*c^3*d^6+35*a*c^2*d^4*e^2+45*a^2*c*d^2*e^4+33*a^3*e^6)*(2*a*d*e+(c*d^2+a*e^2)*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(a^5*d^6*e^5*x^2)],

# p<0
[x^3/((d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,5,3/8*(5*c^2*d^4+2*a*c*d^2*e^2+a^2*e^4)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(5/2)*d^(5/2)*e^(7/2))-3/4*(3*c*d^2+a*e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*e^3)-2*d^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e^3*(c*d^2-a*e^2)*(d+e*x))+1/2*(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*e^3),3/8*(5*c^2*d^4+2*a*c*d^2*e^2+a^2*e^4)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(5/2)*d^(5/2)*e^(7/2))-2*d*x^2*(a*e*(c*d^2-a*e^2)+c*d*(c*d^2-a*e^2)*x)/(e*(c*d^2-a*e^2)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-1/4*((5*c*d^2-3*a*e^2)*(3*c*d^2+a*e^2)-2*c*d*e*(5*c*d^2-a*e^2)*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*e^3*(c*d^2-a*e^2))],
[x^2/((d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,4,-1/2*(3*c*d^2+a*e^2)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(3/2)*d^(3/2)*e^(5/2))+sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*e^2)+2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e^2*(c*d^2-a*e^2)*(d+e*x))],
[x/((d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,3,arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(e^(3/2)*sqrt(c)*sqrt(d))-2*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(e*(c*d^2-a*e^2)*(d+e*x))],
[1/((d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,1,2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d^2-a*e^2)*(d+e*x))],
[1/(x*(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,5,-arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(d^(3/2)*sqrt(a)*sqrt(e))-2*e*(a*e+c*d*x)/(d*(c*d^2-a*e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/(x^2*(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,5,1/2*(c*d^2+3*a*e^2)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(a^(3/2)*d^(5/2)*e^(3/2))-2*e*(a*e+c*d*x)/(d*(c*d^2-a*e^2)*x*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-(c*d^2-3*a*e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(a*d^2*e*(c*d^2-a*e^2)*x)],
[1/(x^3*(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,6,-3/8*(c^2*d^4+2*a*c*d^2*e^2+5*a^2*e^4)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(a^(5/2)*d^(7/2)*e^(5/2))-2*e*(a*e+c*d*x)/(d*(c*d^2-a*e^2)*x^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-1/2*(c*d^2-5*a*e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(a*d^2*e*(c*d^2-a*e^2)*x^2)+1/4*(3*c*d^2-5*a*e^2)*(c*d^2+3*a*e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(a^2*d^3*e^2*(c*d^2-a*e^2)*x)],
[x^5/((d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,6,-2/3*d*x^4*(a*e*(c*d^2-a*e^2)+c*d*(c*d^2-a*e^2)*x)/(e*(c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+5/8*(7*c^2*d^4+6*a*c*d^2*e^2+3*a^2*e^4)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(7/2)*d^(7/2)*e^(9/2))-2/3*x^2*(a*d*e*(c*d^2-a*e^2)*(7*c^2*d^4-12*a*c*d^2*e^2-3*a^2*e^4)+(c*d^2-a*e^2)*(7*c^3*d^6-11*a*c^2*d^4*e^2-a^2*c*d^2*e^4-3*a^3*e^6)*x)/(c*d*e^2*(c*d^2-a*e^2)^4*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-1/12*(105*c^4*d^8-190*a*c^3*d^6*e^2+36*a^2*c^2*d^4*e^4+30*a^3*c*d^2*e^6-45*a^4*e^8-2*c*d*e*(35*c^3*d^6-61*a*c^2*d^4*e^2+9*a^2*c*d^2*e^4-15*a^3*e^6)*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*e^4*(c*d^2-a*e^2)^3)],
[x^4/((d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,6,-2/3*d*x^3*(a*e*(c*d^2-a*e^2)+c*d*(c*d^2-a*e^2)*x)/(e*(c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))-1/2*(5*c*d^2+3*a*e^2)*arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(5/2)*d^(5/2)*e^(7/2))-2/3*x*(a*d*e*(c*d^2-a*e^2)*(5*c^2*d^4-10*a*c*d^2*e^2-3*a^2*e^4)+(c*d^2-a*e^2)*(5*c^3*d^6-9*a*c^2*d^4*e^2-a^2*c*d^2*e^4-3*a^3*e^6)*x)/(c*d*e^2*(c*d^2-a*e^2)^4*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+1/3*(15*c^3*d^6-31*a*c^2*d^4*e^2+9*a^2*c*d^2*e^4-9*a^3*e^6)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*e^3*(c*d^2-a*e^2)^3)],
[x^3/((d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,5,-2/3*d*x^2*(a*e*(c*d^2-a*e^2)+c*d*(c*d^2-a*e^2)*x)/(e*(c*d^2-a*e^2)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+arctanh(1/2*(c*d^2+a*e^2+2*c*d*e*x)/(sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(c^(3/2)*d^(3/2)*e^(5/2))-2/3*(a*d*e*(c*d^2-3*a*e^2)*(3*c*d^2+a*e^2)+(3*c^3*d^6-7*a*c^2*d^4*e^2-a^2*c*d^2*e^4-3*a^3*e^6)*x)/(c*d*e^2*(c*d^2-a*e^2)^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[x^2/((d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,3,2/3*x^2/((c*d^2-a*e^2)*(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-8/3*a*e*(2*a*d*e+(c*d^2+a*e^2)*x)/((c*d^2-a*e^2)^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[x/((d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,2,-2/3*d/(e*(c*d^2-a*e^2)*(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+2/3*(c*d^2+3*a*e^2)*(c*d^2+a*e^2+2*c*d*e*x)/(e*(c*d^2-a*e^2)^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/((d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,2,2/3/((c*d^2-a*e^2)*(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-8/3*c*d*(c*d^2+a*e^2+2*c*d*e*x)/((c*d^2-a*e^2)^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/(x*(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,6,-2/3*e*(a*e+c*d*x)/(d*(c*d^2-a*e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))-arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(a^(3/2)*d^(5/2)*e^(3/2))+2/3*(3*c^3*d^6+a*c^2*d^4*e^2+7*a^2*c*d^2*e^4-3*a^3*e^6+c*d*e*(3*c*d^2-a*e^2)*(c*d^2+3*a*e^2)*x)/(a*d^2*e*(c*d^2-a*e^2)^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[1/(x^2*(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,6,-2/3*e*(a*e+c*d*x)/(d*(c*d^2-a*e^2)*x*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+1/2*(3*c*d^2+5*a*e^2)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(a^(5/2)*d^(7/2)*e^(5/2))+2/3*(3*c^3*d^6+a*c^2*d^4*e^2+9*a^2*c*d^2*e^4-5*a^3*e^6+c*d*e*(3*c^2*d^4+10*a*c*d^2*e^2-5*a^2*e^4)*x)/(a*d^2*e*(c*d^2-a*e^2)^3*x*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-1/3*(9*c^3*d^6-9*a*c^2*d^4*e^2+31*a^2*c*d^2*e^4-15*a^3*e^6)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(a^2*d^3*e^2*(c*d^2-a*e^2)^3*x)],
[1/(x^3*(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,7,-2/3*e*(a*e+c*d*x)/(d*(c*d^2-a*e^2)*x^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))-5/8*(3*c^2*d^4+6*a*c*d^2*e^2+7*a^2*e^4)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(a^(7/2)*d^(9/2)*e^(7/2))+2/3*(3*c^3*d^6+a*c^2*d^4*e^2+11*a^2*c*d^2*e^4-7*a^3*e^6+c*d*e*(3*c^2*d^4+12*a*c*d^2*e^2-7*a^2*e^4)*x)/(a*d^2*e*(c*d^2-a*e^2)^3*x^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-1/6*(15*c^3*d^6-9*a*c^2*d^4*e^2+61*a^2*c*d^2*e^4-35*a^3*e^6)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(a^2*d^3*e^2*(c*d^2-a*e^2)^3*x^2)+1/12*(45*c^4*d^8-30*a*c^3*d^6*e^2-36*a^2*c^2*d^4*e^4+190*a^3*c*d^2*e^6-105*a^4*e^8)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(a^3*d^4*e^3*(c*d^2-a*e^2)^3*x)],
[1/(x^4*(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,8,-2/3*e*(a*e+c*d*x)/(d*(c*d^2-a*e^2)*x^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+5/16*(7*c^3*d^6+15*a*c^2*d^4*e^2+21*a^2*c*d^2*e^4+21*a^3*e^6)*arctanh(1/2*(2*a*d*e+(c*d^2+a*e^2)*x)/(sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)))/(a^(9/2)*d^(11/2)*e^(9/2))+2/3*(3*c^3*d^6+a*c^2*d^4*e^2+13*a^2*c*d^2*e^4-9*a^3*e^6+c*d*e*(3*c^2*d^4+14*a*c*d^2*e^2-9*a^2*e^4)*x)/(a*d^2*e*(c*d^2-a*e^2)^3*x^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-1/3*(7*c^3*d^6-3*a*c^2*d^4*e^2+33*a^2*c*d^2*e^4-21*a^3*e^6)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(a^2*d^3*e^2*(c*d^2-a*e^2)^3*x^3)+1/12*(35*c^4*d^8-16*a*c^3*d^6*e^2-18*a^2*c^2*d^4*e^4+168*a^3*c*d^2*e^6-105*a^4*e^8)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(a^3*d^4*e^3*(c*d^2-a*e^2)^3*x^2)-1/24*(105*c^5*d^10-55*a*c^4*d^8*e^2-54*a^2*c^3*d^6*e^4-78*a^3*c^2*d^4*e^6+525*a^4*c*d^2*e^8-315*a^5*e^10)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(a^4*d^5*e^4*(c*d^2-a*e^2)^3*x)],
[x^2/((d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)),x,3,2/5*x^2/((c*d^2-a*e^2)*(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))-8/15*(a*d*e*(c*d^2-a*e^2)*(c*d^2+3*a*e^2)+(c^3*d^6+a^2*c*d^2*e^4-2*a^3*e^6)*x)/(e*(c*d^2-a*e^2)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+8/15*(c^2*d^4+10*a*c*d^2*e^2+5*a^2*e^4)*(c*d^2+a*e^2+2*c*d*e*x)/(e*(c*d^2-a*e^2)^5*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[x^2/((d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)),x,4,2/7*x^2/((c*d^2-a*e^2)*(d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2))-8/35*(2*a*d*e*(c*d^2+2*a*e^2)+(2*c^2*d^4+a*c*d^2*e^2+3*a^2*e^4)*x)/(e*(c*d^2-a*e^2)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2))+16/105*(3*c^2*d^4+14*a*c*d^2*e^2+7*a^2*e^4)*(c*d^2+a*e^2+2*c*d*e*x)/(e*(c*d^2-a*e^2)^5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))-128/105*c*d*(3*c^2*d^4+14*a*c*d^2*e^2+7*a^2*e^4)*(c*d^2+a*e^2+2*c*d*e*x)/((c*d^2-a*e^2)^7*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],

# Integrands of the form x^m (d+e x)^p (a+b x+c x^2)^p when b d+a e=0 and c d+b e=0

# Integrands of the form x^m (d+e x)^(p/2) (a+b x+c x^2)^(p/2) when b d+a e=0 and c d+b e=0

# p>0
[x^3*sqrt(1+x)*sqrt(1-x+x^2),x,4,6/55*x*sqrt(1+x)*sqrt(1-x+x^2)+2/11*x^4*sqrt(1+x)*sqrt(1-x+x^2)-4/55*3^(3/4)*(1+x)^(3/2)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1-x+x^2)*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/((1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[x^2*sqrt(1+x)*sqrt(1-x+x^2),x,1,2/9*(1+x)^(3/2)*(1-x+x^2)^(3/2)],
[x*sqrt(1+x)*sqrt(1-x+x^2),x,5,2/7*x^2*sqrt(1+x)*sqrt(1-x+x^2)+6/7*sqrt(1+x)*sqrt(1-x+x^2)/(1+x+sqrt(3))+2/7*3^(3/4)*(1+x)^(3/2)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt(1-x+x^2)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/((1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))-3/7*3^(1/4)*(1+x)^(3/2)*EllipticE((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1-x+x^2)*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/((1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[sqrt(1+x)*sqrt(1-x+x^2),x,3,2/5*x*sqrt(1+x)*sqrt(1-x+x^2)+2/5*3^(3/4)*(1+x)^(3/2)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1-x+x^2)*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/((1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[sqrt(1+x)*sqrt(1-x+x^2)/x,x,5,2/3*sqrt(1+x)*sqrt(1-x+x^2)-2/3*arctanh(sqrt(1+x^3))*sqrt(1+x)*sqrt(1-x+x^2)/sqrt(1+x^3)],
[sqrt(1+x)*sqrt(1-x+x^2)/x^2,x,5,-sqrt(1+x)*sqrt(1-x+x^2)/x+3*sqrt(1+x)*sqrt(1-x+x^2)/(1+x+sqrt(3))+3^(3/4)*(1+x)^(3/2)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt(1-x+x^2)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/((1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))-3/2*3^(1/4)*(1+x)^(3/2)*EllipticE((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1-x+x^2)*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/((1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[sqrt(1+x)*sqrt(1-x+x^2)/x^3,x,3,-1/2*sqrt(1+x)*sqrt(1-x+x^2)/x^2+1/2*3^(3/4)*(1+x)^(3/2)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1-x+x^2)*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/((1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[x^3*(1+x)^(3/2)*(1-x+x^2)^(3/2),x,5,54/935*x*sqrt(1+x)*sqrt(1-x+x^2)+18/187*x^4*sqrt(1+x)*sqrt(1-x+x^2)+2/17*x^4*(1+x^3)*sqrt(1+x)*sqrt(1-x+x^2)-36/935*3^(3/4)*(1+x)^(3/2)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1-x+x^2)*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/((1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[x^2*(1+x)^(3/2)*(1-x+x^2)^(3/2),x,1,2/15*(1+x)^(5/2)*(1-x+x^2)^(5/2)],
[x*(1+x)^(3/2)*(1-x+x^2)^(3/2),x,6,18/91*x^2*sqrt(1+x)*sqrt(1-x+x^2)+2/13*x^2*(1+x^3)*sqrt(1+x)*sqrt(1-x+x^2)+54/91*sqrt(1+x)*sqrt(1-x+x^2)/(1+x+sqrt(3))+18/91*3^(3/4)*(1+x)^(3/2)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt(1-x+x^2)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/((1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))-27/91*3^(1/4)*(1+x)^(3/2)*EllipticE((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1-x+x^2)*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/((1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[(1+x)^(3/2)*(1-x+x^2)^(3/2),x,4,18/55*x*sqrt(1+x)*sqrt(1-x+x^2)+2/11*x*(1+x^3)*sqrt(1+x)*sqrt(1-x+x^2)+18/55*3^(3/4)*(1+x)^(3/2)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1-x+x^2)*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/((1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[(1+x)^(3/2)*(1-x+x^2)^(3/2)/x,x,6,2/3*sqrt(1+x)*sqrt(1-x+x^2)+2/9*(1+x^3)*sqrt(1+x)*sqrt(1-x+x^2)-2/3*arctanh(sqrt(1+x^3))*sqrt(1+x)*sqrt(1-x+x^2)/sqrt(1+x^3)],
[(1+x)^(3/2)*(1-x+x^2)^(3/2)/x^2,x,6,9/7*x^2*sqrt(1+x)*sqrt(1-x+x^2)-(1+x^3)*sqrt(1+x)*sqrt(1-x+x^2)/x+27/7*sqrt(1+x)*sqrt(1-x+x^2)/(1+x+sqrt(3))+9/7*3^(3/4)*(1+x)^(3/2)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt(1-x+x^2)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/((1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))-27/14*3^(1/4)*(1+x)^(3/2)*EllipticE((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1-x+x^2)*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/((1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[(1+x)^(3/2)*(1-x+x^2)^(3/2)/x^3,x,4,9/10*x*sqrt(1+x)*sqrt(1-x+x^2)-1/2*(1+x^3)*sqrt(1+x)*sqrt(1-x+x^2)/x^2+9/10*3^(3/4)*(1+x)^(3/2)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1-x+x^2)*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/((1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],

# p<0
[x^3/(sqrt(1+x)*sqrt(1-x+x^2)),x,3,2/5*x*(1+x^3)/(sqrt(1+x)*sqrt(1-x+x^2))-4/5*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1+x)*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[x^2/(sqrt(1+x)*sqrt(1-x+x^2)),x,1,2/3*sqrt(1+x)*sqrt(1-x+x^2)],
[x/(sqrt(1+x)*sqrt(1-x+x^2)),x,4,2*(1+x^3)/((1+x+sqrt(3))*sqrt(1+x)*sqrt(1-x+x^2))+2*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt(1+x)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))-3^(1/4)*EllipticE((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1+x)*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[1/(sqrt(1+x)*sqrt(1-x+x^2)),x,2,2*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1+x)*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[1/(x*sqrt(1+x)*sqrt(1-x+x^2)),x,4,-2/3*arctanh(sqrt(1+x^3))*sqrt(1+x^3)/(sqrt(1+x)*sqrt(1-x+x^2))],
[1/(x^2*sqrt(1+x)*sqrt(1-x+x^2)),x,5,(-1-x^3)/(x*sqrt(1+x)*sqrt(1-x+x^2))+(1+x^3)/((1+x+sqrt(3))*sqrt(1+x)*sqrt(1-x+x^2))+EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt(1+x)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))-1/2*3^(1/4)*EllipticE((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1+x)*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[1/(x^3*sqrt(1+x)*sqrt(1-x+x^2)),x,3,1/2*(-1-x^3)/(x^2*sqrt(1+x)*sqrt(1-x+x^2))-1/2*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1+x)*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[x^3/((1+x)^(3/2)*(1-x+x^2)^(3/2)),x,3,-2/3*x/(sqrt(1+x)*sqrt(1-x+x^2))+4/3*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1+x)*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[x^2/((1+x)^(3/2)*(1-x+x^2)^(3/2)),x,1,(-2/3)/(sqrt(1+x)*sqrt(1-x+x^2))],
[x/((1+x)^(3/2)*(1-x+x^2)^(3/2)),x,5,2/3*x^2/(sqrt(1+x)*sqrt(1-x+x^2))-2/3*(1+x^3)/((1+x+sqrt(3))*sqrt(1+x)*sqrt(1-x+x^2))-2/3*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt(1+x)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))+EllipticE((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1+x)*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(3/4)*sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[1/((1+x)^(3/2)*(1-x+x^2)^(3/2)),x,3,2/3*x/(sqrt(1+x)*sqrt(1-x+x^2))+2/3*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1+x)*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[1/(x*(1+x)^(3/2)*(1-x+x^2)^(3/2)),x,5,2/3/(sqrt(1+x)*sqrt(1-x+x^2))-2/3*arctanh(sqrt(1+x^3))*sqrt(1+x^3)/(sqrt(1+x)*sqrt(1-x+x^2))],
[1/(x^2*(1+x)^(3/2)*(1-x+x^2)^(3/2)),x,6,2/3/(x*sqrt(1+x)*sqrt(1-x+x^2))-5/3*(1+x^3)/(x*sqrt(1+x)*sqrt(1-x+x^2))+5/3*(1+x^3)/((1+x+sqrt(3))*sqrt(1+x)*sqrt(1-x+x^2))+5/3*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt(1+x)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))-5/2*EllipticE((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1+x)*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(3/4)*sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[1/(x^3*(1+x)^(3/2)*(1-x+x^2)^(3/2)),x,4,2/3/(x^2*sqrt(1+x)*sqrt(1-x+x^2))-7/6*(1+x^3)/(x^2*sqrt(1+x)*sqrt(1-x+x^2))-7/6*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1+x)*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[x^3/((1+x)^(5/2)*(1-x+x^2)^(5/2)),x,4,4/27*x/(sqrt(1+x)*sqrt(1-x+x^2))-2/9*x/((1+x^3)*sqrt(1+x)*sqrt(1-x+x^2))+4/27*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1+x)*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[x^2/((1+x)^(5/2)*(1-x+x^2)^(5/2)),x,1,(-2/9)/((1+x)^(3/2)*(1-x+x^2)^(3/2))],
[x/((1+x)^(5/2)*(1-x+x^2)^(5/2)),x,6,10/27*x^2/(sqrt(1+x)*sqrt(1-x+x^2))+2/9*x^2/((1+x^3)*sqrt(1+x)*sqrt(1-x+x^2))-10/27*(1+x^3)/((1+x+sqrt(3))*sqrt(1+x)*sqrt(1-x+x^2))-10/27*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt(1+x)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))+5/9*EllipticE((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1+x)*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(3/4)*sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[1/((1+x)^(5/2)*(1-x+x^2)^(5/2)),x,4,14/27*x/(sqrt(1+x)*sqrt(1-x+x^2))+2/9*x/((1+x^3)*sqrt(1+x)*sqrt(1-x+x^2))+14/27*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1+x)*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[1/(x*(1+x)^(5/2)*(1-x+x^2)^(5/2)),x,6,2/3/(sqrt(1+x)*sqrt(1-x+x^2))+2/9/((1+x^3)*sqrt(1+x)*sqrt(1-x+x^2))-2/3*arctanh(sqrt(1+x^3))*sqrt(1+x^3)/(sqrt(1+x)*sqrt(1-x+x^2))],
[1/(x^2*(1+x)^(5/2)*(1-x+x^2)^(5/2)),x,7,22/27/(x*sqrt(1+x)*sqrt(1-x+x^2))+2/9/(x*(1+x^3)*sqrt(1+x)*sqrt(1-x+x^2))-55/27*(1+x^3)/(x*sqrt(1+x)*sqrt(1-x+x^2))+55/27*(1+x^3)/((1+x+sqrt(3))*sqrt(1+x)*sqrt(1-x+x^2))+55/27*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt(1+x)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))-55/18*EllipticE((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1+x)*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(3/4)*sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[1/(x^3*(1+x)^(5/2)*(1-x+x^2)^(5/2)),x,5,26/27/(x^2*sqrt(1+x)*sqrt(1-x+x^2))+2/9/(x^2*(1+x^3)*sqrt(1+x)*sqrt(1-x+x^2))-91/54*(1+x^3)/(x^2*sqrt(1+x)*sqrt(1-x+x^2))-91/54*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(1+x)*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x+x^2)*sqrt((1+x)/(1+x+sqrt(3))^2))],

# Integrands of the form x^m (d+e x)^n (a+b x+c x^2)^p

# Integrands of the form x^m (d+e x)^n (a+b x+c x^2)^p

# p>0

# p<0
[x/((-1+x)^3*(3+5*x+4*x^2)^2),x,7,(-21/736)/(1-x)^2+(-97/4416)/(1-x)+1/276*(39+44*x)/((1-x)^2*(3+5*x+4*x^2))+11/2304*log(1-x)-11/4608*log(3+5*x+4*x^2)+6023/52992*arctan((5+8*x)/sqrt(23))/sqrt(23)],

# Integrands of the form x^m (d+e x)^n (a+b x+c x^2)^(p/2)

# Integrands of the form x^m (d+e x)^(n/2) (a+b x+c x^2)^p

# p>0

# p<0
[x^4*sqrt(d+e*x)/(a+b*x+c*x^2),x,6,2/3*(c^2*d^2+b^2*e^2+c*e*(b*d-a*e))*(d+e*x)^(3/2)/(c^3*e^3)-2/5*(2*c*d+b*e)*(d+e*x)^(5/2)/(c^2*e^3)+2/7*(d+e*x)^(7/2)/(c*e^3)-2*b*(b^2-2*a*c)*sqrt(d+e*x)/c^4+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*(b^3*c*d-2*a*b*c^2*d-b^4*e+3*a*b^2*c*e-a^2*c^2*e+(-b^4*c*d+4*a*b^2*c^2*d-2*a^2*c^3*d+b^5*e-5*a*b^3*c*e+5*a^2*b*c^2*e)/sqrt(b^2-4*a*c))/(c^(9/2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*(b^3*c*d-2*a*b*c^2*d-b^4*e+3*a*b^2*c*e-a^2*c^2*e+(b^4*c*d-4*a*b^2*c^2*d+2*a^2*c^3*d-b^5*e+5*a*b^3*c*e-5*a^2*b*c^2*e)/sqrt(b^2-4*a*c))/(c^(9/2)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x^3*sqrt(d+e*x)/(a+b*x+c*x^2),x,6,-2/3*(c*d+b*e)*(d+e*x)^(3/2)/(c^2*e^2)+2/5*(d+e*x)^(5/2)/(c*e^2)+2*(b^2-a*c)*sqrt(d+e*x)/c^3+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(b^3-3*a*b*c-(b^2-a*c)*sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(c^(7/2)*sqrt(2)*sqrt(b^2-4*a*c))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(b^3-3*a*b*c+(b^2-a*c)*sqrt(b^2-4*a*c))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(c^(7/2)*sqrt(2)*sqrt(b^2-4*a*c)),-2/3*(c*d+b*e)*(d+e*x)^(3/2)/(c^2*e^2)+2/5*(d+e*x)^(5/2)/(c*e^2)+2*(b^2-a*c)*sqrt(d+e*x)/c^3-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*(b^2*c*d-a*c^2*d-b^3*e+2*a*b*c*e+(-b^3*c*d+3*a*b*c^2*d+b^4*e-4*a*b^2*c*e+2*a^2*c^2*e)/sqrt(b^2-4*a*c))/(c^(7/2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*(b^2*c*d-a*c^2*d-b^3*e+2*a*b*c*e+(b^3*c*d-3*a*b*c^2*d-b^4*e+4*a*b^2*c*e-2*a^2*c^2*e)/sqrt(b^2-4*a*c))/(c^(7/2)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x^2*sqrt(d+e*x)/(a+b*x+c*x^2),x,6,2/3*(d+e*x)^(3/2)/(c*e)-2*b*sqrt(d+e*x)/c^2+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*(b*c*d-b^2*e+a*c*e+(-b^2*c*d+2*a*c^2*d+b^3*e-3*a*b*c*e)/sqrt(b^2-4*a*c))/(c^(5/2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*(b*c*d-b^2*e+a*c*e+(b^2*c*d-2*a*c^2*d-b^3*e+3*a*b*c*e)/sqrt(b^2-4*a*c))/(c^(5/2)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x*sqrt(d+e*x)/(a+b*x+c*x^2),x,5,2*sqrt(d+e*x)/c+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*(b*c*d-b^2*e+2*a*c*e-(c*d-b*e)*sqrt(b^2-4*a*c))/(c^(3/2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*(b*c*d-b^2*e+2*a*c*e+(c*d-b*e)*sqrt(b^2-4*a*c))/(c^(3/2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[sqrt(d+e*x)/(a+b*x+c*x^2),x,4,-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(c)*sqrt(b^2-4*a*c))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(c)*sqrt(b^2-4*a*c))],
[sqrt(d+e*x)/(x*(a+b*x+c*x^2)),x,7,-2*arctanh(sqrt(d+e*x)/sqrt(d))*sqrt(d)/a+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(c)*(b*d-2*a*e+d*sqrt(b^2-4*a*c))/(a*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(c)*(b*d-2*a*e-d*sqrt(b^2-4*a*c))/(a*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[sqrt(d+e*x)/(x^2*(a+b*x+c*x^2)),x,9,e*arctanh(sqrt(d+e*x)/sqrt(d))/(a*sqrt(d))+2*(b*d-a*e)*arctanh(sqrt(d+e*x)/sqrt(d))/(a^2*sqrt(d))-sqrt(d+e*x)/(a*x)-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(c)*(b^2*d-2*a*c*d-a*b*e+(b*d-a*e)*sqrt(b^2-4*a*c))/(a^2*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(c)*(b^2*d-2*a*c*d-a*b*e-(b*d-a*e)*sqrt(b^2-4*a*c))/(a^2*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[sqrt(d+e*x)/(x^3*(a+b*x+c*x^2)),x,12,-3/4*e^2*arctanh(sqrt(d+e*x)/sqrt(d))/(a*d^(3/2))-e*(b*d-a*e)*arctanh(sqrt(d+e*x)/sqrt(d))/(a^2*d^(3/2))-2*(b^2*d-a*c*d-a*b*e)*arctanh(sqrt(d+e*x)/sqrt(d))/(a^3*sqrt(d))-1/2*sqrt(d+e*x)/(a*x^2)+3/4*e*sqrt(d+e*x)/(a*d*x)+(b*d-a*e)*sqrt(d+e*x)/(a^2*d*x)+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(c)*(b^3*d-a*c*(-2*a*e+d*sqrt(b^2-4*a*c))+b^2*(-a*e+d*sqrt(b^2-4*a*c))-a*b*(3*c*d+e*sqrt(b^2-4*a*c)))/(a^3*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(c)*(b^3*d-b^2*(a*e+d*sqrt(b^2-4*a*c))+a*c*(2*a*e+d*sqrt(b^2-4*a*c))-a*b*(3*c*d-e*sqrt(b^2-4*a*c)))/(a^3*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x^4*(d+e*x)^(3/2)/(a+b*x+c*x^2),x,6,-2/3*b*(b^2-2*a*c)*(d+e*x)^(3/2)/c^4+2/5*(c^2*d^2+b^2*e^2+c*e*(b*d-a*e))*(d+e*x)^(5/2)/(c^3*e^3)-2/7*(2*c*d+b*e)*(d+e*x)^(7/2)/(c^2*e^3)+2/9*(d+e*x)^(9/2)/(c*e^3)-2*(b^3*c*d-2*a*b*c^2*d-b^4*e+3*a*b^2*c*e-a^2*c^2*e)*sqrt(d+e*x)/c^5+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*((b*c*d-b^2*e+a*c*e)*(b^2*c*d-2*a*c^2*d-b^3*e+3*a*b*c*e)+(2*b^5*c*d*e-10*a*b^3*c^2*d*e+10*a^2*b*c^3*d*e-b^6*e^2+a*b^2*c^2*(4*c*d^2-9*a*e^2)-b^4*c*(c*d^2-6*a*e^2)-2*a^2*c^3*(c*d^2-a*e^2))/sqrt(b^2-4*a*c))/(c^(11/2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*((b*c*d-b^2*e+a*c*e)*(b^2*c*d-2*a*c^2*d-b^3*e+3*a*b*c*e)+(-2*b^5*c*d*e+10*a*b^3*c^2*d*e-10*a^2*b*c^3*d*e+b^6*e^2-a*b^2*c^2*(4*c*d^2-9*a*e^2)+b^4*c*(c*d^2-6*a*e^2)+2*a^2*c^3*(c*d^2-a*e^2))/sqrt(b^2-4*a*c))/(c^(11/2)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x^3*(d+e*x)^(3/2)/(a+b*x+c*x^2),x,6,2/3*(b^2-a*c)*(d+e*x)^(3/2)/c^3-2/5*(c*d+b*e)*(d+e*x)^(5/2)/(c^2*e^2)+2/7*(d+e*x)^(7/2)/(c*e^2)+2*(b^2*c*d-a*c^2*d-b^3*e+2*a*b*c*e)*sqrt(d+e*x)/c^4+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*(2*b^3*c*d*e-4*a*b*c^2*d*e-b^4*e^2-b^2*c*(c*d^2-3*a*e^2)+a*c^2*(c*d^2-a*e^2)+(-2*b^4*c*d*e+8*a*b^2*c^2*d*e-4*a^2*c^3*d*e+b^5*e^2+b^3*c*(c*d^2-5*a*e^2)-a*b*c^2*(3*c*d^2-5*a*e^2))/sqrt(b^2-4*a*c))/(c^(9/2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*(2*b^3*c*d*e-4*a*b*c^2*d*e-b^4*e^2-b^2*c*(c*d^2-3*a*e^2)+a*c^2*(c*d^2-a*e^2)+(2*b^4*c*d*e-8*a*b^2*c^2*d*e+4*a^2*c^3*d*e-b^5*e^2-b^3*c*(c*d^2-5*a*e^2)+a*b*c^2*(3*c*d^2-5*a*e^2))/sqrt(b^2-4*a*c))/(c^(9/2)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x^2*(d+e*x)^(3/2)/(a+b*x+c*x^2),x,6,-2/3*b*(d+e*x)^(3/2)/c^2+2/5*(d+e*x)^(5/2)/(c*e)-2*(b*c*d-b^2*e+a*c*e)*sqrt(d+e*x)/c^3+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*((c*d-b*e)*(b*c*d-b^2*e+2*a*c*e)+(2*b^3*c*d*e-6*a*b*c^2*d*e-b^4*e^2-b^2*c*(c*d^2-4*a*e^2)+2*a*c^2*(c*d^2-a*e^2))/sqrt(b^2-4*a*c))/(c^(7/2)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*((c*d-b*e)*(b*c*d-b^2*e+2*a*c*e)+(-2*b^3*c*d*e+6*a*b*c^2*d*e+b^4*e^2+b^2*c*(c*d^2-4*a*e^2)-2*a*c^2*(c*d^2-a*e^2))/sqrt(b^2-4*a*c))/(c^(7/2)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[x*(d+e*x)^(3/2)/(a+b*x+c*x^2),x,6,2/3*(d+e*x)^(3/2)/c+2*(c*d-b*e)*sqrt(d+e*x)/c^2+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*(b^3*e^2-b^2*e*(2*c*d+e*sqrt(b^2-4*a*c))+c*(a*e^2*sqrt(b^2-4*a*c)-c*d*(-4*a*e+d*sqrt(b^2-4*a*c)))+b*c*(c*d^2+e*(-3*a*e+2*d*sqrt(b^2-4*a*c))))/(c^(5/2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*(b^3*e^2-b^2*e*(2*c*d-e*sqrt(b^2-4*a*c))-c*(a*e^2*sqrt(b^2-4*a*c)-c*d*(4*a*e+d*sqrt(b^2-4*a*c)))+b*c*(c*d^2-e*(3*a*e+2*d*sqrt(b^2-4*a*c))))/(c^(5/2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x)^(3/2)/(a+b*x+c*x^2),x,5,2*e*sqrt(d+e*x)/c-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*(2*c^2*d^2+b*e^2*(b-sqrt(b^2-4*a*c))-2*c*e*(b*d+a*e-d*sqrt(b^2-4*a*c)))/(c^(3/2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*(2*c^2*d^2+b*e^2*(b+sqrt(b^2-4*a*c))-2*c*e*(b*d+a*e+d*sqrt(b^2-4*a*c)))/(c^(3/2)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x)^(3/2)/(x*(a+b*x+c*x^2)),x,7,-2*d^(3/2)*arctanh(sqrt(d+e*x)/sqrt(d))/a-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*(-b*(c*d^2+a*e^2)+a*e^2*sqrt(b^2-4*a*c)-c*d*(-4*a*e+d*sqrt(b^2-4*a*c)))/(a*sqrt(c)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*(b*(c*d^2+a*e^2)+a*e^2*sqrt(b^2-4*a*c)-c*d*(4*a*e+d*sqrt(b^2-4*a*c)))/(a*sqrt(c)*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x)^(3/2)/(x^2*(a+b*x+c*x^2)),x,9,e*arctanh(sqrt(d+e*x)/sqrt(d))*sqrt(d)/a+2*(b*d-2*a*e)*arctanh(sqrt(d+e*x)/sqrt(d))*sqrt(d)/a^2-d*sqrt(d+e*x)/(a*x)-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(c)*(b^2*d^2+b*d*(-2*a*e+d*sqrt(b^2-4*a*c))-2*a*(c*d^2+e*(-a*e+d*sqrt(b^2-4*a*c))))/(a^2*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(c)*(b^2*d^2-b*d*(2*a*e+d*sqrt(b^2-4*a*c))-2*a*(c*d^2-e*(a*e+d*sqrt(b^2-4*a*c))))/(a^2*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[(d+e*x)^(3/2)/(x^3*(a+b*x+c*x^2)),x,12,-3/4*e^2*arctanh(sqrt(d+e*x)/sqrt(d))/(a*sqrt(d))-e*(b*d-2*a*e)*arctanh(sqrt(d+e*x)/sqrt(d))/(a^2*sqrt(d))-2*(b^2*d^2-2*a*b*d*e-a*(c*d^2-a*e^2))*arctanh(sqrt(d+e*x)/sqrt(d))/(a^3*sqrt(d))-1/2*d*sqrt(d+e*x)/(a*x^2)+3/4*e*sqrt(d+e*x)/(a*x)+(b*d-2*a*e)*sqrt(d+e*x)/(a^2*x)+arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(c)*(b^3*d^2+b^2*d*(-2*a*e+d*sqrt(b^2-4*a*c))+a*(a*e^2*sqrt(b^2-4*a*c)-c*d*(-4*a*e+d*sqrt(b^2-4*a*c)))-a*b*(3*c*d^2+e*(-a*e+2*d*sqrt(b^2-4*a*c))))/(a^3*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c))))-arctanh(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(c)*(b^3*d^2-b^2*d*(2*a*e+d*sqrt(b^2-4*a*c))-a*(a*e^2*sqrt(b^2-4*a*c)-c*d*(4*a*e+d*sqrt(b^2-4*a*c)))-a*b*(3*c*d^2-e*(a*e+2*d*sqrt(b^2-4*a*c))))/(a^3*sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],

# Integrands of the form x^m (d+e x)^n (a+b x+c x^2)^p when n symbolic

# p>0

# p<0
[x^m*(e+f*x)^n/(a+b*x+c*x^2),x,6,2*c*x^(1+m)*(e+f*x)^n*AppellF1(1+m,-n,1,2+m,-f*x/e,-2*c*x/(b-sqrt(b^2-4*a*c)))/((1+m)*(1+f*x/e)^n*(b-sqrt(b^2-4*a*c))*sqrt(b^2-4*a*c))-2*c*x^(1+m)*(e+f*x)^n*AppellF1(1+m,-n,1,2+m,-f*x/e,-2*c*x/(b+sqrt(b^2-4*a*c)))/((1+m)*(1+f*x/e)^n*sqrt(b^2-4*a*c)*(b+sqrt(b^2-4*a*c)))],
[x^3*(e+f*x)^n/(a+b*x+c*x^2),x,4,-(c*e+b*f)*(e+f*x)^(1+n)/(c^2*f^2*(1+n))+(e+f*x)^(2+n)/(c*f^2*(2+n))+(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],2*c*(e+f*x)/(2*c*e-f*(b-sqrt(b^2-4*a*c))))*(a-b^2/c+b*(b^2-3*a*c)/(c*sqrt(b^2-4*a*c)))/(c*(1+n)*(2*c*e-f*(b-sqrt(b^2-4*a*c))))+(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],2*c*(e+f*x)/(2*c*e-f*(b+sqrt(b^2-4*a*c))))*(a-b^2/c-b*(b^2-3*a*c)/(c*sqrt(b^2-4*a*c)))/(c*(1+n)*(2*c*e-f*(b+sqrt(b^2-4*a*c))))],
[x^2*(e+f*x)^n/(a+b*x+c*x^2),x,4,(e+f*x)^(1+n)/(c*f*(1+n))+(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],2*c*(e+f*x)/(2*c*e-f*(b-sqrt(b^2-4*a*c))))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(c*(1+n)*(2*c*e-f*(b-sqrt(b^2-4*a*c))))+(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],2*c*(e+f*x)/(2*c*e-f*(b+sqrt(b^2-4*a*c))))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(c*(1+n)*(2*c*e-f*(b+sqrt(b^2-4*a*c))))],
[x*(e+f*x)^n/(a+b*x+c*x^2),x,4,-(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],2*c*(e+f*x)/(2*c*e-f*(b-sqrt(b^2-4*a*c))))*(1-b/sqrt(b^2-4*a*c))/((1+n)*(2*c*e-f*(b-sqrt(b^2-4*a*c))))-(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],2*c*(e+f*x)/(2*c*e-f*(b+sqrt(b^2-4*a*c))))*(1+b/sqrt(b^2-4*a*c))/((1+n)*(2*c*e-f*(b+sqrt(b^2-4*a*c))))],
[(e+f*x)^n/(a+b*x+c*x^2),x,4,-2*c*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],2*c*(e+f*x)/(2*c*e-f*(b-sqrt(b^2-4*a*c))))/((1+n)*(2*c*e-f*(b-sqrt(b^2-4*a*c)))*sqrt(b^2-4*a*c))+2*c*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],2*c*(e+f*x)/(2*c*e-f*(b+sqrt(b^2-4*a*c))))/((1+n)*sqrt(b^2-4*a*c)*(2*c*e-f*(b+sqrt(b^2-4*a*c))))],
[(e+f*x)^n/(x*(a+b*x+c*x^2)),x,7,-(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],1+f*x/e)/(a*e*(1+n))+c*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],2*c*(e+f*x)/(2*c*e-f*(b-sqrt(b^2-4*a*c))))*(1+b/sqrt(b^2-4*a*c))/(a*(1+n)*(2*c*e-f*(b-sqrt(b^2-4*a*c))))+c*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],2*c*(e+f*x)/(2*c*e-f*(b+sqrt(b^2-4*a*c))))*(1-b/sqrt(b^2-4*a*c))/(a*(1+n)*(2*c*e-f*(b+sqrt(b^2-4*a*c))))],
[(e+f*x)^n/(x^2*(a+b*x+c*x^2)),x,8,b*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],1+f*x/e)/(a^2*e*(1+n))+f*(e+f*x)^(1+n)*hypergeom([2,1+n],[2+n],1+f*x/e)/(a*e^2*(1+n))-c*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],2*c*(e+f*x)/(2*c*e-f*(b-sqrt(b^2-4*a*c))))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(a^2*(1+n)*(2*c*e-f*(b-sqrt(b^2-4*a*c))))-c*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],2*c*(e+f*x)/(2*c*e-f*(b+sqrt(b^2-4*a*c))))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(a^2*(1+n)*(2*c*e-f*(b+sqrt(b^2-4*a*c))))],

# Integrands of the form (d+e x)^m (f+g x)^n (a+b x+c x^2)^p

# Integrands of the form (d+e x)^m (f+g x)^n (a+b x+c x^2)^p when b=0 and c d^2+a e^2=0

# Integrands of the form (d+e x)^m (f+g x)^2 (d^2-e^2 x^2)^p

# p>0

# p<0
[(d+e*x)^4*(f+g*x)^2/(d^2-e^2*x^2),x,3,-d^2*(7*e^2*f^2+16*d*e*f*g+8*d^2*g^2)*x/e^2-d*(2*e^2*f^2+7*d*e*f*g+4*d^2*g^2)*x^2/e-1/3*(e*f+d*g)*(e*f+7*d*g)*x^3-1/2*e*g*(e*f+2*d*g)*x^4-1/5*e^2*g^2*x^5-8*d^3*(e*f+d*g)^2*log(d-e*x)/e^3],
[(d+e*x)^3*(f+g*x)^2/(d^2-e^2*x^2),x,3,-d*(e*f+2*d*g)*(3*e*f+2*d*g)*x/e^2-1/2*(e^2*f^2+6*d*e*f*g+4*d^2*g^2)*x^2/e-1/3*g*(2*e*f+3*d*g)*x^3-1/4*e*g^2*x^4-4*d^2*(e*f+d*g)^2*log(d-e*x)/e^3],
[(d+e*x)^2*(f+g*x)^2/(d^2-e^2*x^2),x,3,-2*d*g*(e*f+d*g)*x/e^2-d*(f+g*x)^2/e-1/3*(f+g*x)^3/g-2*d*(e*f+d*g)^2*log(d-e*x)/e^3],
[(d+e*x)*(f+g*x)^2/(d^2-e^2*x^2),x,3,-g*(e*f+d*g)*x/e^2-1/2*(f+g*x)^2/e-(e*f+d*g)^2*log(d-e*x)/e^3],
[(f+g*x)^2/(d^2-e^2*x^2),x,5,-g^2*x/e^2-1/2*(e*f+d*g)^2*log(d-e*x)/(d*e^3)+1/2*(e*f-d*g)^2*log(d+e*x)/(d*e^3)],
[(f+g*x)^2/((d+e*x)*(d^2-e^2*x^2)),x,3,-1/2*(e*f-d*g)^2/(d*e^3*(d+e*x))-1/4*(e*f+d*g)^2*log(d-e*x)/(d^2*e^3)+1/4*(e*f-d*g)*(e*f+3*d*g)*log(d+e*x)/(d^2*e^3)],
[(f+g*x)^2/((d+e*x)^2*(d^2-e^2*x^2)),x,4,-1/4*(e*f-d*g)^2/(d*e^3*(d+e*x)^2)-1/4*(e*f-d*g)*(e*f+3*d*g)/(d^2*e^3*(d+e*x))+1/4*(e*f+d*g)^2*arctanh(e*x/d)/(d^3*e^3)],
[(f+g*x)^2/((d+e*x)^3*(d^2-e^2*x^2)),x,4,-1/6*(e*f-d*g)^2/(d*e^3*(d+e*x)^3)-1/8*(e*f-d*g)*(e*f+3*d*g)/(d^2*e^3*(d+e*x)^2)-1/8*(e*f+d*g)^2/(d^3*e^3*(d+e*x))+1/8*(e*f+d*g)^2*arctanh(e*x/d)/(d^4*e^3)],
[(f+g*x)^2/((d+e*x)^4*(d^2-e^2*x^2)),x,4,-1/8*(e*f-d*g)^2/(d*e^3*(d+e*x)^4)-1/12*(e*f-d*g)*(e*f+3*d*g)/(d^2*e^3*(d+e*x)^3)-1/16*(e*f+d*g)^2/(d^3*e^3*(d+e*x)^2)-1/16*(e*f+d*g)^2/(d^4*e^3*(d+e*x))+1/16*(e*f+d*g)^2*arctanh(e*x/d)/(d^5*e^3)],
[(d+e*x)^7*(f+g*x)^2/(d^2-e^2*x^2)^2,x,3,d^3*(49*e^2*f^2+160*d*e*f*g+112*d^2*g^2)*x/e^2+1/2*d^2*(23*e^2*f^2+98*d*e*f*g+80*d^2*g^2)*x^2/e+1/3*d*(7*e^2*f^2+46*d*e*f*g+49*d^2*g^2)*x^3+1/4*e*(e^2*f^2+14*d*e*f*g+23*d^2*g^2)*x^4+1/5*e^2*g*(2*e*f+7*d*g)*x^5+1/6*e^3*g^2*x^6+32*d^5*(e*f+d*g)^2/(e^3*(d-e*x))+16*d^4*(e*f+d*g)*(5*e*f+9*d*g)*log(d-e*x)/e^3],
[(d+e*x)^6*(f+g*x)^2/(d^2-e^2*x^2)^2,x,3,d^2*(17*e^2*f^2+64*d*e*f*g+48*d^2*g^2)*x/e^2+d*(3*e^2*f^2+17*d*e*f*g+16*d^2*g^2)*x^2/e+1/3*(e^2*f^2+12*d*e*f*g+17*d^2*g^2)*x^3+1/2*e*g*(e*f+3*d*g)*x^4+1/5*e^2*g^2*x^5+16*d^4*(e*f+d*g)^2/(e^3*(d-e*x))+32*d^3*(e*f+d*g)*(e*f+2*d*g)*log(d-e*x)/e^3],
[(d+e*x)^5*(f+g*x)^2/(d^2-e^2*x^2)^2,x,3,d*(5*e^2*f^2+24*d*e*f*g+20*d^2*g^2)*x/e^2+1/2*(e^2*f^2+10*d*e*f*g+12*d^2*g^2)*x^2/e+1/3*g*(2*e*f+5*d*g)*x^3+1/4*e*g^2*x^4+8*d^3*(e*f+d*g)^2/(e^3*(d-e*x))+4*d^2*(e*f+d*g)*(3*e*f+7*d*g)*log(d-e*x)/e^3],
[(d+e*x)^4*(f+g*x)^2/(d^2-e^2*x^2)^2,x,3,(e^2*f^2+8*d*e*f*g+8*d^2*g^2)*x/e^2+g*(e*f+2*d*g)*x^2/e+1/3*g^2*x^3+4*d^2*(e*f+d*g)^2/(e^3*(d-e*x))+4*d*(e*f+d*g)*(e*f+3*d*g)*log(d-e*x)/e^3],
[(d+e*x)^3*(f+g*x)^2/(d^2-e^2*x^2)^2,x,3,g*(2*e*f+3*d*g)*x/e^2+1/2*g^2*x^2/e+2*d*(e*f+d*g)^2/(e^3*(d-e*x))+(e*f+d*g)*(e*f+5*d*g)*log(d-e*x)/e^3],
[(d+e*x)^2*(f+g*x)^2/(d^2-e^2*x^2)^2,x,3,g^2*x/e^2+(e*f+d*g)^2/(e^3*(d-e*x))+2*g*(e*f+d*g)*log(d-e*x)/e^3],
[(d+e*x)*(f+g*x)^2/(d^2-e^2*x^2)^2,x,3,1/2*(e*f+d*g)^2/(d*e^3*(d-e*x))-1/4*(e*f-3*d*g)*(e*f+d*g)*log(d-e*x)/(d^2*e^3)+1/4*(e*f-d*g)^2*log(d+e*x)/(d^2*e^3)],
[(f+g*x)^2/(d^2-e^2*x^2)^2,x,2,1/2*(d^2*g+e^2*f*x)*(f+g*x)/(d^2*e^2*(d^2-e^2*x^2))+1/2*(e*f-d*g)*(e*f+d*g)*arctanh(e*x/d)/(d^3*e^3)],
[(f+g*x)^2/((d+e*x)*(d^2-e^2*x^2)^2),x,4,1/8*(e*f+d*g)^2/(d^3*e^3*(d-e*x))-1/8*(e*f-d*g)^2/(d^2*e^3*(d+e*x)^2)+1/4*(-e^2*f^2+d^2*g^2)/(d^3*e^3*(d+e*x))+1/8*(3*e*f-d*g)*(e*f+d*g)*arctanh(e*x/d)/(d^4*e^3)],
[(f+g*x)^2/((d+e*x)^2*(d^2-e^2*x^2)^2),x,4,1/16*(e*f+d*g)^2/(d^4*e^3*(d-e*x))-1/12*(e*f-d*g)^2/(d^2*e^3*(d+e*x)^3)+1/8*(-e^2*f^2+d^2*g^2)/(d^3*e^3*(d+e*x)^2)-1/16*(3*e*f-d*g)*(e*f+d*g)/(d^4*e^3*(d+e*x))+1/4*f*(e*f+d*g)*arctanh(e*x/d)/(d^5*e^2)],
[(f+g*x)^2/((d+e*x)^3*(d^2-e^2*x^2)^2),x,4,1/32*(e*f+d*g)^2/(d^5*e^3*(d-e*x))-1/16*(e*f-d*g)^2/(d^2*e^3*(d+e*x)^4)+1/12*(-e^2*f^2+d^2*g^2)/(d^3*e^3*(d+e*x)^3)-1/32*(3*e*f-d*g)*(e*f+d*g)/(d^4*e^3*(d+e*x)^2)-1/8*f*(e*f+d*g)/(d^5*e^2*(d+e*x))+1/32*(e*f+d*g)*(5*e*f+d*g)*arctanh(e*x/d)/(d^6*e^3)],
[(f+g*x)^2/((d+e*x)^4*(d^2-e^2*x^2)^2),x,4,1/64*(e*f+d*g)^2/(d^6*e^3*(d-e*x))-1/20*(e*f-d*g)^2/(d^2*e^3*(d+e*x)^5)+1/16*(-e^2*f^2+d^2*g^2)/(d^3*e^3*(d+e*x)^4)-1/48*(3*e*f-d*g)*(e*f+d*g)/(d^4*e^3*(d+e*x)^3)-1/16*f*(e*f+d*g)/(d^5*e^2*(d+e*x)^2)-1/64*(e*f+d*g)*(5*e*f+d*g)/(d^6*e^3*(d+e*x))+1/32*(e*f+d*g)*(3*e*f+d*g)*arctanh(e*x/d)/(d^7*e^3)],
[(d+e*x)^7*(f+g*x)^2/(d^2-e^2*x^2)^3,x,3,-d*(7*e^2*f^2+48*d*e*f*g+56*d^2*g^2)*x/e^2-1/2*(e*f+2*d*g)*(e*f+12*d*g)*x^2/e-1/3*g*(2*e*f+7*d*g)*x^3-1/4*e*g^2*x^4+8*d^4*(e*f+d*g)^2/(e^3*(d-e*x)^2)-32*d^3*(e*f+d*g)*(e*f+2*d*g)/(e^3*(d-e*x))-8*d^2*(3*e^2*f^2+14*d*e*f*g+13*d^2*g^2)*log(d-e*x)/e^3],
[(d+e*x)^6*(f+g*x)^2/(d^2-e^2*x^2)^3,x,3,-(e^2*f^2+12*d*e*f*g+18*d^2*g^2)*x/e^2-g*(e*f+3*d*g)*x^2/e-1/3*g^2*x^3+4*d^3*(e*f+d*g)^2/(e^3*(d-e*x)^2)-4*d^2*(e*f+d*g)*(3*e*f+7*d*g)/(e^3*(d-e*x))-2*d*(3*e^2*f^2+18*d*e*f*g+19*d^2*g^2)*log(d-e*x)/e^3],
[(d+e*x)^5*(f+g*x)^2/(d^2-e^2*x^2)^3,x,3,-g*(2*e*f+5*d*g)*x/e^2-1/2*g^2*x^2/e+2*d^2*(e*f+d*g)^2/(e^3*(d-e*x)^2)-4*d*(e*f+d*g)*(e*f+3*d*g)/(e^3*(d-e*x))-(e^2*f^2+10*d*e*f*g+13*d^2*g^2)*log(d-e*x)/e^3],
[(d+e*x)^4*(f+g*x)^2/(d^2-e^2*x^2)^3,x,3,-g^2*x/e^2+d*(e*f+d*g)^2/(e^3*(d-e*x)^2)-(e*f+d*g)*(e*f+5*d*g)/(e^3*(d-e*x))-2*g*(e*f+2*d*g)*log(d-e*x)/e^3],
[(d+e*x)^3*(f+g*x)^2/(d^2-e^2*x^2)^3,x,3,1/2*(e*f+d*g)^2/(e^3*(d-e*x)^2)-2*g*(e*f+d*g)/(e^3*(d-e*x))-g^2*log(d-e*x)/e^3],
[(d+e*x)^2*(f+g*x)^2/(d^2-e^2*x^2)^3,x,4,1/4*(e*f+d*g)^2/(d*e^3*(d-e*x)^2)+1/4*(e*f-3*d*g)*(e*f+d*g)/(d^2*e^3*(d-e*x))+1/4*(e*f-d*g)^2*arctanh(e*x/d)/(d^3*e^3)],
[(d+e*x)*(f+g*x)^2/(d^2-e^2*x^2)^3,x,4,1/8*(e*f+d*g)^2/(d^2*e^3*(d-e*x)^2)+1/4*(e^2*f^2-d^2*g^2)/(d^3*e^3*(d-e*x))-1/8*(e*f-d*g)^2/(d^3*e^3*(d+e*x))+1/8*(e*f-d*g)*(3*e*f+d*g)*arctanh(e*x/d)/(d^4*e^3)],
[(f+g*x)^2/(d^2-e^2*x^2)^3,x,3,1/4*(d^2*g+e^2*f*x)*(f+g*x)/(d^2*e^2*(d^2-e^2*x^2)^2)+1/8*(2*d^2*f*g+(3*e^2*f^2-d^2*g^2)*x)/(d^4*e^2*(d^2-e^2*x^2))+1/8*(3*e^2*f^2-d^2*g^2)*arctanh(e*x/d)/(d^5*e^3)],
[(f+g*x)^2/((d+e*x)*(d^2-e^2*x^2)^3),x,4,1/32*(e*f+d*g)^2/(d^4*e^3*(d-e*x)^2)+1/8*f*(e*f+d*g)/(d^5*e^2*(d-e*x))-1/24*(e*f-d*g)^2/(d^3*e^3*(d+e*x)^3)-1/32*(e*f-d*g)*(3*e*f+d*g)/(d^4*e^3*(d+e*x)^2)+1/16*(-3*e^2*f^2+d^2*g^2)/(d^5*e^3*(d+e*x))+1/16*(5*e^2*f^2+2*d*e*f*g-d^2*g^2)*arctanh(e*x/d)/(d^6*e^3)],
[(f+g*x)^2/((d+e*x)^2*(d^2-e^2*x^2)^3),x,4,1/64*(e*f+d*g)^2/(d^5*e^3*(d-e*x)^2)+1/64*(e*f+d*g)*(5*e*f+d*g)/(d^6*e^3*(d-e*x))-1/32*(e*f-d*g)^2/(d^3*e^3*(d+e*x)^4)-1/48*(e*f-d*g)*(3*e*f+d*g)/(d^4*e^3*(d+e*x)^3)+1/32*(-3*e^2*f^2+d^2*g^2)/(d^5*e^3*(d+e*x)^2)+1/32*(-5*e^2*f^2-2*d*e*f*g+d^2*g^2)/(d^6*e^3*(d+e*x))+1/64*(15*e^2*f^2+10*d*e*f*g-d^2*g^2)*arctanh(e*x/d)/(d^7*e^3)],

# Integrands of the form (d+e x)^m (f+g x)^n (d^2-e^2 x^2)^(p/2)
[(d+e*x)^3*(f+g*x)^5/(d^2-e^2*x^2)^(7/2),x,7,1/5*(e*f+d*g)^5*(d+e*x)^3/(d*e^6*(d^2-e^2*x^2)^(5/2))+1/15*(2*e*f-23*d*g)*(e*f+d*g)^4*(d+e*x)^2/(d^2*e^6*(d^2-e^2*x^2)^(3/2))-1/2*g^3*(20*e^2*f^2+30*d*e*f*g+13*d^2*g^2)*arctan(e*x/sqrt(d^2-e^2*x^2))/e^6+1/15*(e*f+d*g)^3*(2*e^2*f^2-21*d*e*f*g+127*d^2*g^2)*(d+e*x)/(d^3*e^6*sqrt(d^2-e^2*x^2))+g^4*(5*e*f+3*d*g)*sqrt(d^2-e^2*x^2)/e^6+1/2*g^5*x*sqrt(d^2-e^2*x^2)/e^5],
[(d+e*x)^3*(f+g*x)^4/(d^2-e^2*x^2)^(7/2),x,6,1/5*(e*f+d*g)^4*(d+e*x)^3/(d*e^5*(d^2-e^2*x^2)^(5/2))+2/15*(e*f-9*d*g)*(e*f+d*g)^3*(d+e*x)^2/(d^2*e^5*(d^2-e^2*x^2)^(3/2))-g^3*(4*e*f+3*d*g)*arctan(e*x/sqrt(d^2-e^2*x^2))/e^5+2/15*(e*f+d*g)^2*(e^2*f^2-8*d*e*f*g+36*d^2*g^2)*(d+e*x)/(d^3*e^5*sqrt(d^2-e^2*x^2))+g^4*sqrt(d^2-e^2*x^2)/e^5],
[(d+e*x)^3*(f+g*x)^3/(d^2-e^2*x^2)^(7/2),x,5,1/5*(e*f+d*g)^3*(d+e*x)^3/(d*e^4*(d^2-e^2*x^2)^(5/2))+1/15*(2*e*f-13*d*g)*(e*f+d*g)^2*(d+e*x)^2/(d^2*e^4*(d^2-e^2*x^2)^(3/2))-g^3*arctan(e*x/sqrt(d^2-e^2*x^2))/e^4+1/15*(e*f+d*g)*(2*e^2*f^2-11*d*e*f*g+32*d^2*g^2)*(d+e*x)/(d^3*e^4*sqrt(d^2-e^2*x^2))],
[(d+e*x)^3*(f+g*x)^2/(d^2-e^2*x^2)^(7/2),x,3,1/5*(e*f+d*g)^2*(d+e*x)^3/(d*e^3*(d^2-e^2*x^2)^(5/2))+2/15*(e*f-4*d*g)*(e*f+d*g)*(d+e*x)^2/(d^2*e^3*(d^2-e^2*x^2)^(3/2))+1/15*(2*e^2*f^2-6*d*e*f*g+7*d^2*g^2)*(d+e*x)/(d^3*e^3*sqrt(d^2-e^2*x^2))],
[(d+e*x)^3*(f+g*x)/(d^2-e^2*x^2)^(7/2),x,3,1/5*(e*f+d*g)*(d+e*x)^3/(d*e^2*(d^2-e^2*x^2)^(5/2))+2/15*(2*e*f-3*d*g)*(d+e*x)/(d*e^2*(d^2-e^2*x^2)^(3/2))+1/15*(2*e*f-3*d*g)*x/(d^3*e*sqrt(d^2-e^2*x^2))],
[(d+e*x)^3/(d^2-e^2*x^2)^(7/2),x,4,1/5*sqrt(d^2-e^2*x^2)/(d*e*(d-e*x)^3)+2/15*sqrt(d^2-e^2*x^2)/(d^2*e*(d-e*x)^2)+2/15*sqrt(d^2-e^2*x^2)/(d^3*e*(d-e*x))],
[(d+e*x)^3/((f+g*x)*(d^2-e^2*x^2)^(7/2)),x,6,4/5*d*(d+e*x)/((e*f+d*g)*(d^2-e^2*x^2)^(5/2))+1/15*(-5*d*(e*f-d*g)+e*(e*f+11*d*g)*x)/(d*(e*f+d*g)^2*(d^2-e^2*x^2)^(3/2))+g^3*arctan((d^2*g+e^2*f*x)/(sqrt(e^2*f^2-d^2*g^2)*sqrt(d^2-e^2*x^2)))/((e*f+d*g)^3*sqrt(e^2*f^2-d^2*g^2))+1/15*(15*d^3*g^2+e*(2*e^2*f^2+9*d*e*f*g+22*d^2*g^2)*x)/(d^3*(e*f+d*g)^3*sqrt(d^2-e^2*x^2))],
[(d+e*x)^3/((f+g*x)^2*(d^2-e^2*x^2)^(7/2)),x,6,4/5*d*e*(d+e*x)/((e*f+d*g)^2*(d^2-e^2*x^2)^(5/2))-1/15*e*(5*d*(e*f-3*d*g)-e*(e*f+21*d*g)*x)/(d*(e*f+d*g)^3*(d^2-e^2*x^2)^(3/2))+e*g^3*(4*e*f-3*d*g)*arctan((d^2*g+e^2*f*x)/(sqrt(e^2*f^2-d^2*g^2)*sqrt(d^2-e^2*x^2)))/((e*f-d*g)*(e*f+d*g)^4*sqrt(e^2*f^2-d^2*g^2))+1/15*e*(45*d^3*g^2+e*(2*e^2*f^2+14*d*e*f*g+57*d^2*g^2)*x)/(d^3*(e*f+d*g)^4*sqrt(d^2-e^2*x^2))+g^4*sqrt(d^2-e^2*x^2)/((e*f-d*g)*(e*f+d*g)^4*(f+g*x))],
[(d+e*x)^3/((f+g*x)^3*(d^2-e^2*x^2)^(7/2)),x,7,4/5*d*e^2*(d+e*x)/((e*f+d*g)^3*(d^2-e^2*x^2)^(5/2))-1/15*e^2*(5*d*(e*f-5*d*g)-e*(e*f+31*d*g)*x)/(d*(e*f+d*g)^4*(d^2-e^2*x^2)^(3/2))+1/2*e^2*g^3*(20*e^2*f^2-30*d*e*f*g+13*d^2*g^2)*arctan((d^2*g+e^2*f*x)/(sqrt(e^2*f^2-d^2*g^2)*sqrt(d^2-e^2*x^2)))/((e*f-d*g)^2*(e*f+d*g)^5*sqrt(e^2*f^2-d^2*g^2))+1/15*e^2*(90*d^3*g^2+e*(2*e^2*f^2+19*d*e*f*g+107*d^2*g^2)*x)/(d^3*(e*f+d*g)^5*sqrt(d^2-e^2*x^2))+1/2*g^4*sqrt(d^2-e^2*x^2)/((e*f-d*g)*(e*f+d*g)^4*(f+g*x)^2)+3/2*e*g^4*(3*e*f-2*d*g)*sqrt(d^2-e^2*x^2)/((e*f-d*g)^2*(e*f+d*g)^5*(f+g*x))],

# Integrands of the form (d+e x)^m (f+g x)^n (a+b x+c x^2)^p when b=0

# Integrands of the form (d+e x)^m (f+g x)^(n/2) (a+c x^2)^p

# p>0 & n>0

# p>0 & n<0
[(a+c*x^2)/((d+e*x)^(3/2)*(f+g*x)),x,4,-2*(c*f^2+a*g^2)*arctan(sqrt(g)*sqrt(d+e*x)/sqrt(e*f-d*g))/(g^(3/2)*(e*f-d*g)^(3/2))-2*(c*d^2+a*e^2)/(e^2*(e*f-d*g)*sqrt(d+e*x))+2*c*sqrt(d+e*x)/(e^2*g)],
[(d+e*x)^3*(a+c*x^2)/sqrt(f+g*x),x,3,2/3*(e*f-d*g)^2*(3*a*e*g^2+c*f*(5*e*f-2*d*g))*(f+g*x)^(3/2)/g^6-2/5*(e*f-d*g)*(3*a*e^2*g^2+c*(10*e^2*f^2-8*d*e*f*g+d^2*g^2))*(f+g*x)^(5/2)/g^6+2/7*e*(a*e^2*g^2+c*(10*e^2*f^2-12*d*e*f*g+3*d^2*g^2))*(f+g*x)^(7/2)/g^6-2/9*c*e^2*(5*e*f-3*d*g)*(f+g*x)^(9/2)/g^6+2/11*c*e^3*(f+g*x)^(11/2)/g^6-2*(e*f-d*g)^3*(c*f^2+a*g^2)*sqrt(f+g*x)/g^6],
[(d+e*x)^2*(a+c*x^2)/sqrt(f+g*x),x,3,-4/3*(e*f-d*g)*(a*e*g^2+c*f*(2*e*f-d*g))*(f+g*x)^(3/2)/g^5+2/5*(a*e^2*g^2+c*(6*e^2*f^2-6*d*e*f*g+d^2*g^2))*(f+g*x)^(5/2)/g^5-4/7*c*e*(2*e*f-d*g)*(f+g*x)^(7/2)/g^5+2/9*c*e^2*(f+g*x)^(9/2)/g^5+2*(e*f-d*g)^2*(c*f^2+a*g^2)*sqrt(f+g*x)/g^5],
[(d+e*x)*(a+c*x^2)/sqrt(f+g*x),x,2,2/3*(a*e*g^2+c*f*(3*e*f-2*d*g))*(f+g*x)^(3/2)/g^4-2/5*c*(3*e*f-d*g)*(f+g*x)^(5/2)/g^4+2/7*c*e*(f+g*x)^(7/2)/g^4-2*(e*f-d*g)*(c*f^2+a*g^2)*sqrt(f+g*x)/g^4],
[(a+c*x^2)/sqrt(f+g*x),x,2,-4/3*c*f*(f+g*x)^(3/2)/g^3+2/5*c*(f+g*x)^(5/2)/g^3+2*(c*f^2+a*g^2)*sqrt(f+g*x)/g^3],
[(a+c*x^2)/((d+e*x)*sqrt(f+g*x)),x,4,2/3*c*(f+g*x)^(3/2)/(e*g^2)-2*(c*d^2+a*e^2)*arctanh(sqrt(e)*sqrt(f+g*x)/sqrt(e*f-d*g))/(e^(5/2)*sqrt(e*f-d*g))-2*c*(e*f+d*g)*sqrt(f+g*x)/(e^2*g^2)],
[(a+c*x^2)/((d+e*x)^2*sqrt(f+g*x)),x,4,(a*e^2*g+c*d*(4*e*f-3*d*g))*arctanh(sqrt(e)*sqrt(f+g*x)/sqrt(e*f-d*g))/(e^(5/2)*(e*f-d*g)^(3/2))+2*c*sqrt(f+g*x)/(e^2*g)-(a+c*d^2/e^2)*sqrt(f+g*x)/((e*f-d*g)*(d+e*x))],
[(a+c*x^2)/((d+e*x)^3*sqrt(f+g*x)),x,4,-1/4*(3*a*e^2*g^2+c*(8*e^2*f^2-8*d*e*f*g+3*d^2*g^2))*arctanh(sqrt(e)*sqrt(f+g*x)/sqrt(e*f-d*g))/(e^(5/2)*(e*f-d*g)^(5/2))-1/2*(a+c*d^2/e^2)*sqrt(f+g*x)/((e*f-d*g)*(d+e*x)^2)+1/4*(3*a*e^2*g+c*d*(8*e*f-5*d*g))*sqrt(f+g*x)/(e^2*(e*f-d*g)^2*(d+e*x))],
[(d+e*x)^3*(a+c*x^2)/(f+g*x)^(3/2),x,3,-2/3*(e*f-d*g)*(3*a*e^2*g^2+c*(10*e^2*f^2-8*d*e*f*g+d^2*g^2))*(f+g*x)^(3/2)/g^6+2/5*e*(a*e^2*g^2+c*(10*e^2*f^2-12*d*e*f*g+3*d^2*g^2))*(f+g*x)^(5/2)/g^6-2/7*c*e^2*(5*e*f-3*d*g)*(f+g*x)^(7/2)/g^6+2/9*c*e^3*(f+g*x)^(9/2)/g^6+2*(e*f-d*g)^3*(c*f^2+a*g^2)/(g^6*sqrt(f+g*x))+2*(e*f-d*g)^2*(3*a*e*g^2+c*f*(5*e*f-2*d*g))*sqrt(f+g*x)/g^6],
[(d+e*x)^2*(a+c*x^2)/(f+g*x)^(3/2),x,3,2/3*(a*e^2*g^2+c*(6*e^2*f^2-6*d*e*f*g+d^2*g^2))*(f+g*x)^(3/2)/g^5-4/5*c*e*(2*e*f-d*g)*(f+g*x)^(5/2)/g^5+2/7*c*e^2*(f+g*x)^(7/2)/g^5-2*(e*f-d*g)^2*(c*f^2+a*g^2)/(g^5*sqrt(f+g*x))-4*(e*f-d*g)*(a*e*g^2+c*f*(2*e*f-d*g))*sqrt(f+g*x)/g^5],
[(d+e*x)*(a+c*x^2)/(f+g*x)^(3/2),x,2,-2/3*c*(3*e*f-d*g)*(f+g*x)^(3/2)/g^4+2/5*c*e*(f+g*x)^(5/2)/g^4+2*(e*f-d*g)*(c*f^2+a*g^2)/(g^4*sqrt(f+g*x))+2*(a*e*g^2+c*f*(3*e*f-2*d*g))*sqrt(f+g*x)/g^4],
[(a+c*x^2)/(f+g*x)^(3/2),x,2,2/3*c*(f+g*x)^(3/2)/g^3-2*(c*f^2+a*g^2)/(g^3*sqrt(f+g*x))-4*c*f*sqrt(f+g*x)/g^3],
[(a+c*x^2)/((d+e*x)*(f+g*x)^(3/2)),x,4,-2*(c*d^2+a*e^2)*arctanh(sqrt(e)*sqrt(f+g*x)/sqrt(e*f-d*g))/(e^(3/2)*(e*f-d*g)^(3/2))+2*(c*f^2+a*g^2)/(g^2*(e*f-d*g)*sqrt(f+g*x))+2*c*sqrt(f+g*x)/(e*g^2)],
[(a+c*x^2)/((d+e*x)^2*(f+g*x)^(3/2)),x,4,(3*a*e^2*g+c*d*(4*e*f-d*g))*arctanh(sqrt(e)*sqrt(f+g*x)/sqrt(e*f-d*g))/(e^(3/2)*(e*f-d*g)^(5/2))-2*(c*f^2+a*g^2)/(g*(e*f-d*g)^2*sqrt(f+g*x))-(c*d^2+a*e^2)*sqrt(f+g*x)/(e*(e*f-d*g)^2*(d+e*x))],
[(a+c*x^2)/((d+e*x)^3*(f+g*x)^(3/2)),x,5,-1/4*(15*a*e^2*g^2+c*(8*e^2*f^2+8*d*e*f*g-d^2*g^2))*arctanh(sqrt(e)*sqrt(f+g*x)/sqrt(e*f-d*g))/(e^(3/2)*(e*f-d*g)^(7/2))+2*(c*f^2+a*g^2)/((e*f-d*g)^3*sqrt(f+g*x))-1/2*(c*d^2+a*e^2)*sqrt(f+g*x)/(e*(e*f-d*g)^2*(d+e*x)^2)+1/4*(7*a*e^2*g+c*d*(8*e*f-d*g))*sqrt(f+g*x)/(e*(e*f-d*g)^3*(d+e*x))],

# p<0

# Integrands of the form (d+e x)^(m/2) (f+g x)^(n/2) (a+c x^2)^p

# p>0
[(a+c*x^2)/(sqrt(d+e*x)*sqrt(f+g*x)),x,5,1/4*(8*a*e^2*g^2+c*(3*e^2*f^2+2*d*e*f*g+3*d^2*g^2))*arctanh(sqrt(g)*sqrt(d+e*x)/(sqrt(e)*sqrt(f+g*x)))/(e^(5/2)*g^(5/2))+1/2*c*(d+e*x)^(3/2)*sqrt(f+g*x)/(e^2*g)-1/4*c*(3*e*f+5*d*g)*sqrt(d+e*x)*sqrt(f+g*x)/(e^2*g^2)],
[(-1+2*x^2)/(sqrt(-1+x)*sqrt(1+x)),x,1,x*sqrt(-1+x)*sqrt(1+x)],

# p<0
[(d+e*x)^(3/2)*sqrt(f+g*x)/(a+c*x^2),x,11,(e*f+3*d*g)*arctanh(sqrt(g)*sqrt(d+e*x)/(sqrt(e)*sqrt(f+g*x)))*sqrt(e)/(c*sqrt(g))+e*sqrt(d+e*x)*sqrt(f+g*x)/c+arctanh(sqrt(d+e*x)*sqrt(-g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(-e*sqrt(-a)+d*sqrt(c))))*(-(c*d^2*f-a*e*(e*f+2*d*g))*sqrt(-a)+a*(a*e^2*g-c*d*(2*e*f+d*g))/sqrt(c))/(a*c*sqrt(-e*sqrt(-a)+d*sqrt(c))*sqrt(-g*sqrt(-a)+f*sqrt(c)))+arctanh(sqrt(d+e*x)*sqrt(g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(e*sqrt(-a)+d*sqrt(c))))*((c*d^2*f-a*e*(e*f+2*d*g))*sqrt(-a)+a*(a*e^2*g-c*d*(2*e*f+d*g))/sqrt(c))/(a*c*sqrt(e*sqrt(-a)+d*sqrt(c))*sqrt(g*sqrt(-a)+f*sqrt(c)))],
[(d+e*x)^(1/2)*sqrt(f+g*x)/(a+c*x^2),x,10,2*arctanh(sqrt(g)*sqrt(d+e*x)/(sqrt(e)*sqrt(f+g*x)))*sqrt(e)*sqrt(g)/c+arctanh(sqrt(d+e*x)*sqrt(-g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(-e*sqrt(-a)+d*sqrt(c))))*(c*d*f-a*e*g-(e*f+d*g)*sqrt(-a)*sqrt(c))/(c*sqrt(-a)*sqrt(-e*sqrt(-a)+d*sqrt(c))*sqrt(-g*sqrt(-a)+f*sqrt(c)))-arctanh(sqrt(d+e*x)*sqrt(g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(e*sqrt(-a)+d*sqrt(c))))*(c*d*f-a*e*g+(e*f+d*g)*sqrt(-a)*sqrt(c))/(c*sqrt(-a)*sqrt(e*sqrt(-a)+d*sqrt(c))*sqrt(g*sqrt(-a)+f*sqrt(c)))],
[sqrt(f+g*x)/((d+e*x)^(1/2)*(a+c*x^2)),x,6,arctanh(sqrt(d+e*x)*sqrt(-g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(-e*sqrt(-a)+d*sqrt(c))))*sqrt(-g*sqrt(-a)+f*sqrt(c))/(sqrt(-a)*sqrt(c)*sqrt(-e*sqrt(-a)+d*sqrt(c)))-arctanh(sqrt(d+e*x)*sqrt(g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(e*sqrt(-a)+d*sqrt(c))))*sqrt(g*sqrt(-a)+f*sqrt(c))/(sqrt(-a)*sqrt(c)*sqrt(e*sqrt(-a)+d*sqrt(c)))],
[sqrt(f+g*x)/((d+e*x)^(3/2)*(a+c*x^2)),x,8,-2*e*sqrt(f+g*x)/((c*d^2+a*e^2)*sqrt(d+e*x))+arctanh(sqrt(d+e*x)*sqrt(-g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(-e*sqrt(-a)+d*sqrt(c))))*(c*d*f+a*e*g+(e*f-d*g)*sqrt(-a)*sqrt(c))/((c*d^2+a*e^2)*sqrt(-a)*sqrt(-e*sqrt(-a)+d*sqrt(c))*sqrt(-g*sqrt(-a)+f*sqrt(c)))-arctanh(sqrt(d+e*x)*sqrt(g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(e*sqrt(-a)+d*sqrt(c))))*(c*d*f+a*e*g-(e*f-d*g)*sqrt(-a)*sqrt(c))/((c*d^2+a*e^2)*sqrt(-a)*sqrt(e*sqrt(-a)+d*sqrt(c))*sqrt(g*sqrt(-a)+f*sqrt(c)))],
[sqrt(f+g*x)/((d+e*x)^(5/2)*(a+c*x^2)),x,11,-2/3*e*sqrt(f+g*x)/((c*d^2+a*e^2)*(d+e*x)^(3/2))+4/3*e*g*sqrt(f+g*x)/((c*d^2+a*e^2)*(e*f-d*g)*sqrt(d+e*x))+e*(c*d*f+a*e*g-(e*f-d*g)*sqrt(-a)*sqrt(c))*sqrt(f+g*x)/((c*d^2+a*e^2)*(e*f-d*g)*sqrt(-a)*(e*sqrt(-a)+d*sqrt(c))*sqrt(d+e*x))-e*(c*d*f+a*e*g+(e*f-d*g)*sqrt(-a)*sqrt(c))*sqrt(f+g*x)/((c*d^2+a*e^2)*(e*f-d*g)*sqrt(-a)*(-e*sqrt(-a)+d*sqrt(c))*sqrt(d+e*x))+arctanh(sqrt(d+e*x)*sqrt(-g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(-e*sqrt(-a)+d*sqrt(c))))*sqrt(c)*(c*d*f+a*e*g+(e*f-d*g)*sqrt(-a)*sqrt(c))/((c*d^2+a*e^2)*sqrt(-a)*(-e*sqrt(-a)+d*sqrt(c))^(3/2)*sqrt(-g*sqrt(-a)+f*sqrt(c)))+arctanh(sqrt(d+e*x)*sqrt(g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(e*sqrt(-a)+d*sqrt(c))))*sqrt(c)*(c*d*f*sqrt(-a)+a*e*g*sqrt(-a)+a*(e*f-d*g)*sqrt(c))/(a*(c*d^2+a*e^2)*(e*sqrt(-a)+d*sqrt(c))^(3/2)*sqrt(g*sqrt(-a)+f*sqrt(c)))],
[(d+e*x)^(3/2)/((a+c*x^2)*sqrt(f+g*x)),x,11,2*e^(3/2)*arctanh(sqrt(g)*sqrt(d+e*x)/(sqrt(e)*sqrt(f+g*x)))/(c*sqrt(g))+arctanh(sqrt(d+e*x)*sqrt(-g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(-e*sqrt(-a)+d*sqrt(c))))*(c*d^2-a*e^2-2*d*e*sqrt(-a)*sqrt(c))/(c*sqrt(-a)*sqrt(-e*sqrt(-a)+d*sqrt(c))*sqrt(-g*sqrt(-a)+f*sqrt(c)))-arctanh(sqrt(d+e*x)*sqrt(g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(e*sqrt(-a)+d*sqrt(c))))*(c*d^2-a*e^2+2*d*e*sqrt(-a)*sqrt(c))/(c*sqrt(-a)*sqrt(e*sqrt(-a)+d*sqrt(c))*sqrt(g*sqrt(-a)+f*sqrt(c)))],
[(d+e*x)^(1/2)/((a+c*x^2)*sqrt(f+g*x)),x,6,arctanh(sqrt(d+e*x)*sqrt(-g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(-e*sqrt(-a)+d*sqrt(c))))*sqrt(-e*sqrt(-a)+d*sqrt(c))/(sqrt(-a)*sqrt(c)*sqrt(-g*sqrt(-a)+f*sqrt(c)))-arctanh(sqrt(d+e*x)*sqrt(g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(e*sqrt(-a)+d*sqrt(c))))*sqrt(e*sqrt(-a)+d*sqrt(c))/(sqrt(-a)*sqrt(c)*sqrt(g*sqrt(-a)+f*sqrt(c)))],
[1/((d+e*x)^(1/2)*(a+c*x^2)*sqrt(f+g*x)),x,6,arctanh(sqrt(d+e*x)*sqrt(-g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(-e*sqrt(-a)+d*sqrt(c))))/(sqrt(-a)*sqrt(-e*sqrt(-a)+d*sqrt(c))*sqrt(-g*sqrt(-a)+f*sqrt(c)))-arctanh(sqrt(d+e*x)*sqrt(g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(e*sqrt(-a)+d*sqrt(c))))/(sqrt(-a)*sqrt(e*sqrt(-a)+d*sqrt(c))*sqrt(g*sqrt(-a)+f*sqrt(c)))],
[1/((d+e*x)^(3/2)*(a+c*x^2)*sqrt(f+g*x)),x,8,-e*sqrt(f+g*x)/((e*f-d*g)*sqrt(-a)*(-e*sqrt(-a)+d*sqrt(c))*sqrt(d+e*x))+e*sqrt(f+g*x)/((e*f-d*g)*sqrt(-a)*(e*sqrt(-a)+d*sqrt(c))*sqrt(d+e*x))+arctanh(sqrt(d+e*x)*sqrt(-g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(-e*sqrt(-a)+d*sqrt(c))))*sqrt(c)/(sqrt(-a)*(-e*sqrt(-a)+d*sqrt(c))^(3/2)*sqrt(-g*sqrt(-a)+f*sqrt(c)))-arctanh(sqrt(d+e*x)*sqrt(g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(e*sqrt(-a)+d*sqrt(c))))*sqrt(c)/(sqrt(-a)*(e*sqrt(-a)+d*sqrt(c))^(3/2)*sqrt(g*sqrt(-a)+f*sqrt(c)))],
[(d+e*x)^(3/2)/((f+g*x)^(3/2)*(a+c*x^2)),x,21,-2*(e*f-d*g)*arctanh(sqrt(g)*sqrt(d+e*x)/(sqrt(e)*sqrt(f+g*x)))*sqrt(e)/((c*f^2+a*g^2)*sqrt(g))-arctanh(sqrt(g)*sqrt(d+e*x)/(sqrt(e)*sqrt(f+g*x)))*(c*d*f+a*e*g-(e*f-d*g)*sqrt(-a)*sqrt(c))*sqrt(e)/((c*f^2+a*g^2)*sqrt(-a)*sqrt(c)*sqrt(g))+arctanh(sqrt(g)*sqrt(d+e*x)/(sqrt(e)*sqrt(f+g*x)))*(c*d*f+a*e*g+(e*f-d*g)*sqrt(-a)*sqrt(c))*sqrt(e)/((c*f^2+a*g^2)*sqrt(-a)*sqrt(c)*sqrt(g))+2*(e*f-d*g)*sqrt(d+e*x)/((c*f^2+a*g^2)*sqrt(f+g*x))+arctanh(sqrt(d+e*x)*sqrt(-g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(-e*sqrt(-a)+d*sqrt(c))))*(c*d*f+a*e*g-(e*f-d*g)*sqrt(-a)*sqrt(c))*sqrt(-e*sqrt(-a)+d*sqrt(c))/((c*f^2+a*g^2)*sqrt(-a)*sqrt(c)*sqrt(-g*sqrt(-a)+f*sqrt(c)))-arctanh(sqrt(d+e*x)*sqrt(g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(e*sqrt(-a)+d*sqrt(c))))*(c*d*f+a*e*g+(e*f-d*g)*sqrt(-a)*sqrt(c))*sqrt(e*sqrt(-a)+d*sqrt(c))/((c*f^2+a*g^2)*sqrt(-a)*sqrt(c)*sqrt(g*sqrt(-a)+f*sqrt(c)))],
[(d+e*x)^(1/2)/((f+g*x)^(3/2)*(a+c*x^2)),x,8,-2*g*sqrt(d+e*x)/((c*f^2+a*g^2)*sqrt(f+g*x))+arctanh(sqrt(d+e*x)*sqrt(-g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(-e*sqrt(-a)+d*sqrt(c))))*(c*d*f+a*e*g-(e*f-d*g)*sqrt(-a)*sqrt(c))/((c*f^2+a*g^2)*sqrt(-a)*sqrt(-e*sqrt(-a)+d*sqrt(c))*sqrt(-g*sqrt(-a)+f*sqrt(c)))-arctanh(sqrt(d+e*x)*sqrt(g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(e*sqrt(-a)+d*sqrt(c))))*(c*d*f+a*e*g+(e*f-d*g)*sqrt(-a)*sqrt(c))/((c*f^2+a*g^2)*sqrt(-a)*sqrt(e*sqrt(-a)+d*sqrt(c))*sqrt(g*sqrt(-a)+f*sqrt(c)))],
[1/((d+e*x)^(1/2)*(f+g*x)^(3/2)*(a+c*x^2)),x,8,g*sqrt(d+e*x)/((e*f-d*g)*sqrt(-a)*(-g*sqrt(-a)+f*sqrt(c))*sqrt(f+g*x))-g*sqrt(d+e*x)/((e*f-d*g)*sqrt(-a)*(g*sqrt(-a)+f*sqrt(c))*sqrt(f+g*x))+arctanh(sqrt(d+e*x)*sqrt(-g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(-e*sqrt(-a)+d*sqrt(c))))*sqrt(c)/(sqrt(-a)*(-g*sqrt(-a)+f*sqrt(c))^(3/2)*sqrt(-e*sqrt(-a)+d*sqrt(c)))-arctanh(sqrt(d+e*x)*sqrt(g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(e*sqrt(-a)+d*sqrt(c))))*sqrt(c)/(sqrt(-a)*(g*sqrt(-a)+f*sqrt(c))^(3/2)*sqrt(e*sqrt(-a)+d*sqrt(c)))],
[1/((d+e*x)^(3/2)*(f+g*x)^(3/2)*(a+c*x^2)),x,12,c*arctanh(sqrt(d+e*x)*sqrt(-g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(-e*sqrt(-a)+d*sqrt(c))))/(sqrt(-a)*(-e*sqrt(-a)+d*sqrt(c))^(3/2)*(-g*sqrt(-a)+f*sqrt(c))^(3/2))-c*arctanh(sqrt(d+e*x)*sqrt(g*sqrt(-a)+f*sqrt(c))/(sqrt(f+g*x)*sqrt(e*sqrt(-a)+d*sqrt(c))))/(sqrt(-a)*(e*sqrt(-a)+d*sqrt(c))^(3/2)*(g*sqrt(-a)+f*sqrt(c))^(3/2))-e/((e*f-d*g)*sqrt(-a)*(-e*sqrt(-a)+d*sqrt(c))*sqrt(d+e*x)*sqrt(f+g*x))+e/((e*f-d*g)*sqrt(-a)*(e*sqrt(-a)+d*sqrt(c))*sqrt(d+e*x)*sqrt(f+g*x))+g*(2*e*g*sqrt(-a)-(e*f+d*g)*sqrt(c))*sqrt(d+e*x)/((e*f-d*g)^2*sqrt(-a)*(-e*sqrt(-a)+d*sqrt(c))*(-g*sqrt(-a)+f*sqrt(c))*sqrt(f+g*x))+g*(2*e*g*sqrt(-a)+(e*f+d*g)*sqrt(c))*sqrt(d+e*x)/((e*f-d*g)^2*sqrt(-a)*(e*sqrt(-a)+d*sqrt(c))*(g*sqrt(-a)+f*sqrt(c))*sqrt(f+g*x))],
[sqrt(x)/((1+x^2)*sqrt(1+x)),x,6,-1/2*(1-I)^(3/2)*arctanh(sqrt(1-I)*sqrt(x)/sqrt(1+x))-1/2*(1+I)^(3/2)*arctanh(sqrt(1+I)*sqrt(x)/sqrt(1+x))],

# Integrands of the form (d+e x)^m (f+g x)^n (a+c x^2)^(p/2)

# p>0
[(f+g*x)^2*sqrt(1-x^2)/(1-x)^4,x,5,1/5*(f+g)^2*(1+x)^4/(1-x^2)^(5/2)+1/15*(f-9*g)*(f+g)*(1+x)^3/(1-x^2)^(3/2)-g^2*arcsin(x)+2*g^2*(1+x)/sqrt(1-x^2)],
[(1-a^2*x^2)^(3/2)/((1-a*x)^2*(c+d*x)),x,6,-(a*c-2*d)*arcsin(a*x)/d^2+(a*c-d)^2*arctan((d+a^2*c*x)/(sqrt(a^2*c^2-d^2)*sqrt(1-a^2*x^2)))/(d^2*sqrt(a^2*c^2-d^2))-sqrt(1-a^2*x^2)/d],

# p<0
[(1+a*x)^2/((c+d*x)*sqrt(1-a^2*x^2)),x,5,-(a*c-2*d)*arcsin(a*x)/d^2+(a*c-d)^2*arctan((d+a^2*c*x)/(sqrt(a^2*c^2-d^2)*sqrt(1-a^2*x^2)))/(d^2*sqrt(a^2*c^2-d^2))-sqrt(1-a^2*x^2)/d],

# Integrands of the form (d+e x)^m (f+g x)^(n/2) (a+c x^2)^(p/2)

# p>0 & n>0
[(d+e*x)^3*sqrt(f+g*x)*sqrt(a+c*x^2),x,10,-2/3465*(2*a*e^2*g^2*(74*e*f-231*d*g)-c*(233*e^3*f^3-843*d*e^2*f^2*g+1107*d^2*e*f*g^2-567*d^3*g^3))*(f+g*x)^(3/2)*sqrt(a+c*x^2)/(c*g^4)+2/693*e*(18*a*e^2*g^2-c*(29*e^2*f^2-96*d*e*f*g+81*d^2*g^2))*(f+g*x)^(5/2)*sqrt(a+c*x^2)/(c*g^4)+2/99*e^2*(e*f-3*d*g)*(f+g*x)^(7/2)*sqrt(a+c*x^2)/g^4-2/3465*(150*a^2*e^4*g^4-6*a*c*e^2*g^2*(2*e^2*f^2-33*d*e*f*g+165*d^2*g^2)+c^2*(187*e^4*f^4-732*d*e^3*f^3*g+1098*d^2*e^2*f^2*g^2-798*d^3*e*f*g^3+315*d^4*g^4))*sqrt(f+g*x)*sqrt(a+c*x^2)/(c^2*e*g^4)+2/11*(d+e*x)^4*sqrt(f+g*x)*sqrt(a+c*x^2)/e+4/3465*(3*a^2*e^2*g^4*(26*e*f+231*d*g)-c^2*f^2*(64*e^3*f^3-264*d*e^2*f^2*g+396*d^2*e*f*g^2-231*d^3*g^3)-9*a*c*g^2*(6*e^3*f^3-33*d*e^2*f^2*g+88*d^2*e*f*g^2+77*d^3*g^3))*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(c^(3/2)*g^5*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))-4/3465*(c*f^2+a*g^2)*(75*a^2*e^3*g^4-3*a*c*e*g^2*(2*e^2*f^2-33*d*e*f*g+165*d^2*g^2)-c^2*f*(64*e^3*f^3-264*d*e^2*f^2*g+396*d^2*e*f*g^2-231*d^3*g^3))*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(c^(5/2)*g^5*sqrt(f+g*x)*sqrt(a+c*x^2))],
[(d+e*x)^2*sqrt(f+g*x)*sqrt(a+c*x^2),x,9,4/315*(7*a*e^2*g^2-c*(8*e^2*f^2-24*d*e*f*g+21*d^2*g^2))*(f+g*x)^(3/2)*sqrt(a+c*x^2)/(c*g^3)+2/63*e*(e*f-3*d*g)*(f+g*x)^(5/2)*sqrt(a+c*x^2)/g^3-2/315*(6*a*e^2*g^2*(e*f-10*d*g)-c*(19*e^3*f^3-57*d*e^2*f^2*g+63*d^2*e*f*g^2-35*d^3*g^3))*sqrt(f+g*x)*sqrt(a+c*x^2)/(c*e*g^3)+2/9*(d+e*x)^3*sqrt(f+g*x)*sqrt(a+c*x^2)/e+4/315*(21*a^2*e^2*g^4+3*a*c*g^2*(3*e^2*f^2-16*d*e*f*g-21*d^2*g^2)+c^2*f^2*(8*e^2*f^2-24*d*e*f*g+21*d^2*g^2))*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(c^(3/2)*g^4*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))-4/315*(c*f^2+a*g^2)*(3*a*e*g^2*(e*f-10*d*g)+c*f*(8*e^2*f^2-24*d*e*f*g+21*d^2*g^2))*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(c^(3/2)*g^4*sqrt(f+g*x)*sqrt(a+c*x^2))],
[(d+e*x)*sqrt(f+g*x)*sqrt(a+c*x^2),x,7,2/7*e*(a+c*x^2)^(3/2)*sqrt(f+g*x)/c-2/105*(5*a*e*g^2+c*f*(4*e*f-7*d*g)-3*c*g*(e*f+7*d*g)*x)*sqrt(f+g*x)*sqrt(a+c*x^2)/(c*g^2)-4/105*(c*f^2*(4*e*f-7*d*g)+a*g^2*(8*e*f+21*d*g))*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(g^3*sqrt(c)*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))+4/105*(c*f^2+a*g^2)*(5*a*e*g^2+c*f*(4*e*f-7*d*g))*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(c^(3/2)*g^3*sqrt(f+g*x)*sqrt(a+c*x^2))],
[sqrt(f+g*x)*sqrt(a+c*x^2),x,7,2/5*(f+g*x)^(3/2)*sqrt(a+c*x^2)/g-4/15*f*sqrt(f+g*x)*sqrt(a+c*x^2)/g+4/15*(c*f^2-3*a*g^2)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(g^2*sqrt(c)*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))-4/15*f*(c*f^2+a*g^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(g^2*sqrt(c)*sqrt(f+g*x)*sqrt(a+c*x^2))],
[sqrt(f+g*x)*sqrt(a+c*x^2)/(d+e*x),x,14,2/3*sqrt(f+g*x)*sqrt(a+c*x^2)/e-2/3*(e*f-3*d*g)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(e^2*g*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))-2/3*(2*a*e^2*g-3*c*d*(e*f-d*g))*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^3*sqrt(c)*sqrt(f+g*x)*sqrt(a+c*x^2))+2/3*f*(e*f-3*d*g)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^2*g*sqrt(f+g*x)*sqrt(a+c*x^2))-2*(c*d^2+a*e^2)*(e*f-d*g)*EllipticPi(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),2*e/(e+d*sqrt(c)/sqrt(-a)),sqrt(2*g*sqrt(-a)/(g*sqrt(-a)+f*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^3*(e+d*sqrt(c)/sqrt(-a))*sqrt(f+g*x)*sqrt(a+c*x^2))],
[sqrt(f+g*x)*sqrt(a+c*x^2)/(d+e*x)^2,x,14,-sqrt(f+g*x)*sqrt(a+c*x^2)/(e*(d+e*x))-3*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(e^2*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))+3*f*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^2*sqrt(f+g*x)*sqrt(a+c*x^2))-(2*e*f-3*d*g)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^3*sqrt(f+g*x)*sqrt(a+c*x^2))-(a*e^2*g-c*d*(2*e*f-3*d*g))*EllipticPi(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),2*e/(e+d*sqrt(c)/sqrt(-a)),sqrt(2*g*sqrt(-a)/(g*sqrt(-a)+f*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^3*(e+d*sqrt(c)/sqrt(-a))*sqrt(f+g*x)*sqrt(a+c*x^2))],
[sqrt(f+g*x)*sqrt(a+c*x^2)/(d+e*x)^3,x,23,-1/2*sqrt(f+g*x)*sqrt(a+c*x^2)/(e*(d+e*x)^2)-1/4*(a*e^2*g-c*d*(2*e*f-3*d*g))*sqrt(f+g*x)*sqrt(a+c*x^2)/(e*(c*d^2+a*e^2)*(e*f-d*g)*(d+e*x))-1/4*(a*e^2*g-c*d*(2*e*f-3*d*g))*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(e^2*(c*d^2+a*e^2)*(e*f-d*g)*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))-3/2*g*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^3*sqrt(f+g*x)*sqrt(a+c*x^2))+1/4*f*(a*e^2*g-c*d*(2*e*f-3*d*g))*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^2*(c*d^2+a*e^2)*(e*f-d*g)*sqrt(f+g*x)*sqrt(a+c*x^2))-1/4*d*g*(a*e^2*g-c*d*(2*e*f-3*d*g))*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^3*(c*d^2+a*e^2)*(e*f-d*g)*sqrt(f+g*x)*sqrt(a+c*x^2))-c*(e*f-3*d*g)*EllipticPi(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),2*e/(e+d*sqrt(c)/sqrt(-a)),sqrt(2*g*sqrt(-a)/(g*sqrt(-a)+f*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^3*(e+d*sqrt(c)/sqrt(-a))*sqrt(f+g*x)*sqrt(a+c*x^2))+1/4*(a*e^2*g-c*d*(2*e*f-3*d*g))^2*EllipticPi(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),2*e/(e+d*sqrt(c)/sqrt(-a)),sqrt(2*g*sqrt(-a)/(g*sqrt(-a)+f*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^3*(c*d^2+a*e^2)*(e*f-d*g)*(e+d*sqrt(c)/sqrt(-a))*sqrt(f+g*x)*sqrt(a+c*x^2))],

# p>0 & n<0
[(d+e*x)^3*sqrt(a+c*x^2)/sqrt(f+g*x),x,9,4/315*e*(7*a*e^2*g^2+c*(64*e^2*f^2-111*d*e*f*g+42*d^2*g^2))*(f+g*x)^(3/2)*sqrt(a+c*x^2)/(c*g^4)-4/63*e^2*(4*e*f-3*d*g)*(f+g*x)^(5/2)*sqrt(a+c*x^2)/g^4-4/315*(9*a*e^2*g^2*(2*e*f-5*d*g)+c*(76*e^3*f^3-204*d*e^2*f^2*g+168*d^2*e*f*g^2-35*d^3*g^3))*sqrt(f+g*x)*sqrt(a+c*x^2)/(c*g^4)+2/9*(d+e*x)^3*sqrt(f+g*x)*sqrt(a+c*x^2)/g+4/315*(21*a^2*e^3*g^4-3*a*c*e*g^2*(10*e^2*f^2-39*d*e*f*g+63*d^2*g^2)-c^2*f*(64*e^3*f^3-216*d*e^2*f^2*g+252*d^2*e*f*g^2-105*d^3*g^3))*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(c^(3/2)*g^5*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))-4/315*(c*f^2+a*g^2)*(9*a*e^2*g^2*(2*e*f-5*d*g)-c*(64*e^3*f^3-216*d*e^2*f^2*g+252*d^2*e*f*g^2-105*d^3*g^3))*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(c^(3/2)*g^5*sqrt(f+g*x)*sqrt(a+c*x^2))],
[(d+e*x)^2*sqrt(a+c*x^2)/sqrt(f+g*x),x,8,-4/35*e*(3*e*f-2*d*g)*(f+g*x)^(3/2)*sqrt(a+c*x^2)/g^3+4/105*(5*a*e^2*g^2+c*(21*e^2*f^2-34*d*e*f*g+10*d^2*g^2))*sqrt(f+g*x)*sqrt(a+c*x^2)/(c*g^3)+2/7*(d+e*x)^2*sqrt(f+g*x)*sqrt(a+c*x^2)/g+4/105*(a*e*g^2*(13*e*f-42*d*g)+c*f*(24*e^2*f^2-56*d*e*f*g+35*d^2*g^2))*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(g^4*sqrt(c)*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))+4/105*(c*f^2+a*g^2)*(5*a*e^2*g^2-c*(24*e^2*f^2-56*d*e*f*g+35*d^2*g^2))*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(c^(3/2)*g^4*sqrt(f+g*x)*sqrt(a+c*x^2))],
[(d+e*x)*sqrt(a+c*x^2)/sqrt(f+g*x),x,6,-2/15*(4*e*f-5*d*g-3*e*g*x)*sqrt(f+g*x)*sqrt(a+c*x^2)/g^2-4/15*(3*a*e*g^2+c*f*(4*e*f-5*d*g))*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(g^3*sqrt(c)*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))+4/15*(4*e*f-5*d*g)*(c*f^2+a*g^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(g^3*sqrt(c)*sqrt(f+g*x)*sqrt(a+c*x^2))],
[sqrt(a+c*x^2)/sqrt(f+g*x),x,6,2/3*sqrt(f+g*x)*sqrt(a+c*x^2)/g+4/3*f*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(g^2*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))-4/3*(c*f^2+a*g^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(g^2*sqrt(c)*sqrt(f+g*x)*sqrt(a+c*x^2))],
[sqrt(a+c*x^2)/((d+e*x)*sqrt(f+g*x)),x,10,-2*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(e*g*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))+2*(e*f+d*g)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^2*g*sqrt(f+g*x)*sqrt(a+c*x^2))-2*(c*d^2+a*e^2)*EllipticPi(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),2*e/(e+d*sqrt(c)/sqrt(-a)),sqrt(2*g*sqrt(-a)/(g*sqrt(-a)+f*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^2*(e+d*sqrt(c)/sqrt(-a))*sqrt(f+g*x)*sqrt(a+c*x^2))],
[sqrt(a+c*x^2)/((d+e*x)^2*sqrt(f+g*x)),x,14,-sqrt(f+g*x)*sqrt(a+c*x^2)/((e*f-d*g)*(d+e*x))-EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(e*(e*f-d*g)*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))+f*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e*(e*f-d*g)*sqrt(f+g*x)*sqrt(a+c*x^2))-(2*e*f-d*g)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^2*(e*f-d*g)*sqrt(f+g*x)*sqrt(a+c*x^2))+(a*e^2*g+c*d*(2*e*f-d*g))*EllipticPi(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),2*e/(e+d*sqrt(c)/sqrt(-a)),sqrt(2*g*sqrt(-a)/(g*sqrt(-a)+f*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^2*(e*f-d*g)*(e+d*sqrt(c)/sqrt(-a))*sqrt(f+g*x)*sqrt(a+c*x^2))],
[sqrt(a+c*x^2)/((d+e*x)^3*sqrt(f+g*x)),x,23,-1/2*sqrt(f+g*x)*sqrt(a+c*x^2)/((e*f-d*g)*(d+e*x)^2)+1/4*(3*a*e^2*g+c*d*(2*e*f+d*g))*sqrt(f+g*x)*sqrt(a+c*x^2)/((c*d^2+a*e^2)*(e*f-d*g)^2*(d+e*x))+1/4*(3*a*e^2*g+c*d*(2*e*f+d*g))*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(e*(c*d^2+a*e^2)*(e*f-d*g)^2*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))+1/2*g*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^2*(e*f-d*g)*sqrt(f+g*x)*sqrt(a+c*x^2))-1/4*f*(3*a*e^2*g+c*d*(2*e*f+d*g))*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e*(c*d^2+a*e^2)*(e*f-d*g)^2*sqrt(f+g*x)*sqrt(a+c*x^2))+1/4*d*g*(3*a*e^2*g+c*d*(2*e*f+d*g))*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^2*(c*d^2+a*e^2)*(e*f-d*g)^2*sqrt(f+g*x)*sqrt(a+c*x^2))-c*(e*f+d*g)*EllipticPi(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),2*e/(e+d*sqrt(c)/sqrt(-a)),sqrt(2*g*sqrt(-a)/(g*sqrt(-a)+f*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^2*(e*f-d*g)*(e+d*sqrt(c)/sqrt(-a))*sqrt(f+g*x)*sqrt(a+c*x^2))-1/4*(a*e^2*g-c*d*(2*e*f-3*d*g))*(3*a*e^2*g+c*d*(2*e*f+d*g))*EllipticPi(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),2*e/(e+d*sqrt(c)/sqrt(-a)),sqrt(2*g*sqrt(-a)/(g*sqrt(-a)+f*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^2*(c*d^2+a*e^2)*(e*f-d*g)^2*(e+d*sqrt(c)/sqrt(-a))*sqrt(f+g*x)*sqrt(a+c*x^2))],

# p<0 & n>0
[(d+e*x)^3*sqrt(f+g*x)/sqrt(a+c*x^2),x,8,2/35*e^2*(e*f+11*d*g)*(f+g*x)^(3/2)*sqrt(a+c*x^2)/(c*g^2)-2/105*e*(25*a*e^2*g^2+c*(7*e^2*f^2+12*d*e*f*g-90*d^2*g^2))*sqrt(f+g*x)*sqrt(a+c*x^2)/(c^2*g^2)+2/7*e*(d+e*x)^2*sqrt(f+g*x)*sqrt(a+c*x^2)/c+2/105*(a*e^2*g^2*(19*e*f+189*d*g)-c*(8*e^3*f^3-42*d*e^2*f^2*g+105*d^2*e*f*g^2+105*d^3*g^3))*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(c^(3/2)*g^3*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))-2/105*e*(c*f^2+a*g^2)*(25*a*e^2*g^2-c*(8*e^2*f^2-42*d*e*f*g+105*d^2*g^2))*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(c^(5/2)*g^3*sqrt(f+g*x)*sqrt(a+c*x^2))],
[(d+e*x)^2*sqrt(f+g*x)/sqrt(a+c*x^2),x,7,2/15*e*(e*f+7*d*g)*sqrt(f+g*x)*sqrt(a+c*x^2)/(c*g)+2/5*e*(d+e*x)*sqrt(f+g*x)*sqrt(a+c*x^2)/c+2/15*(9*a*e^2*g^2+c*(2*e^2*f^2-10*d*e*f*g-15*d^2*g^2))*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(c^(3/2)*g^2*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))-4/15*e*(e*f-5*d*g)*(c*f^2+a*g^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(c^(3/2)*g^2*sqrt(f+g*x)*sqrt(a+c*x^2))],
[(d+e*x)*sqrt(f+g*x)/sqrt(a+c*x^2),x,6,2/3*e*sqrt(f+g*x)*sqrt(a+c*x^2)/c-2/3*(e*f+3*d*g)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(g*sqrt(c)*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))+2/3*e*(c*f^2+a*g^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(c^(3/2)*g*sqrt(f+g*x)*sqrt(a+c*x^2))],
[sqrt(f+g*x)/sqrt(a+c*x^2),x,2,-2*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(sqrt(c)*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))],
[sqrt(f+g*x)/((d+e*x)*sqrt(a+c*x^2)),x,7,-2*g*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e*sqrt(c)*sqrt(f+g*x)*sqrt(a+c*x^2))-2*(e*f-d*g)*EllipticPi(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),2*e/(e+d*sqrt(c)/sqrt(-a)),sqrt(2*g*sqrt(-a)/(g*sqrt(-a)+f*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e*(e+d*sqrt(c)/sqrt(-a))*sqrt(f+g*x)*sqrt(a+c*x^2))],
[sqrt(f+g*x)/((d+e*x)^2*sqrt(a+c*x^2)),x,14,-e*sqrt(f+g*x)*sqrt(a+c*x^2)/((c*d^2+a*e^2)*(d+e*x))-EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/((c*d^2+a*e^2)*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))+f*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/((c*d^2+a*e^2)*sqrt(f+g*x)*sqrt(a+c*x^2))-d*g*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e*(c*d^2+a*e^2)*sqrt(f+g*x)*sqrt(a+c*x^2))-(a*e^2*g+c*d*(2*e*f-d*g))*EllipticPi(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),2*e/(e+d*sqrt(c)/sqrt(-a)),sqrt(2*g*sqrt(-a)/(g*sqrt(-a)+f*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e*(c*d^2+a*e^2)*(e+d*sqrt(c)/sqrt(-a))*sqrt(f+g*x)*sqrt(a+c*x^2))],
[sqrt(f+g*x)/((d+e*x)^3*sqrt(a+c*x^2)),x,23,-1/2*e*sqrt(f+g*x)*sqrt(a+c*x^2)/((c*d^2+a*e^2)*(d+e*x)^2)-1/4*e*(a*e^2*g+c*d*(6*e*f-5*d*g))*sqrt(f+g*x)*sqrt(a+c*x^2)/((c*d^2+a*e^2)^2*(e*f-d*g)*(d+e*x))-1/4*(a*e^2*g+c*d*(6*e*f-5*d*g))*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/((c*d^2+a*e^2)^2*(e*f-d*g)*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))+1/2*g*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e*(c*d^2+a*e^2)*sqrt(f+g*x)*sqrt(a+c*x^2))+1/4*f*(a*e^2*g+c*d*(6*e*f-5*d*g))*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/((c*d^2+a*e^2)^2*(e*f-d*g)*sqrt(f+g*x)*sqrt(a+c*x^2))-1/4*d*g*(a*e^2*g+c*d*(6*e*f-5*d*g))*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e*(c*d^2+a*e^2)^2*(e*f-d*g)*sqrt(f+g*x)*sqrt(a+c*x^2))+c*(e*f-3*d*g)*EllipticPi(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),2*e/(e+d*sqrt(c)/sqrt(-a)),sqrt(2*g*sqrt(-a)/(g*sqrt(-a)+f*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e*(c*d^2+a*e^2)*(e+d*sqrt(c)/sqrt(-a))*sqrt(f+g*x)*sqrt(a+c*x^2))+1/4*(a*e^2*g+c*d*(6*e*f-5*d*g))*(a*e^2*g-c*d*(2*e*f-3*d*g))*EllipticPi(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),2*e/(e+d*sqrt(c)/sqrt(-a)),sqrt(2*g*sqrt(-a)/(g*sqrt(-a)+f*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e*(c*d^2+a*e^2)^2*(e*f-d*g)*(e+d*sqrt(c)/sqrt(-a))*sqrt(f+g*x)*sqrt(a+c*x^2))],
[(f+g*x)^(5/2)/((d+e*x)*sqrt(a+c*x^2)),x,16,2/3*g^2*sqrt(f+g*x)*sqrt(a+c*x^2)/(c*e)-2/3*g*(7*e*f-3*d*g)*EllipticE(sqrt(1+a*x*sqrt(c)/(-a)^(3/2))/sqrt(2),sqrt(2*a*g/(a*g-f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(e^2*sqrt(c)*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))+2/3*g*(a*e^2*g^2+c*(-2*e^2*f^2+6*d*e*f*g-3*d^2*g^2))*EllipticF(sqrt(1+a*x*sqrt(c)/(-a)^(3/2))/sqrt(2),sqrt(2*a*g/(a*g-f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(c^(3/2)*e^3*sqrt(f+g*x)*sqrt(a+c*x^2))-2*(e*f-d*g)^2*EllipticPi(sqrt(f+g*x)*sqrt(c/(c*f+g*sqrt(-a)*sqrt(c))),e*(f+g*sqrt(-a)/sqrt(c))/(e*f-d*g),sqrt((g*sqrt(-a)+f*sqrt(c))/(-g*sqrt(-a)+f*sqrt(c))))*sqrt(g*(sqrt(-a)-x*sqrt(c))/(g*sqrt(-a)+f*sqrt(c)))*sqrt(-g*(sqrt(-a)+x*sqrt(c))/(-g*sqrt(-a)+f*sqrt(c)))/(e^3*sqrt(a+c*x^2)*sqrt(c/(c*f+g*sqrt(-a)*sqrt(c)))),2/3*g^2*sqrt(f+g*x)*sqrt(a+c*x^2)/(c*e)-8/3*f*g*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(e*sqrt(c)*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))-2*g*(e*f-d*g)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(e^2*sqrt(c)*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))+2/3*g*(c*f^2+a*g^2)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(c^(3/2)*e*sqrt(f+g*x)*sqrt(a+c*x^2))-2*g*(e*f-d*g)^2*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^3*sqrt(c)*sqrt(f+g*x)*sqrt(a+c*x^2))-2*(e*f-d*g)^3*EllipticPi(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),2*e/(e+d*sqrt(c)/sqrt(-a)),sqrt(2*g*sqrt(-a)/(g*sqrt(-a)+f*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^3*(e+d*sqrt(c)/sqrt(-a))*sqrt(f+g*x)*sqrt(a+c*x^2))],
[(f+g*x)^(3/2)/((d+e*x)*sqrt(a+c*x^2)),x,10,-2*g*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(e*sqrt(c)*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))-2*g*(e*f-d*g)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^2*sqrt(c)*sqrt(f+g*x)*sqrt(a+c*x^2))-2*(e*f-d*g)^2*EllipticPi(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),2*e/(e+d*sqrt(c)/sqrt(-a)),sqrt(2*g*sqrt(-a)/(g*sqrt(-a)+f*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(e^2*(e+d*sqrt(c)/sqrt(-a))*sqrt(f+g*x)*sqrt(a+c*x^2))],

# p<0 & n<0
[(d+e*x)^3/(sqrt(f+g*x)*sqrt(a+c*x^2)),x,7,-8/15*e^2*(e*f-3*d*g)*sqrt(f+g*x)*sqrt(a+c*x^2)/(c*g^2)+2/5*e^2*(d+e*x)*sqrt(f+g*x)*sqrt(a+c*x^2)/(c*g)+2/15*e*(9*a*e^2*g^2-c*(8*e^2*f^2-30*d*e*f*g+45*d^2*g^2))*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(c^(3/2)*g^3*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))-2/15*(a*e^2*g^2*(7*e*f-15*d*g)-c*(8*e^3*f^3-30*d*e^2*f^2*g+45*d^2*e*f*g^2-15*d^3*g^3))*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(c^(3/2)*g^3*sqrt(f+g*x)*sqrt(a+c*x^2))],
[(d+e*x)^2/(sqrt(f+g*x)*sqrt(a+c*x^2)),x,7,2/3*e^2*sqrt(f+g*x)*sqrt(a+c*x^2)/(c*g)+4/3*e*(e*f-3*d*g)*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(g^2*sqrt(c)*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))-2/3*((3*c*d^2-a*e^2)*g^2+2*c*e*f*(e*f-3*d*g))*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(c^(3/2)*g^2*sqrt(f+g*x)*sqrt(a+c*x^2))],
[(d+e*x)/(sqrt(f+g*x)*sqrt(a+c*x^2)),x,5,-2*e*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/(g*sqrt(c)*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))+2*(e*f-d*g)*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(g*sqrt(c)*sqrt(f+g*x)*sqrt(a+c*x^2))],
[1/(sqrt(f+g*x)*sqrt(a+c*x^2)),x,2,-2*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/(sqrt(c)*sqrt(f+g*x)*sqrt(a+c*x^2))],
[1/((d+e*x)*sqrt(f+g*x)*sqrt(a+c*x^2)),x,4,-2*EllipticPi(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),2*e/(e+d*sqrt(c)/sqrt(-a)),sqrt(2*g*sqrt(-a)/(g*sqrt(-a)+f*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/((e+d*sqrt(c)/sqrt(-a))*sqrt(f+g*x)*sqrt(a+c*x^2))],
[1/((d+e*x)^2*sqrt(f+g*x)*sqrt(a+c*x^2)),x,14,-e^2*sqrt(f+g*x)*sqrt(a+c*x^2)/((c*d^2+a*e^2)*(e*f-d*g)*(d+e*x))-e*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/((c*d^2+a*e^2)*(e*f-d*g)*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))+e*f*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/((c*d^2+a*e^2)*(e*f-d*g)*sqrt(f+g*x)*sqrt(a+c*x^2))-d*g*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/((c*d^2+a*e^2)*(e*f-d*g)*sqrt(f+g*x)*sqrt(a+c*x^2))+(a*e^2*g-c*d*(2*e*f-3*d*g))*EllipticPi(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),2*e/(e+d*sqrt(c)/sqrt(-a)),sqrt(2*g*sqrt(-a)/(g*sqrt(-a)+f*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/((c*d^2+a*e^2)*(e*f-d*g)*(e+d*sqrt(c)/sqrt(-a))*sqrt(f+g*x)*sqrt(a+c*x^2))],
[1/((d+e*x)^3*sqrt(f+g*x)*sqrt(a+c*x^2)),x,23,-1/2*e^2*sqrt(f+g*x)*sqrt(a+c*x^2)/((c*d^2+a*e^2)*(e*f-d*g)*(d+e*x)^2)+3/4*e^2*(a*e^2*g-c*d*(2*e*f-3*d*g))*sqrt(f+g*x)*sqrt(a+c*x^2)/((c*d^2+a*e^2)^2*(e*f-d*g)^2*(d+e*x))+3/4*e*(a*e^2*g-c*d*(2*e*f-3*d*g))*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/((c*d^2+a*e^2)^2*(e*f-d*g)^2*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))+1/2*g*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/((c*d^2+a*e^2)*(e*f-d*g)*sqrt(f+g*x)*sqrt(a+c*x^2))-3/4*e*f*(a*e^2*g-c*d*(2*e*f-3*d*g))*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/((c*d^2+a*e^2)^2*(e*f-d*g)^2*sqrt(f+g*x)*sqrt(a+c*x^2))+3/4*d*g*(a*e^2*g-c*d*(2*e*f-3*d*g))*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/((c*d^2+a*e^2)^2*(e*f-d*g)^2*sqrt(f+g*x)*sqrt(a+c*x^2))+c*(e*f-3*d*g)*EllipticPi(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),2*e/(e+d*sqrt(c)/sqrt(-a)),sqrt(2*g*sqrt(-a)/(g*sqrt(-a)+f*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/((c*d^2+a*e^2)*(e*f-d*g)*(e+d*sqrt(c)/sqrt(-a))*sqrt(f+g*x)*sqrt(a+c*x^2))-3/4*(a*e^2*g-c*d*(2*e*f-3*d*g))^2*EllipticPi(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),2*e/(e+d*sqrt(c)/sqrt(-a)),sqrt(2*g*sqrt(-a)/(g*sqrt(-a)+f*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/((c*d^2+a*e^2)^2*(e*f-d*g)^2*(e+d*sqrt(c)/sqrt(-a))*sqrt(f+g*x)*sqrt(a+c*x^2))],
[1/((d+e*x)*(f+g*x)^(3/2)*sqrt(a+c*x^2)),x,10,2*g^2*sqrt(a+c*x^2)/((e*f-d*g)*(c*f^2+a*g^2)*sqrt(f+g*x))+2*g*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/((e*f-d*g)*(c*f^2+a*g^2)*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))-2*e*EllipticPi(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),2*e/(e+d*sqrt(c)/sqrt(-a)),sqrt(2*g*sqrt(-a)/(g*sqrt(-a)+f*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/((e*f-d*g)*(e+d*sqrt(c)/sqrt(-a))*sqrt(f+g*x)*sqrt(a+c*x^2))],
[1/((d+e*x)*(f+g*x)^(5/2)*sqrt(a+c*x^2)),x,17,2/3*g^2*sqrt(a+c*x^2)/((e*f-d*g)*(c*f^2+a*g^2)*(f+g*x)^(3/2))+8/3*c*f*g^2*sqrt(a+c*x^2)/((e*f-d*g)*(c*f^2+a*g^2)^2*sqrt(f+g*x))+2*e*g^2*sqrt(a+c*x^2)/((e*f-d*g)^2*(c*f^2+a*g^2)*sqrt(f+g*x))+8/3*c^(3/2)*f*g*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/((e*f-d*g)*(c*f^2+a*g^2)^2*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))+2*e*g*EllipticE(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(f+g*x)*sqrt(1+c*x^2/a)/((e*f-d*g)^2*(c*f^2+a*g^2)*sqrt(a+c*x^2)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c))))-2/3*g*EllipticF(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),sqrt(-2*a*g/(-a*g+f*sqrt(-a)*sqrt(c))))*sqrt(-a)*sqrt(c)*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/((e*f-d*g)*(c*f^2+a*g^2)*sqrt(f+g*x)*sqrt(a+c*x^2))-2*e^2*EllipticPi(sqrt(1-x*sqrt(c)/sqrt(-a))/sqrt(2),2*e/(e+d*sqrt(c)/sqrt(-a)),sqrt(2*g*sqrt(-a)/(g*sqrt(-a)+f*sqrt(c))))*sqrt(1+c*x^2/a)*sqrt((f+g*x)*sqrt(c)/(g*sqrt(-a)+f*sqrt(c)))/((e*f-d*g)^2*(e+d*sqrt(c)/sqrt(-a))*sqrt(f+g*x)*sqrt(a+c*x^2))],
[1/((d+e*x)*sqrt(f+g*x)*sqrt(1+c*x^2)),x,4,-2*EllipticPi(sqrt(1-x*sqrt(-c))/sqrt(2),2*e/(e+d*sqrt(-c)),sqrt(2*g/(g+f*sqrt(-c))))*sqrt((f+g*x)*sqrt(-c)/(g+f*sqrt(-c)))/((e+d*sqrt(-c))*sqrt(f+g*x))],

# Integrands of the form (d+e x)^(m/2) (f+g x)^(n/2) (a+c x^2)^(p/2)
[1/(sqrt(d+e*x)*sqrt(f+g*x)*sqrt(a+c*x^2)),x,2,-(c*f^2+a*g^2)^(1/4)*(d+e*x)*sqrt(cos(2*arctan((c*d^2+a*e^2)^(1/4)*sqrt(f+g*x)/((c*f^2+a*g^2)^(1/4)*sqrt(d+e*x))))^2)/cos(2*arctan((c*d^2+a*e^2)^(1/4)*sqrt(f+g*x)/((c*f^2+a*g^2)^(1/4)*sqrt(d+e*x))))*EllipticF(sin(2*arctan((c*d^2+a*e^2)^(1/4)*sqrt(f+g*x)/((c*f^2+a*g^2)^(1/4)*sqrt(d+e*x)))),sqrt(1/2*(1+(c*d*f+a*e*g)/(sqrt(c*d^2+a*e^2)*sqrt(c*f^2+a*g^2)))))*(1+(f+g*x)*sqrt(c*d^2+a*e^2)/((d+e*x)*sqrt(c*f^2+a*g^2)))*sqrt((e*f-d*g)^2*(a+c*x^2)/((c*f^2+a*g^2)*(d+e*x)^2))*sqrt((1-2*(c*d*f+a*e*g)*(f+g*x)/((c*f^2+a*g^2)*(d+e*x))+(c*d^2+a*e^2)*(f+g*x)^2/((c*f^2+a*g^2)*(d+e*x)^2))/(1+(f+g*x)*sqrt(c*d^2+a*e^2)/((d+e*x)*sqrt(c*f^2+a*g^2)))^2)/((c*d^2+a*e^2)^(1/4)*(e*f-d*g)*sqrt(a+c*x^2)*sqrt(1-2*(c*d*f+a*e*g)*(f+g*x)/((c*f^2+a*g^2)*(d+e*x))+(c*d^2+a*e^2)*(f+g*x)^2/((c*f^2+a*g^2)*(d+e*x)^2)))],
[1/(sqrt(-1+x)*sqrt(1+x)*sqrt(-1+2*x^2)),x,4,EllipticF(x,sqrt(2))*sqrt(1-2*x^2)*sqrt(1-x^2)/(sqrt(-1+x)*sqrt(1+x)*sqrt(-1+2*x^2))],

# Integrands of the form (d+e x)^m (f+g x)^n (a+b x+c x^2)^p when c d^2-b d e+a e^2=0 and m+p=0

# Integrands of the form (d+e x)^(m/2) / (a d e+(c d^2+a e^2) x+c d e x^2)^(m/2) (f+g x)^n

# m>0
[(f+g*x)^3*sqrt(d+e*x)/sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,4,-16/35*(c*d*f-a*e*g)^2*(2*a*e^2*g-c*d*(3*e*f-d*g))*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^4*d^4*e*sqrt(d+e*x))+12/35*(c*d*f-a*e*g)*(f+g*x)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*sqrt(d+e*x))+2/7*(f+g*x)^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*sqrt(d+e*x))+16/35*g*(c*d*f-a*e*g)^2*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*e)],
[(f+g*x)^2*sqrt(d+e*x)/sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,3,-8/15*(c*d*f-a*e*g)*(2*a*e^2*g-c*d*(3*e*f-d*g))*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*e*sqrt(d+e*x))+2/5*(f+g*x)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*sqrt(d+e*x))+8/15*g*(c*d*f-a*e*g)*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*e)],
[(f+g*x)*sqrt(d+e*x)/sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,2,-2/3*(2*a*e^2*g-c*d*(3*e*f-d*g))*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*e*sqrt(d+e*x))+2/3*g*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*e)],
[sqrt(d+e*x)/sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,1,2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*sqrt(d+e*x))],
[sqrt(d+e*x)/((f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,2,2*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/(sqrt(g)*sqrt(c*d*f-a*e*g))],
[sqrt(d+e*x)/((f+g*x)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,3,c*d*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/((c*d*f-a*e*g)^(3/2)*sqrt(g))+sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)*(f+g*x)*sqrt(d+e*x))],
[sqrt(d+e*x)/((f+g*x)^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,4,3/4*c^2*d^2*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/((c*d*f-a*e*g)^(5/2)*sqrt(g))+1/2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)*(f+g*x)^2*sqrt(d+e*x))+3/4*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^2*(f+g*x)*sqrt(d+e*x))],
[sqrt(d+e*x)/((f+g*x)^4*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,5,5/8*c^3*d^3*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/((c*d*f-a*e*g)^(7/2)*sqrt(g))+1/3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)*(f+g*x)^3*sqrt(d+e*x))+5/12*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^2*(f+g*x)^2*sqrt(d+e*x))+5/8*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^3*(f+g*x)*sqrt(d+e*x))],
[(d+e*x)^(3/2)*(f+g*x)^3/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,4,-2*(f+g*x)^3*sqrt(d+e*x)/(c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-16/5*g*(c*d*f-a*e*g)*(2*a*e^2*g-c*d*(3*e*f-d*g))*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^4*d^4*e*sqrt(d+e*x))+12/5*g*(f+g*x)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*sqrt(d+e*x))+16/5*g^2*(c*d*f-a*e*g)*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*e)],
[(d+e*x)^(3/2)*(f+g*x)^2/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,3,-2*(f+g*x)^2*sqrt(d+e*x)/(c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-8/3*g*(2*a*e^2*g-c*d*(3*e*f-d*g))*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*e*sqrt(d+e*x))+8/3*g^2*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*e)],
[(d+e*x)^(3/2)*(f+g*x)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,2,-2*(c*d*f-a*e*g)*(d+e*x)^(3/2)/(c*d*(c*d^2-a*e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-2*(2*a*e^2*g-c*d*(e*f+d*g))*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*(c*d^2-a*e^2)*sqrt(d+e*x))],
[(d+e*x)^(3/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,1,-2*sqrt(d+e*x)/(c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[(d+e*x)^(3/2)/((f+g*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,3,-2*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))*sqrt(g)/(c*d*f-a*e*g)^(3/2)-2*sqrt(d+e*x)/((c*d*f-a*e*g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[(d+e*x)^(3/2)/((f+g*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,4,-3*c*d*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))*sqrt(g)/(c*d*f-a*e*g)^(5/2)-2*sqrt(d+e*x)/((c*d*f-a*e*g)*(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-3*g*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^2*(f+g*x)*sqrt(d+e*x))],
[(d+e*x)^(3/2)/((f+g*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,5,-15/4*c^2*d^2*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))*sqrt(g)/(c*d*f-a*e*g)^(7/2)-2*sqrt(d+e*x)/((c*d*f-a*e*g)*(f+g*x)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-5/2*g*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^2*(f+g*x)^2*sqrt(d+e*x))-15/4*c*d*g*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^3*(f+g*x)*sqrt(d+e*x))],
[(d+e*x)^(5/2)*(f+g*x)^3/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,4,-2/3*(d+e*x)^(3/2)*(f+g*x)^3/(c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))-4*g*(f+g*x)^2*sqrt(d+e*x)/(c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-16/3*g^2*(2*a*e^2*g-c*d*(3*e*f-d*g))*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^4*d^4*e*sqrt(d+e*x))+16/3*g^3*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*e)],
[(d+e*x)^(5/2)*(f+g*x)^2/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,3,-2/3*(d+e*x)^(3/2)*(f+g*x)^2/(c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))-8/3*g*(c*d*f-a*e*g)*(d+e*x)^(3/2)/(c^2*d^2*(c*d^2-a*e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-8/3*g*(2*a*e^2*g-c*d*(e*f+d*g))*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*(c*d^2-a*e^2)*sqrt(d+e*x))],
[(d+e*x)^(5/2)*(f+g*x)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,2,-2/3*(c*d*f-a*e*g)*(d+e*x)^(5/2)/(c*d*(c*d^2-a*e^2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+2/3*(2*a*e^2*g+c*d*(e*f-3*d*g))*sqrt(d+e*x)/(c^2*d^2*(c*d^2-a*e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[(d+e*x)^(5/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,1,-2/3*(d+e*x)^(3/2)/(c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))],
[(d+e*x)^(5/2)/((f+g*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)),x,4,-2/3*(d+e*x)^(3/2)/((c*d*f-a*e*g)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+2*g^(3/2)*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/(c*d*f-a*e*g)^(5/2)+2*g*sqrt(d+e*x)/((c*d*f-a*e*g)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[(d+e*x)^(5/2)/((f+g*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)),x,5,-2/3*(d+e*x)^(3/2)/((c*d*f-a*e*g)*(f+g*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+5*c*d*g^(3/2)*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/(c*d*f-a*e*g)^(7/2)+10/3*g*sqrt(d+e*x)/((c*d*f-a*e*g)^2*(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+5*g^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^3*(f+g*x)*sqrt(d+e*x))],
[(d+e*x)^(5/2)/((f+g*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)),x,6,-2/3*(d+e*x)^(3/2)/((c*d*f-a*e*g)*(f+g*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+35/4*c^2*d^2*g^(3/2)*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/(c*d*f-a*e*g)^(9/2)+14/3*g*sqrt(d+e*x)/((c*d*f-a*e*g)^2*(f+g*x)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+35/6*g^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^3*(f+g*x)^2*sqrt(d+e*x))+35/4*c*d*g^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^4*(f+g*x)*sqrt(d+e*x))],

# m<0
[(f+g*x)^4*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/sqrt(d+e*x),x,5,-128/3465*(c*d*f-a*e*g)^3*(2*a*e^2*g-c*d*(5*e*f-3*d*g))*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^5*d^5*e*(d+e*x)^(3/2))+32/231*(c*d*f-a*e*g)^2*(f+g*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^3*d^3*(d+e*x)^(3/2))+16/99*(c*d*f-a*e*g)*(f+g*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^2*d^2*(d+e*x)^(3/2))+2/11*(f+g*x)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c*d*(d+e*x)^(3/2))+128/1155*g*(c*d*f-a*e*g)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^4*d^4*e*sqrt(d+e*x))],
[(f+g*x)^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/sqrt(d+e*x),x,4,-16/315*(c*d*f-a*e*g)^2*(2*a*e^2*g-c*d*(5*e*f-3*d*g))*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^4*d^4*e*(d+e*x)^(3/2))+4/21*(c*d*f-a*e*g)*(f+g*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^2*d^2*(d+e*x)^(3/2))+2/9*(f+g*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c*d*(d+e*x)^(3/2))+16/105*g*(c*d*f-a*e*g)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^3*d^3*e*sqrt(d+e*x))],
[(f+g*x)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/sqrt(d+e*x),x,3,-8/105*(c*d*f-a*e*g)*(2*a*e^2*g-c*d*(5*e*f-3*d*g))*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^3*d^3*e*(d+e*x)^(3/2))+2/7*(f+g*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c*d*(d+e*x)^(3/2))+8/35*g*(c*d*f-a*e*g)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^2*d^2*e*sqrt(d+e*x))],
[(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/sqrt(d+e*x),x,2,-2/15*(2*a*e^2*g-c*d*(5*e*f-3*d*g))*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c^2*d^2*e*(d+e*x)^(3/2))+2/5*g*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c*d*e*sqrt(d+e*x))],
[sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/sqrt(d+e*x),x,1,2/3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(c*d*(d+e*x)^(3/2))],
[sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((f+g*x)*sqrt(d+e*x)),x,3,-2*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))*sqrt(c*d*f-a*e*g)/g^(3/2)+2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*sqrt(d+e*x))],
[sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((f+g*x)^2*sqrt(d+e*x)),x,3,c*d*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/(g^(3/2)*sqrt(c*d*f-a*e*g))-sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*(f+g*x)*sqrt(d+e*x))],
[sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((f+g*x)^3*sqrt(d+e*x)),x,4,1/4*c^2*d^2*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/(g^(3/2)*(c*d*f-a*e*g)^(3/2))-1/2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*(f+g*x)^2*sqrt(d+e*x))+1/4*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*(c*d*f-a*e*g)*(f+g*x)*sqrt(d+e*x))],
[sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((f+g*x)^4*sqrt(d+e*x)),x,5,1/8*c^3*d^3*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/(g^(3/2)*(c*d*f-a*e*g)^(5/2))-1/3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*(f+g*x)^3*sqrt(d+e*x))+1/12*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*(c*d*f-a*e*g)*(f+g*x)^2*sqrt(d+e*x))+1/8*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*(c*d*f-a*e*g)^2*(f+g*x)*sqrt(d+e*x))],
[sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((f+g*x)^5*sqrt(d+e*x)),x,6,5/64*c^4*d^4*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/(g^(3/2)*(c*d*f-a*e*g)^(7/2))-1/4*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*(f+g*x)^4*sqrt(d+e*x))+1/24*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*(c*d*f-a*e*g)*(f+g*x)^3*sqrt(d+e*x))+5/96*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*(c*d*f-a*e*g)^2*(f+g*x)^2*sqrt(d+e*x))+5/64*c^3*d^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*(c*d*f-a*e*g)^3*(f+g*x)*sqrt(d+e*x))],
[(f+g*x)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^(3/2),x,5,-128/15015*(c*d*f-a*e*g)^3*(2*a*e^2*g-c*d*(7*e*f-5*d*g))*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^5*d^5*e*(d+e*x)^(5/2))+128/3003*g*(c*d*f-a*e*g)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^4*d^4*e*(d+e*x)^(3/2))+32/429*(c*d*f-a*e*g)^2*(f+g*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^3*d^3*(d+e*x)^(5/2))+16/143*(c*d*f-a*e*g)*(f+g*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^2*d^2*(d+e*x)^(5/2))+2/13*(f+g*x)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c*d*(d+e*x)^(5/2))],
[(f+g*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^(3/2),x,4,-16/1155*(c*d*f-a*e*g)^2*(2*a*e^2*g-c*d*(7*e*f-5*d*g))*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^4*d^4*e*(d+e*x)^(5/2))+16/231*g*(c*d*f-a*e*g)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^3*d^3*e*(d+e*x)^(3/2))+4/33*(c*d*f-a*e*g)*(f+g*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^2*d^2*(d+e*x)^(5/2))+2/11*(f+g*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c*d*(d+e*x)^(5/2))],
[(f+g*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^(3/2),x,3,-8/315*(c*d*f-a*e*g)*(2*a*e^2*g-c*d*(7*e*f-5*d*g))*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^3*d^3*e*(d+e*x)^(5/2))+8/63*g*(c*d*f-a*e*g)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^2*d^2*e*(d+e*x)^(3/2))+2/9*(f+g*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c*d*(d+e*x)^(5/2))],
[(f+g*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^(3/2),x,2,-2/35*(2*a*e^2*g-c*d*(7*e*f-5*d*g))*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c^2*d^2*e*(d+e*x)^(5/2))+2/7*g*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c*d*e*(d+e*x)^(3/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^(3/2),x,1,2/5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(c*d*(d+e*x)^(5/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((d+e*x)^(3/2)*(f+g*x)),x,4,2/3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g*(d+e*x)^(3/2))+2*(c*d*f-a*e*g)^(3/2)*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/g^(5/2)-2*(c*d*f-a*e*g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^2*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((d+e*x)^(3/2)*(f+g*x)^2),x,4,-(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g*(d+e*x)^(3/2)*(f+g*x))-3*c*d*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))*sqrt(c*d*f-a*e*g)/g^(5/2)+3*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^2*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((d+e*x)^(3/2)*(f+g*x)^3),x,4,-1/2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g*(d+e*x)^(3/2)*(f+g*x)^2)+3/4*c^2*d^2*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/(g^(5/2)*sqrt(c*d*f-a*e*g))-3/4*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^2*(f+g*x)*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((d+e*x)^(3/2)*(f+g*x)^4),x,5,-1/3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g*(d+e*x)^(3/2)*(f+g*x)^3)+1/8*c^3*d^3*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/(g^(5/2)*(c*d*f-a*e*g)^(3/2))-1/4*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^2*(f+g*x)^2*sqrt(d+e*x))+1/8*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^2*(c*d*f-a*e*g)*(f+g*x)*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((d+e*x)^(3/2)*(f+g*x)^5),x,6,-1/4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g*(d+e*x)^(3/2)*(f+g*x)^4)+3/64*c^4*d^4*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/(g^(5/2)*(c*d*f-a*e*g)^(5/2))-1/8*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^2*(f+g*x)^3*sqrt(d+e*x))+1/32*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^2*(c*d*f-a*e*g)*(f+g*x)^2*sqrt(d+e*x))+3/64*c^3*d^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^2*(c*d*f-a*e*g)^2*(f+g*x)*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((d+e*x)^(3/2)*(f+g*x)^6),x,7,-1/5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g*(d+e*x)^(3/2)*(f+g*x)^5)+3/128*c^5*d^5*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/(g^(5/2)*(c*d*f-a*e*g)^(7/2))-3/40*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^2*(f+g*x)^4*sqrt(d+e*x))+1/80*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^2*(c*d*f-a*e*g)*(f+g*x)^3*sqrt(d+e*x))+1/64*c^3*d^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^2*(c*d*f-a*e*g)^2*(f+g*x)^2*sqrt(d+e*x))+3/128*c^4*d^4*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^2*(c*d*f-a*e*g)^3*(f+g*x)*sqrt(d+e*x))],
[(f+g*x)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^(5/2),x,5,-128/45045*(c*d*f-a*e*g)^3*(2*a*e^2*g-c*d*(9*e*f-7*d*g))*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^5*d^5*e*(d+e*x)^(7/2))+128/6435*g*(c*d*f-a*e*g)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^4*d^4*e*(d+e*x)^(5/2))+32/715*(c*d*f-a*e*g)^2*(f+g*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^3*d^3*(d+e*x)^(7/2))+16/195*(c*d*f-a*e*g)*(f+g*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^2*d^2*(d+e*x)^(7/2))+2/15*(f+g*x)^4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c*d*(d+e*x)^(7/2))],
[(f+g*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^(5/2),x,4,-16/3003*(c*d*f-a*e*g)^2*(2*a*e^2*g-c*d*(9*e*f-7*d*g))*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^4*d^4*e*(d+e*x)^(7/2))+16/429*g*(c*d*f-a*e*g)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^3*d^3*e*(d+e*x)^(5/2))+12/143*(c*d*f-a*e*g)*(f+g*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^2*d^2*(d+e*x)^(7/2))+2/13*(f+g*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c*d*(d+e*x)^(7/2))],
[(f+g*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^(5/2),x,3,-8/693*(c*d*f-a*e*g)*(2*a*e^2*g-c*d*(9*e*f-7*d*g))*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^3*d^3*e*(d+e*x)^(7/2))+8/99*g*(c*d*f-a*e*g)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^2*d^2*e*(d+e*x)^(5/2))+2/11*(f+g*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c*d*(d+e*x)^(7/2))],
[(f+g*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^(5/2),x,2,-2/63*(2*a*e^2*g-c*d*(9*e*f-7*d*g))*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c^2*d^2*e*(d+e*x)^(7/2))+2/9*g*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c*d*e*(d+e*x)^(5/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^(5/2),x,1,2/7*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/(c*d*(d+e*x)^(7/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((d+e*x)^(5/2)*(f+g*x)),x,5,-2/3*(c*d*f-a*e*g)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g^2*(d+e*x)^(3/2))+2/5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(g*(d+e*x)^(5/2))-2*(c*d*f-a*e*g)^(5/2)*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/g^(7/2)+2*(c*d*f-a*e*g)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^3*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((d+e*x)^(5/2)*(f+g*x)^2),x,5,5/3*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g^2*(d+e*x)^(3/2))-(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(g*(d+e*x)^(5/2)*(f+g*x))+5*c*d*(c*d*f-a*e*g)^(3/2)*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/g^(7/2)-5*c*d*(c*d*f-a*e*g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^3*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((d+e*x)^(5/2)*(f+g*x)^3),x,5,-5/4*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g^2*(d+e*x)^(3/2)*(f+g*x))-1/2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(g*(d+e*x)^(5/2)*(f+g*x)^2)-15/4*c^2*d^2*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))*sqrt(c*d*f-a*e*g)/g^(7/2)+15/4*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^3*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((d+e*x)^(5/2)*(f+g*x)^4),x,5,-5/12*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g^2*(d+e*x)^(3/2)*(f+g*x)^2)-1/3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(g*(d+e*x)^(5/2)*(f+g*x)^3)+5/8*c^3*d^3*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/(g^(7/2)*sqrt(c*d*f-a*e*g))-5/8*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^3*(f+g*x)*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((d+e*x)^(5/2)*(f+g*x)^5),x,6,-5/24*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g^2*(d+e*x)^(3/2)*(f+g*x)^3)-1/4*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(g*(d+e*x)^(5/2)*(f+g*x)^4)+5/64*c^4*d^4*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/(g^(7/2)*(c*d*f-a*e*g)^(3/2))-5/32*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^3*(f+g*x)^2*sqrt(d+e*x))+5/64*c^3*d^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^3*(c*d*f-a*e*g)*(f+g*x)*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((d+e*x)^(5/2)*(f+g*x)^6),x,7,-1/8*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g^2*(d+e*x)^(3/2)*(f+g*x)^4)-1/5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(g*(d+e*x)^(5/2)*(f+g*x)^5)+3/128*c^5*d^5*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/(g^(7/2)*(c*d*f-a*e*g)^(5/2))-1/16*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^3*(f+g*x)^3*sqrt(d+e*x))+1/64*c^3*d^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^3*(c*d*f-a*e*g)*(f+g*x)^2*sqrt(d+e*x))+3/128*c^4*d^4*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^3*(c*d*f-a*e*g)^2*(f+g*x)*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((d+e*x)^(5/2)*(f+g*x)^7),x,8,-1/12*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g^2*(d+e*x)^(3/2)*(f+g*x)^5)-1/6*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(g*(d+e*x)^(5/2)*(f+g*x)^6)+5/512*c^6*d^6*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/(g^(7/2)*(c*d*f-a*e*g)^(7/2))-1/32*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^3*(f+g*x)^4*sqrt(d+e*x))+1/192*c^3*d^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^3*(c*d*f-a*e*g)*(f+g*x)^3*sqrt(d+e*x))+5/768*c^4*d^4*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^3*(c*d*f-a*e*g)^2*(f+g*x)^2*sqrt(d+e*x))+5/512*c^5*d^5*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^3*(c*d*f-a*e*g)^3*(f+g*x)*sqrt(d+e*x))],

# Integrands of the form (d+e x)^(m/2) / (a d e+(c d^2+a e^2) x+c d e x^2)^(m/2) (f+g x)^(n/2)

# m>0
[(f+g*x)^(5/2)*sqrt(d+e*x)/sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,7,5/8*(c*d*f-a*e*g)^3*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(c^(7/2)*d^(7/2)*sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+5/12*(c*d*f-a*e*g)*(f+g*x)^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*sqrt(d+e*x))+1/3*(f+g*x)^(5/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*sqrt(d+e*x))+5/8*(c*d*f-a*e*g)^2*sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*sqrt(d+e*x))],
[(f+g*x)^(3/2)*sqrt(d+e*x)/sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,6,3/4*(c*d*f-a*e*g)^2*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(c^(5/2)*d^(5/2)*sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+1/2*(f+g*x)^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*sqrt(d+e*x))+3/4*(c*d*f-a*e*g)*sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*sqrt(d+e*x))],
[(f+g*x)^(1/2)*sqrt(d+e*x)/sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,5,(c*d*f-a*e*g)*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(c^(3/2)*d^(3/2)*sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*sqrt(d+e*x))],
[sqrt(d+e*x)/((f+g*x)^(1/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,4,2*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(sqrt(c)*sqrt(d)*sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[sqrt(d+e*x)/((f+g*x)^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,1,2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)*sqrt(d+e*x)*sqrt(f+g*x))],
[sqrt(d+e*x)/((f+g*x)^(5/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,2,2/3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)*(f+g*x)^(3/2)*sqrt(d+e*x))+4/3*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^2*sqrt(d+e*x)*sqrt(f+g*x))],
[sqrt(d+e*x)/((f+g*x)^(7/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,3,2/5*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)*(f+g*x)^(5/2)*sqrt(d+e*x))+8/15*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^2*(f+g*x)^(3/2)*sqrt(d+e*x))+16/15*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^3*sqrt(d+e*x)*sqrt(f+g*x))],
[sqrt(d+e*x)/((f+g*x)^(9/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,4,2/7*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)*(f+g*x)^(7/2)*sqrt(d+e*x))+12/35*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^2*(f+g*x)^(5/2)*sqrt(d+e*x))+16/35*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^3*(f+g*x)^(3/2)*sqrt(d+e*x))+32/35*c^3*d^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^4*sqrt(d+e*x)*sqrt(f+g*x))],
[(d+e*x)^(3/2)*(f+g*x)^(5/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,7,-2*(f+g*x)^(5/2)*sqrt(d+e*x)/(c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+15/4*(c*d*f-a*e*g)^2*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(g)*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(c^(7/2)*d^(7/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+5/2*g*(f+g*x)^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*sqrt(d+e*x))+15/4*g*(c*d*f-a*e*g)*sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*sqrt(d+e*x))],
[(d+e*x)^(3/2)*(f+g*x)^(3/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,6,-2*(f+g*x)^(3/2)*sqrt(d+e*x)/(c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+3*(c*d*f-a*e*g)*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(g)*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(c^(5/2)*d^(5/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+3*g*sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*sqrt(d+e*x))],
[(d+e*x)^(3/2)*(f+g*x)^(1/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,5,2*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(g)*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(c^(3/2)*d^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-2*sqrt(d+e*x)*sqrt(f+g*x)/(c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[(d+e*x)^(3/2)/((f+g*x)^(1/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,1,-2*sqrt(d+e*x)*sqrt(f+g*x)/((c*d*f-a*e*g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[(d+e*x)^(3/2)/((f+g*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,2,-2*sqrt(d+e*x)/((c*d*f-a*e*g)*sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-4*g*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^2*sqrt(d+e*x)*sqrt(f+g*x))],
[(d+e*x)^(3/2)/((f+g*x)^(5/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,3,-2*sqrt(d+e*x)/((c*d*f-a*e*g)*(f+g*x)^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-8/3*g*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^2*(f+g*x)^(3/2)*sqrt(d+e*x))-16/3*c*d*g*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^3*sqrt(d+e*x)*sqrt(f+g*x))],
[(d+e*x)^(3/2)/((f+g*x)^(7/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),x,4,-2*sqrt(d+e*x)/((c*d*f-a*e*g)*(f+g*x)^(5/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-12/5*g*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^2*(f+g*x)^(5/2)*sqrt(d+e*x))-16/5*c*d*g*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^3*(f+g*x)^(3/2)*sqrt(d+e*x))-32/5*c^2*d^2*g*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^4*sqrt(d+e*x)*sqrt(f+g*x))],
[(d+e*x)^(5/2)*(f+g*x)^(5/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,7,-2/3*(d+e*x)^(3/2)*(f+g*x)^(5/2)/(c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))-10/3*g*(f+g*x)^(3/2)*sqrt(d+e*x)/(c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+5*g^(3/2)*(c*d*f-a*e*g)*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(c^(7/2)*d^(7/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+5*g^2*sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*sqrt(d+e*x))],
[(d+e*x)^(5/2)*(f+g*x)^(3/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,6,-2/3*(d+e*x)^(3/2)*(f+g*x)^(3/2)/(c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+2*g^(3/2)*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(c^(5/2)*d^(5/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-2*g*sqrt(d+e*x)*sqrt(f+g*x)/(c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[(d+e*x)^(5/2)*(f+g*x)^(1/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,1,-2/3*(d+e*x)^(3/2)*(f+g*x)^(3/2)/((c*d*f-a*e*g)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))],
[(d+e*x)^(5/2)/((f+g*x)^(1/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)),x,2,-2/3*(d+e*x)^(3/2)*sqrt(f+g*x)/((c*d*f-a*e*g)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+4/3*g*sqrt(d+e*x)*sqrt(f+g*x)/((c*d*f-a*e*g)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[(d+e*x)^(5/2)/((f+g*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)),x,3,-2/3*(d+e*x)^(3/2)/((c*d*f-a*e*g)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)*sqrt(f+g*x))+8/3*g*sqrt(d+e*x)/((c*d*f-a*e*g)^2*sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+16/3*g^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^3*sqrt(d+e*x)*sqrt(f+g*x))],
[(d+e*x)^(5/2)/((f+g*x)^(5/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)),x,4,-2/3*(d+e*x)^(3/2)/((c*d*f-a*e*g)*(f+g*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2))+4*g*sqrt(d+e*x)/((c*d*f-a*e*g)^2*(f+g*x)^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+16/3*g^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^3*(f+g*x)^(3/2)*sqrt(d+e*x))+32/3*c*d*g^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)^4*sqrt(d+e*x)*sqrt(f+g*x))],

# m<0
[(f+g*x)^(5/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/sqrt(d+e*x),x,8,-5/64*(c*d*f-a*e*g)^4*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(c^(7/2)*d^(7/2)*g^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-5/96*(c*d*f-a*e*g)^2*(f+g*x)^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*g*sqrt(d+e*x))+1/24*(a*e/(c*d)-f/g)*(f+g*x)^(5/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/sqrt(d+e*x)+1/4*(f+g*x)^(7/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*sqrt(d+e*x))-5/64*(c*d*f-a*e*g)^3*sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*g*sqrt(d+e*x))],
[(f+g*x)^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/sqrt(d+e*x),x,7,-1/8*(c*d*f-a*e*g)^3*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(c^(5/2)*d^(5/2)*g^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+1/12*(a*e/(c*d)-f/g)*(f+g*x)^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/sqrt(d+e*x)+1/3*(f+g*x)^(5/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*sqrt(d+e*x))-1/8*(c*d*f-a*e*g)^2*sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*g*sqrt(d+e*x))],
[(f+g*x)^(1/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/sqrt(d+e*x),x,6,-1/4*(c*d*f-a*e*g)^2*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(c^(3/2)*d^(3/2)*g^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+1/2*(f+g*x)^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*sqrt(d+e*x))+1/4*(a*e/(c*d)-f/g)*sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/sqrt(d+e*x)],
[sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((f+g*x)^(1/2)*sqrt(d+e*x)),x,5,-(c*d*f-a*e*g)*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(g^(3/2)*sqrt(c)*sqrt(d)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*sqrt(d+e*x))],
[sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((f+g*x)^(3/2)*sqrt(d+e*x)),x,5,2*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(c)*sqrt(d)*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(g^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*sqrt(d+e*x)*sqrt(f+g*x))],
[sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((f+g*x)^(5/2)*sqrt(d+e*x)),x,1,2/3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((c*d*f-a*e*g)*(d+e*x)^(3/2)*(f+g*x)^(3/2))],
[sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((f+g*x)^(7/2)*sqrt(d+e*x)),x,2,2/5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((c*d*f-a*e*g)*(d+e*x)^(3/2)*(f+g*x)^(5/2))+4/15*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((c*d*f-a*e*g)^2*(d+e*x)^(3/2)*(f+g*x)^(3/2))],
[sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((f+g*x)^(9/2)*sqrt(d+e*x)),x,3,2/7*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((c*d*f-a*e*g)*(d+e*x)^(3/2)*(f+g*x)^(7/2))+8/35*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((c*d*f-a*e*g)^2*(d+e*x)^(3/2)*(f+g*x)^(5/2))+16/105*c^2*d^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((c*d*f-a*e*g)^3*(d+e*x)^(3/2)*(f+g*x)^(3/2))],
[sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((f+g*x)^(11/2)*sqrt(d+e*x)),x,4,2/9*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((c*d*f-a*e*g)*(d+e*x)^(3/2)*(f+g*x)^(9/2))+4/21*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((c*d*f-a*e*g)^2*(d+e*x)^(3/2)*(f+g*x)^(7/2))+16/105*c^2*d^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((c*d*f-a*e*g)^3*(d+e*x)^(3/2)*(f+g*x)^(5/2))+32/315*c^3*d^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((c*d*f-a*e*g)^4*(d+e*x)^(3/2)*(f+g*x)^(3/2))],
[(f+g*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^(3/2),x,8,1/4*(f+g*x)^(5/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g*(d+e*x)^(3/2))+3/64*(c*d*f-a*e*g)^4*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(c^(5/2)*d^(5/2)*g^(5/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+1/32*(c*d*f-a*e*g)^2*(f+g*x)^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*g^2*sqrt(d+e*x))-1/8*(c*d*f-a*e*g)*(f+g*x)^(5/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^2*sqrt(d+e*x))+3/64*(c*d*f-a*e*g)^3*sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*g^2*sqrt(d+e*x))],
[(f+g*x)^(1/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^(3/2),x,7,1/3*(f+g*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g*(d+e*x)^(3/2))+1/8*(c*d*f-a*e*g)^3*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(c^(3/2)*d^(3/2)*g^(5/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-1/4*(c*d*f-a*e*g)*(f+g*x)^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^2*sqrt(d+e*x))+1/8*(c*d*f-a*e*g)^2*sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*g^2*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((d+e*x)^(3/2)*(f+g*x)^(1/2)),x,6,1/2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)*sqrt(f+g*x)/(g*(d+e*x)^(3/2))+3/4*(c*d*f-a*e*g)^2*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(g^(5/2)*sqrt(c)*sqrt(d)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-3/4*(c*d*f-a*e*g)*sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^2*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((d+e*x)^(3/2)*(f+g*x)^(3/2)),x,6,-2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g*(d+e*x)^(3/2)*sqrt(f+g*x))-3*(c*d*f-a*e*g)*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(c)*sqrt(d)*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(g^(5/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+3*c*d*sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^2*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((d+e*x)^(3/2)*(f+g*x)^(5/2)),x,6,-2/3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g*(d+e*x)^(3/2)*(f+g*x)^(3/2))+2*c^(3/2)*d^(3/2)*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(g^(5/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-2*c*d*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^2*sqrt(d+e*x)*sqrt(f+g*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((d+e*x)^(3/2)*(f+g*x)^(7/2)),x,1,2/5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((c*d*f-a*e*g)*(d+e*x)^(5/2)*(f+g*x)^(5/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((d+e*x)^(3/2)*(f+g*x)^(9/2)),x,2,2/7*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((c*d*f-a*e*g)*(d+e*x)^(5/2)*(f+g*x)^(7/2))+4/35*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((c*d*f-a*e*g)^2*(d+e*x)^(5/2)*(f+g*x)^(5/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((d+e*x)^(3/2)*(f+g*x)^(11/2)),x,3,2/9*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((c*d*f-a*e*g)*(d+e*x)^(5/2)*(f+g*x)^(9/2))+8/63*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((c*d*f-a*e*g)^2*(d+e*x)^(5/2)*(f+g*x)^(7/2))+16/315*c^2*d^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((c*d*f-a*e*g)^3*(d+e*x)^(5/2)*(f+g*x)^(5/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/((d+e*x)^(3/2)*(f+g*x)^(13/2)),x,4,2/11*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((c*d*f-a*e*g)*(d+e*x)^(5/2)*(f+g*x)^(11/2))+4/33*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((c*d*f-a*e*g)^2*(d+e*x)^(5/2)*(f+g*x)^(9/2))+16/231*c^2*d^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((c*d*f-a*e*g)^3*(d+e*x)^(5/2)*(f+g*x)^(7/2))+32/1155*c^3*d^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((c*d*f-a*e*g)^4*(d+e*x)^(5/2)*(f+g*x)^(5/2))],
[(f+g*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^(5/2),x,9,-1/8*(c*d*f-a*e*g)*(f+g*x)^(5/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g^2*(d+e*x)^(3/2))+1/5*(f+g*x)^(5/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(g*(d+e*x)^(5/2))-3/128*(c*d*f-a*e*g)^5*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(c^(5/2)*d^(5/2)*g^(7/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-1/64*(c*d*f-a*e*g)^3*(f+g*x)^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*g^3*sqrt(d+e*x))+1/16*(c*d*f-a*e*g)^2*(f+g*x)^(5/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^3*sqrt(d+e*x))-3/128*(c*d*f-a*e*g)^4*sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*g^3*sqrt(d+e*x))],
[(f+g*x)^(1/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^(5/2),x,8,-5/24*(c*d*f-a*e*g)*(f+g*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g^2*(d+e*x)^(3/2))+1/4*(f+g*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(g*(d+e*x)^(5/2))-5/64*(c*d*f-a*e*g)^4*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(c^(3/2)*d^(3/2)*g^(7/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+5/32*(c*d*f-a*e*g)^2*(f+g*x)^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^3*sqrt(d+e*x))-5/64*(c*d*f-a*e*g)^3*sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*g^3*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((d+e*x)^(5/2)*(f+g*x)^(1/2)),x,7,-5/12*(c*d*f-a*e*g)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)*sqrt(f+g*x)/(g^2*(d+e*x)^(3/2))+1/3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)*sqrt(f+g*x)/(g*(d+e*x)^(5/2))-5/8*(c*d*f-a*e*g)^3*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(g^(7/2)*sqrt(c)*sqrt(d)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+5/8*(c*d*f-a*e*g)^2*sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^3*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((d+e*x)^(5/2)*(f+g*x)^(3/2)),x,7,-2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(g*(d+e*x)^(5/2)*sqrt(f+g*x))+5/2*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)*sqrt(f+g*x)/(g^2*(d+e*x)^(3/2))+15/4*(c*d*f-a*e*g)^2*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(c)*sqrt(d)*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(g^(7/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-15/4*c*d*(c*d*f-a*e*g)*sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^3*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((d+e*x)^(5/2)*(f+g*x)^(5/2)),x,7,-2/3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(g*(d+e*x)^(5/2)*(f+g*x)^(3/2))-10/3*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g^2*(d+e*x)^(3/2)*sqrt(f+g*x))-5*c^(3/2)*d^(3/2)*(c*d*f-a*e*g)*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(g^(7/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+5*c^2*d^2*sqrt(f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^3*sqrt(d+e*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((d+e*x)^(5/2)*(f+g*x)^(7/2)),x,7,-2/3*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(g^2*(d+e*x)^(3/2)*(f+g*x)^(3/2))-2/5*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(g*(d+e*x)^(5/2)*(f+g*x)^(5/2))+2*c^(5/2)*d^(5/2)*arctanh(sqrt(g)*sqrt(a*e+c*d*x)/(sqrt(c)*sqrt(d)*sqrt(f+g*x)))*sqrt(a*e+c*d*x)*sqrt(d+e*x)/(g^(7/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))-2*c^2*d^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g^3*sqrt(d+e*x)*sqrt(f+g*x))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((d+e*x)^(5/2)*(f+g*x)^(9/2)),x,1,2/7*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/((c*d*f-a*e*g)*(d+e*x)^(7/2)*(f+g*x)^(7/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((d+e*x)^(5/2)*(f+g*x)^(11/2)),x,2,2/9*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/((c*d*f-a*e*g)*(d+e*x)^(7/2)*(f+g*x)^(9/2))+4/63*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/((c*d*f-a*e*g)^2*(d+e*x)^(7/2)*(f+g*x)^(7/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((d+e*x)^(5/2)*(f+g*x)^(13/2)),x,3,2/11*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/((c*d*f-a*e*g)*(d+e*x)^(7/2)*(f+g*x)^(11/2))+8/99*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/((c*d*f-a*e*g)^2*(d+e*x)^(7/2)*(f+g*x)^(9/2))+16/693*c^2*d^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/((c*d*f-a*e*g)^3*(d+e*x)^(7/2)*(f+g*x)^(7/2))],
[(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/((d+e*x)^(5/2)*(f+g*x)^(15/2)),x,4,2/13*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/((c*d*f-a*e*g)*(d+e*x)^(7/2)*(f+g*x)^(13/2))+12/143*c*d*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/((c*d*f-a*e*g)^2*(d+e*x)^(7/2)*(f+g*x)^(11/2))+16/429*c^2*d^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/((c*d*f-a*e*g)^3*(d+e*x)^(7/2)*(f+g*x)^(9/2))+32/3003*c^3*d^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(7/2)/((c*d*f-a*e*g)^4*(d+e*x)^(7/2)*(f+g*x)^(7/2))],

# Integrands of the form (d+e x)^(m/2) / (a d e+(c d^2+a e^2) x+c d e x^2)^(m/2) (f+g x)^n when n symbolic
[(d+e*x)^(5/2)*(f+g*x)^n/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2),x,3,-(a*e+c*d*x)*(d+e*x)^(5/2)*(f+g*x)^(1+n)*hypergeom([1,-1/2+n],[2+n],c*d*(f+g*x)/(c*d*f-a*e*g))/((c*d*f-a*e*g)*(1+n)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)),-2/3*(f+g*x)^n*hypergeom([-3/2,-n],[-1/2],-g*(a*e+c*d*x)/(c*d*f-a*e*g))*sqrt(d+e*x)/(c*d*(a*e+c*d*x)*(c*d*(f+g*x)/(c*d*f-a*e*g))^n*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[(d+e*x)^(3/2)*(f+g*x)^n/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2),x,3,-(a*e+c*d*x)*(d+e*x)^(3/2)*(f+g*x)^(1+n)*hypergeom([1,1/2+n],[2+n],c*d*(f+g*x)/(c*d*f-a*e*g))/((c*d*f-a*e*g)*(1+n)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)),-2*(f+g*x)^n*hypergeom([-1/2,-n],[1/2],-g*(a*e+c*d*x)/(c*d*f-a*e*g))*sqrt(d+e*x)/(c*d*(c*d*(f+g*x)/(c*d*f-a*e*g))^n*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[(d+e*x)^(1/2)*(f+g*x)^n/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2),x,3,-(a*e+c*d*x)*(f+g*x)^(1+n)*hypergeom([1,3/2+n],[2+n],c*d*(f+g*x)/(c*d*f-a*e*g))*sqrt(d+e*x)/((c*d*f-a*e*g)*(1+n)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),2*(a*e+c*d*x)*(f+g*x)^n*hypergeom([1/2,-n],[3/2],-g*(a*e+c*d*x)/(c*d*f-a*e*g))*sqrt(d+e*x)/(c*d*(c*d*(f+g*x)/(c*d*f-a*e*g))^n*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))],
[(f+g*x)^n*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1/2)/(d+e*x)^(1/2),x,3,-(a*e+c*d*x)*(f+g*x)^(1+n)*hypergeom([1,5/2+n],[2+n],c*d*(f+g*x)/(c*d*f-a*e*g))*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/((c*d*f-a*e*g)*(1+n)*sqrt(d+e*x)),2/3*(a*e+c*d*x)*(f+g*x)^n*hypergeom([3/2,-n],[5/2],-g*(a*e+c*d*x)/(c*d*f-a*e*g))*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*(c*d*(f+g*x)/(c*d*f-a*e*g))^n*sqrt(d+e*x))],
[(f+g*x)^n*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)/(d+e*x)^(3/2),x,3,-(a*e+c*d*x)*(f+g*x)^(1+n)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(3/2)*hypergeom([1,7/2+n],[2+n],c*d*(f+g*x)/(c*d*f-a*e*g))/((c*d*f-a*e*g)*(1+n)*(d+e*x)^(3/2)),2/5*(a*e+c*d*x)^2*(f+g*x)^n*hypergeom([5/2,-n],[7/2],-g*(a*e+c*d*x)/(c*d*f-a*e*g))*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*(c*d*(f+g*x)/(c*d*f-a*e*g))^n*sqrt(d+e*x))],
[(f+g*x)^n*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)/(d+e*x)^(5/2),x,3,-(a*e+c*d*x)*(f+g*x)^(1+n)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(5/2)*hypergeom([1,9/2+n],[2+n],c*d*(f+g*x)/(c*d*f-a*e*g))/((c*d*f-a*e*g)*(1+n)*(d+e*x)^(5/2)),2/7*(a*e+c*d*x)^3*(f+g*x)^n*hypergeom([7/2,-n],[9/2],-g*(a*e+c*d*x)/(c*d*f-a*e*g))*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*(c*d*(f+g*x)/(c*d*f-a*e*g))^n*sqrt(d+e*x))],

# Integrands of the form (d+e x)^m / (a d e+(c d^2+a e^2) x+c d e x^2)^m (f+g x)^n when m symbolic
[(d+e*x)^m*(f+g*x)^n/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m,x,3,-(a*e+c*d*x)*(d+e*x)^m*(f+g*x)^(1+n)*hypergeom([1,2-m+n],[2+n],c*d*(f+g*x)/(c*d*f-a*e*g))/((c*d*f-a*e*g)*(1+n)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m),(-g*(a*e+c*d*x)/(c*d*f-a*e*g))^m*(d+e*x)^m*(f+g*x)^(1+n)*hypergeom([m,1+n],[2+n],c*d*(f+g*x)/(c*d*f-a*e*g))/(g*(1+n)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m)],
[(d+e*x)^m*(f+g*x)^3/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m,x,4,-6*(c*d*f-a*e*g)^2*(a*e^2*g+c*d*(d*g*(1-m)-e*f*(2-m)))*(d+e*x)^(-1+m)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1-m)/(c^4*d^4*e*(12-7*m+m^2)*(2-3*m+m^2))+6*g*(c*d*f-a*e*g)^2*(d+e*x)^m*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1-m)/(c^3*d^3*e*(2-m)*(3-m)*(4-m))+3*(c*d*f-a*e*g)*(d+e*x)^(-1+m)*(f+g*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1-m)/(c^2*d^2*(3-m)*(4-m))+(d+e*x)^(-1+m)*(f+g*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1-m)/(c*d*(4-m))],
[(d+e*x)^m*(f+g*x)^2/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m,x,3,-2*(c*d*f-a*e*g)*(a*e^2*g+c*d*(d*g*(1-m)-e*f*(2-m)))*(d+e*x)^(-1+m)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1-m)/(c^3*d^3*e*(1-m)*(2-m)*(3-m))+2*g*(c*d*f-a*e*g)*(d+e*x)^m*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1-m)/(c^2*d^2*e*(2-m)*(3-m))+(d+e*x)^(-1+m)*(f+g*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1-m)/(c*d*(3-m))],
[(d+e*x)^m*(f+g*x)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m,x,2,-(a*e^2*g+c*d*(d*g*(1-m)-e*f*(2-m)))*(d+e*x)^(-1+m)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1-m)/(c^2*d^2*e*(1-m)*(2-m))+g*(d+e*x)^m*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1-m)/(c*d*e*(2-m))],
[(d+e*x)^m/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m,x,1,(d+e*x)^(-1+m)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1-m)/(c*d*(1-m))],
[(d+e*x)^m/((f+g*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m),x,2,(a*e+c*d*x)*(d+e*x)^m*hypergeom([1,1-m],[2-m],-g*(a*e+c*d*x)/(c*d*f-a*e*g))/((c*d*f-a*e*g)*(1-m)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m)],
[(d+e*x)^m/((f+g*x)^2*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m),x,2,c*d*(a*e+c*d*x)*(d+e*x)^m*hypergeom([2,1-m],[2-m],-g*(a*e+c*d*x)/(c*d*f-a*e*g))/((c*d*f-a*e*g)^2*(1-m)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m)],
[(d+e*x)^m/((f+g*x)^3*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m),x,2,c^2*d^2*(a*e+c*d*x)*(d+e*x)^m*hypergeom([3,1-m],[2-m],-g*(a*e+c*d*x)/(c*d*f-a*e*g))/((c*d*f-a*e*g)^3*(1-m)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m)],
[(d+e*x)^m*(f+g*x)^(3/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m,x,3,2/5*(-g*(a*e+c*d*x)/(c*d*f-a*e*g))^m*(d+e*x)^m*(f+g*x)^(5/2)*hypergeom([5/2,m],[7/2],c*d*(f+g*x)/(c*d*f-a*e*g))/(g*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m)],
[(d+e*x)^m*(f+g*x)^(1/2)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m,x,3,2/3*(-g*(a*e+c*d*x)/(c*d*f-a*e*g))^m*(d+e*x)^m*(f+g*x)^(3/2)*hypergeom([3/2,m],[5/2],c*d*(f+g*x)/(c*d*f-a*e*g))/(g*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m)],
[(d+e*x)^m/((f+g*x)^(1/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m),x,3,2*(-g*(a*e+c*d*x)/(c*d*f-a*e*g))^m*(d+e*x)^m*hypergeom([1/2,m],[3/2],c*d*(f+g*x)/(c*d*f-a*e*g))*sqrt(f+g*x)/(g*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m)],
[(d+e*x)^m/((f+g*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m),x,3,-2*(-g*(a*e+c*d*x)/(c*d*f-a*e*g))^m*(d+e*x)^m*hypergeom([-1/2,m],[1/2],c*d*(f+g*x)/(c*d*f-a*e*g))/(g*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m*sqrt(f+g*x))],
[(d+e*x)^m/((f+g*x)^(5/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m),x,3,-2/3*(-g*(a*e+c*d*x)/(c*d*f-a*e*g))^m*(d+e*x)^m*hypergeom([-3/2,m],[-1/2],c*d*(f+g*x)/(c*d*f-a*e*g))/(g*(f+g*x)^(3/2)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m)],
[(a*e+c*d*x)^n*(d+e*x)^m/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m,x,1,(a*e+c*d*x)^n*(d+e*x)^(-1+m)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(1-m)/(c*d*(1-m+n))],
[(d+e*x)^m*(c*d^2*e*g-e*(c*d^2+a*e^2)*g-c*d*e^2*g*x)^(-1+m)/(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m,x,3,-(d+e*x)^m*(-a*e^3*g-c*d*e^2*g*x)^m*log(a*e+c*d*x)/(c*d*e^2*g*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m)],

# Integrands of the form (d+e x)^m (f+g x)^n (a+b x+c x^2)^p when c d^2-b d e+a e^2=0 and m+p-1=0
[(d+e*x)^(3/2)*(f+g*x)^n/sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,4,(2*a*e^2*g*(1+n)+c*d*(e*f-d*g*(3+2*n)))*(a*e+c*d*x)*(f+g*x)^(1+n)*hypergeom([1,3/2+n],[2+n],c*d*(f+g*x)/(c*d*f-a*e*g))*sqrt(d+e*x)/(c*d*g*(c*d*f-a*e*g)*(1+n)*(3+2*n)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+2*e*(f+g*x)^(1+n)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*g*(3+2*n)*sqrt(d+e*x)),-2*(2*a*e^2*g*(1+n)+c*d*(e*f-d*g*(3+2*n)))*(a*e+c*d*x)*(f+g*x)^n*hypergeom([1/2,-n],[3/2],-g*(a*e+c*d*x)/(c*d*f-a*e*g))*sqrt(d+e*x)/(c^2*d^2*g*(3+2*n)*(c*d*(f+g*x)/(c*d*f-a*e*g))^n*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2))+2*e*(f+g*x)^(1+n)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*g*(3+2*n)*sqrt(d+e*x))],
[(d+e*x)^(3/2)*(f+g*x)^4/sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,6,128/3465*(c*d*f-a*e*g)^3*(10*a*e^2*g+c*d*(e*f-11*d*g))*(2*a*e^2*g-c*d*(3*e*f-d*g))*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^6*d^6*e*g*sqrt(d+e*x))-32/1155*(c*d*f-a*e*g)^2*(10*a*e^2*g+c*d*(e*f-11*d*g))*(f+g*x)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^4*d^4*g*sqrt(d+e*x))-16/693*(c*d*f-a*e*g)*(10*a*e^2*g+c*d*(e*f-11*d*g))*(f+g*x)^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*g*sqrt(d+e*x))-2/99*(10*a*e^2*g+c*d*(e*f-11*d*g))*(f+g*x)^4*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*g*sqrt(d+e*x))+2/11*e*(f+g*x)^5*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*g*sqrt(d+e*x))-128/3465*(c*d*f-a*e*g)^3*(10*a*e^2*g+c*d*(e*f-11*d*g))*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^5*d^5*e)],
[(d+e*x)^(3/2)*(f+g*x)^3/sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,5,16/315*(c*d*f-a*e*g)^2*(8*a*e^2*g+c*d*(e*f-9*d*g))*(2*a*e^2*g-c*d*(3*e*f-d*g))*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^5*d^5*e*g*sqrt(d+e*x))-4/105*(c*d*f-a*e*g)*(8*a*e^2*g+c*d*(e*f-9*d*g))*(f+g*x)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*g*sqrt(d+e*x))-2/63*(8*a*e^2*g+c*d*(e*f-9*d*g))*(f+g*x)^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*g*sqrt(d+e*x))+2/9*e*(f+g*x)^4*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*g*sqrt(d+e*x))-16/315*(c*d*f-a*e*g)^2*(8*a*e^2*g+c*d*(e*f-9*d*g))*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^4*d^4*e)],
[(d+e*x)^(3/2)*(f+g*x)^2/sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,4,8/105*(c*d*f-a*e*g)*(6*a*e^2*g+c*d*(e*f-7*d*g))*(2*a*e^2*g-c*d*(3*e*f-d*g))*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^4*d^4*e*g*sqrt(d+e*x))-2/35*(6*a*e^2*g+c*d*(e*f-7*d*g))*(f+g*x)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*g*sqrt(d+e*x))+2/7*e*(f+g*x)^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*g*sqrt(d+e*x))-8/105*(c*d*f-a*e*g)*(6*a*e^2*g+c*d*(e*f-7*d*g))*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*e)],
[(d+e*x)^(3/2)*(f+g*x)/sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,3,2/5*g*(d+e*x)^(3/2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*e)-4/15*(c*d^2-a*e^2)*(4*a*e^2*g-c*d*(5*e*f-d*g))*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^3*d^3*e*sqrt(d+e*x))-2/15*(4*a*e^2*g-c*d*(5*e*f-d*g))*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*e)],
[(d+e*x)^(3/2)/sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2),x,2,4/3*(c*d^2-a*e^2)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c^2*d^2*sqrt(d+e*x))+2/3*sqrt(d+e*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d)],
[(d+e*x)^(3/2)/((f+g*x)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,3,-2*(e*f-d*g)*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/(g^(3/2)*sqrt(c*d*f-a*e*g))+2*e*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(c*d*g*sqrt(d+e*x))],
[(d+e*x)^(3/2)/((f+g*x)^2*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,3,-(2*a*e^2*g-c*d*(e*f+d*g))*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/(g^(3/2)*(c*d*f-a*e*g)^(3/2))-(e*f-d*g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*(c*d*f-a*e*g)*(f+g*x)*sqrt(d+e*x))],
[(d+e*x)^(3/2)/((f+g*x)^3*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,4,-1/4*c*d*(4*a*e^2*g-c*d*(e*f+3*d*g))*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/(g^(3/2)*(c*d*f-a*e*g)^(5/2))-1/2*(e*f-d*g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*(c*d*f-a*e*g)*(f+g*x)^2*sqrt(d+e*x))-1/4*(4*a*e^2*g-c*d*(e*f+3*d*g))*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*(c*d*f-a*e*g)^2*(f+g*x)*sqrt(d+e*x))],
[(d+e*x)^(3/2)/((f+g*x)^4*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)),x,5,-1/8*c^2*d^2*(6*a*e^2*g-c*d*(e*f+5*d*g))*arctan(sqrt(g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(sqrt(c*d*f-a*e*g)*sqrt(d+e*x)))/(g^(3/2)*(c*d*f-a*e*g)^(7/2))-1/3*(e*f-d*g)*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*(c*d*f-a*e*g)*(f+g*x)^3*sqrt(d+e*x))-1/12*(6*a*e^2*g-c*d*(e*f+5*d*g))*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*(c*d*f-a*e*g)^2*(f+g*x)^2*sqrt(d+e*x))-1/8*c*d*(6*a*e^2*g-c*d*(e*f+5*d*g))*sqrt(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)/(g*(c*d*f-a*e*g)^3*(f+g*x)*sqrt(d+e*x))],

# Integrands of the form (d+e x)^m (f+g x)^m (a+b x+c x^2)^p when e f+d g=0

# Integrands of the form (1+d x)^(m/2) (1-d x)^(m/2) (a+b x+c x^2)^p

# m>0

# m<0
[(a+b*x+c*x^2)^3/(sqrt(1-d*x)*sqrt(1+d*x)),x,8,1/16*(5*c^3+18*b^2*c*d^2+18*a*c^2*d^2+24*a*b^2*d^4+24*a^2*c*d^4+16*a^3*d^6)*arcsin(d*x)/d^7-1/15*b*(24*c^2+10*b^2*d^2+60*a*c*d^2+45*a^2*d^4)*sqrt(1-d^2*x^2)/d^6-1/16*(5*c^3+18*b^2*c*d^2+18*a*c^2*d^2+24*a*b^2*d^4+24*a^2*c*d^4)*x*sqrt(1-d^2*x^2)/d^6-1/15*b*(12*c^2+5*b^2*d^2+30*a*c*d^2)*x^2*sqrt(1-d^2*x^2)/d^4-1/24*c*(5*c^2+18*b^2*d^2+18*a*c*d^2)*x^3*sqrt(1-d^2*x^2)/d^4-3/5*b*c^2*x^4*sqrt(1-d^2*x^2)/d^2-1/6*c^3*x^5*sqrt(1-d^2*x^2)/d^2],
[(a+b*x+c*x^2)^2/(sqrt(1-d*x)*sqrt(1+d*x)),x,6,1/8*(3*c^2+4*b^2*d^2+8*a*c*d^2+8*a^2*d^4)*arcsin(d*x)/d^5-2/3*b*(2*c+3*a*d^2)*sqrt(1-d^2*x^2)/d^4-1/8*(4*b^2+c*(8*a+3*c/d^2))*x*sqrt(1-d^2*x^2)/d^2-2/3*b*c*x^2*sqrt(1-d^2*x^2)/d^2-1/4*c^2*x^3*sqrt(1-d^2*x^2)/d^2],
[(a+b*x+c*x^2)/(sqrt(1-d*x)*sqrt(1+d*x)),x,4,1/2*(c+2*a*d^2)*arcsin(d*x)/d^3-b*sqrt(1-d^2*x^2)/d^2-1/2*c*x*sqrt(1-d^2*x^2)/d^2],
[1/((a+b*x+c*x^2)*sqrt(1-d*x)*sqrt(1+d*x)),x,6,-c*arctanh((2*c+d^2*x*(b-sqrt(b^2-4*a*c)))/(sqrt(2)*sqrt(1-d^2*x^2)*sqrt(2*c^2+2*a*c*d^2-b*d^2*(b-sqrt(b^2-4*a*c)))))*sqrt(2)/(sqrt(b^2-4*a*c)*sqrt(2*c^2+2*a*c*d^2-b*d^2*(b-sqrt(b^2-4*a*c))))+c*arctanh((2*c+d^2*x*(b+sqrt(b^2-4*a*c)))/(sqrt(2)*sqrt(1-d^2*x^2)*sqrt(2*c^2+2*a*c*d^2-b*d^2*(b+sqrt(b^2-4*a*c)))))*sqrt(2)/(sqrt(b^2-4*a*c)*sqrt(2*c^2+2*a*c*d^2-b*d^2*(b+sqrt(b^2-4*a*c))))],
[1/((a+b*x+c*x^2)^2*sqrt(1-d*x)*sqrt(1+d*x)),x,7,-(b*(b^2*d^2-c*(c+3*a*d^2))-c*(2*c^2-b^2*d^2+2*a*c*d^2)*x)*sqrt(1-d^2*x^2)/((b^2-4*a*c)*(b^2*d^2-(c+a*d^2)^2)*(a+b*x+c*x^2))-c*arctanh((2*c+d^2*x*(b-sqrt(b^2-4*a*c)))/(sqrt(2)*sqrt(1-d^2*x^2)*sqrt(2*c^2+2*a*c*d^2-b*d^2*(b-sqrt(b^2-4*a*c)))))*(4*c^3+12*a*c^2*d^2-a*b*d^4*(b+sqrt(b^2-4*a*c))-c*d^2*(5*b^2-8*a^2*d^2-b*sqrt(b^2-4*a*c)))/((b^2-4*a*c)^(3/2)*(b^2*d^2-(c+a*d^2)^2)*sqrt(2)*sqrt(2*c^2+2*a*c*d^2-b*d^2*(b-sqrt(b^2-4*a*c))))+c*arctanh((2*c+d^2*x*(b+sqrt(b^2-4*a*c)))/(sqrt(2)*sqrt(1-d^2*x^2)*sqrt(2*c^2+2*a*c*d^2-b*d^2*(b+sqrt(b^2-4*a*c)))))*(4*c^3+12*a*c^2*d^2-2*a*b^2*d^4-4*c*d^2*(b^2-2*a^2*d^2)-b*d^2*(c-a*d^2)*(b+sqrt(b^2-4*a*c)))/((b^2-4*a*c)^(3/2)*(b^2*d^2-(c+a*d^2)^2)*sqrt(2)*sqrt(2*c^2+2*a*c*d^2-b*d^2*(b+sqrt(b^2-4*a*c))))],
[(a+b*x+c*x^2)^3/((1-d*x)^(3/2)*(1+d*x)^(3/2)),x,7,-3/8*(5*c^3+12*b^2*c*d^2+12*a*c^2*d^2+8*a*b^2*d^4+8*a^2*c*d^4)*arcsin(d*x)/d^7+(b*(3*a^2+3*c^2/d^4+b^2/d^2+6*a*c/d^2)*d^4+(c+a*d^2)*(c^2+3*b^2*d^2+2*a*c*d^2+a^2*d^4)*x)/(d^6*sqrt(1-d^2*x^2))+b*(5*c^2+b^2*d^2+6*a*c*d^2)*sqrt(1-d^2*x^2)/d^6+1/8*c*(7*c^2+12*b^2*d^2+12*a*c*d^2)*x*sqrt(1-d^2*x^2)/d^6+b*c^2*x^2*sqrt(1-d^2*x^2)/d^4+1/4*c^3*x^3*sqrt(1-d^2*x^2)/d^4],
[(a+b*x+c*x^2)^2/((1-d*x)^(3/2)*(1+d*x)^(3/2)),x,5,-1/2*(2*b^2+c*(4*a+3*c/d^2))*arcsin(d*x)/d^3+(2*b*(a+c/d^2)*d^2+(c^2+b^2*d^2+2*a*c*d^2+a^2*d^4)*x)/(d^4*sqrt(1-d^2*x^2))+2*b*c*sqrt(1-d^2*x^2)/d^4+1/2*c^2*x*sqrt(1-d^2*x^2)/d^4],
[(a+b*x+c*x^2)/((1-d*x)^(3/2)*(1+d*x)^(3/2)),x,4,-c*arcsin(d*x)/d^3+(b+(c+a*d^2)*x)/(d^2*sqrt(1-d^2*x^2))],
[1/((1-d*x)^(3/2)*(1+d*x)^(3/2)*(a+b*x+c*x^2)),x,7,d^2*(b-(c+a*d^2)*x)/((b^2*d^2-(c+a*d^2)^2)*sqrt(1-d^2*x^2))+c*arctanh((2*c+d^2*x*(b-sqrt(b^2-4*a*c)))/(sqrt(2)*sqrt(1-d^2*x^2)*sqrt(2*c^2+2*a*c*d^2-b*d^2*(b-sqrt(b^2-4*a*c)))))*(2*c^2+2*a*c*d^2-b*d^2*(b+sqrt(b^2-4*a*c)))/((b^2*d^2-(c+a*d^2)^2)*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(2*c^2+2*a*c*d^2-b*d^2*(b-sqrt(b^2-4*a*c))))-c*arctanh((2*c+d^2*x*(b+sqrt(b^2-4*a*c)))/(sqrt(2)*sqrt(1-d^2*x^2)*sqrt(2*c^2+2*a*c*d^2-b*d^2*(b+sqrt(b^2-4*a*c)))))*(2*c^2+2*a*c*d^2-b*d^2*(b-sqrt(b^2-4*a*c)))/((b^2*d^2-(c+a*d^2)^2)*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(2*c^2+2*a*c*d^2-b*d^2*(b+sqrt(b^2-4*a*c))))],
[1/((1-d*x)^(3/2)*(1+d*x)^(3/2)*(a+b*x+c*x^2)^2),x,8,-d^2*(b*(c^3+2*b^2*c*d^2-10*a*c^2*d^2+3*a*b^2*d^4-11*a^2*c*d^4)-(2*c^4+b^2*d^4*(2*b^2+a^2*d^2)-c^2*d^2*(b^2+6*a^2*d^2)-c*(6*a*b^2*d^4+4*a^3*d^6))*x)/((b^2-4*a*c)*(c-b*d+a*d^2)^2*(c+b*d+a*d^2)^2*sqrt(1-d^2*x^2))+(-b*(b^2*d^2-c*(c+3*a*d^2))+c*(2*c^2-b^2*d^2+2*a*c*d^2)*x)/((b^2-4*a*c)*(b^2*d^2-(c+a*d^2)^2)*(a+b*x+c*x^2)*sqrt(1-d^2*x^2))+c*arctanh((2*c+d^2*x*(b-sqrt(b^2-4*a*c)))/(sqrt(2)*sqrt(1-d^2*x^2)*sqrt(2*c^2+2*a*c*d^2-b*d^2*(b-sqrt(b^2-4*a*c)))))*(4*c^5+24*a*c^4*d^2+3*a*b^3*d^6*(b+sqrt(b^2-4*a*c))-c^3*d^2*(9*b^2-36*a^2*d^2-b*sqrt(b^2-4*a*c))-2*a*c^2*d^4*(7*b^2-8*a^2*d^2+5*b*sqrt(b^2-4*a*c))+b*c*d^4*(2*b^3-17*a^2*b*d^2+2*b^2*sqrt(b^2-4*a*c)-11*a^2*d^2*sqrt(b^2-4*a*c)))/((b^2-4*a*c)^(3/2)*(c^2-b^2*d^2+2*a*c*d^2+a^2*d^4)^2*sqrt(2)*sqrt(2*c^2+2*a*c*d^2-b*d^2*(b-sqrt(b^2-4*a*c))))+c*arctanh((2*c+d^2*x*(b+sqrt(b^2-4*a*c)))/(sqrt(2)*sqrt(1-d^2*x^2)*sqrt(2*c^2+2*a*c*d^2-b*d^2*(b+sqrt(b^2-4*a*c)))))*(-2*(2*c^5*d^2+12*a*c^4*d^4+3*a*b^4*d^8+2*b^2*c*d^6*(b^2-7*a^2*d^2)-c^3*(4*b^2*d^4-18*a^2*d^6)-4*c^2*(3*a*b^2*d^6-2*a^3*d^8))+b*d^4*(c^3+2*b^2*c*d^2-10*a*c^2*d^2+3*a*b^2*d^4-11*a^2*c*d^4)*(b+sqrt(b^2-4*a*c)))/((b^2-4*a*c)^(3/2)*d^2*(c^2-b^2*d^2+2*a*c*d^2+a^2*d^4)^2*sqrt(2)*sqrt(2*c^2+2*a*c*d^2-b*d^2*(b+sqrt(b^2-4*a*c))))],

# Integrands of the form (1+d x)^m (1-d x)^m (a+b x+c x^2)^p when m symbolic
[(1-e*x)^m*(1+e*x)^m*(a+c*x^2)^p,x,3,x*(a+c*x^2)^p*AppellF1(1/2,-p,-m,3/2,-c*x^2/a,e^2*x^2)/(1+c*x^2/a)^p],
[(d-e*x)^m*(d+e*x)^m*(a+c*x^2)^p,x,4,x*(d-e*x)^m*(d+e*x)^m*(a+c*x^2)^p*AppellF1(1/2,-p,-m,3/2,-c*x^2/a,e^2*x^2/d^2)/((1+c*x^2/a)^p*(1-e^2*x^2/d^2)^m)],
[(d+e*x)^m*(d*f-e*f*x)^m*(a+c*x^2)^p,x,4,x*(d+e*x)^m*(d*f-e*f*x)^m*(a+c*x^2)^p*AppellF1(1/2,-p,-m,3/2,-c*x^2/a,e^2*x^2/d^2)/((1+c*x^2/a)^p*(1-e^2*x^2/d^2)^m)],

# Integrands of the form (d+e x)^m (f+g x)^n (a+b x+c x^2)^p when 2 c d-b e=0
[(d+e*x)^3*(f+g*x)^n*(a+2*c*d*x+c*e*x^2),x,2,-(e*f-d*g)^3*(a*g^2+c*f*(e*f-2*d*g))*(f+g*x)^(1+n)/(g^6*(1+n))+(e*f-d*g)^2*(3*a*e*g^2+c*(5*e^2*f^2-10*d*e*f*g+2*d^2*g^2))*(f+g*x)^(2+n)/(g^6*(2+n))-e*(e*f-d*g)*(3*a*e*g^2+c*(10*e^2*f^2-20*d*e*f*g+7*d^2*g^2))*(f+g*x)^(3+n)/(g^6*(3+n))+e^2*(a*e*g^2+c*(10*e^2*f^2-20*d*e*f*g+9*d^2*g^2))*(f+g*x)^(4+n)/(g^6*(4+n))-5*c*e^3*(e*f-d*g)*(f+g*x)^(5+n)/(g^6*(5+n))+c*e^4*(f+g*x)^(6+n)/(g^6*(6+n))],
[(d+e*x)^2*(f+g*x)^n*(a+2*c*d*x+c*e*x^2),x,2,(e*f-d*g)^2*(a*g^2+c*f*(e*f-2*d*g))*(f+g*x)^(1+n)/(g^5*(1+n))-2*(e*f-d*g)*(a*e*g^2+c*(2*e^2*f^2-4*d*e*f*g+d^2*g^2))*(f+g*x)^(2+n)/(g^5*(2+n))+e*(a*e*g^2+c*(6*e^2*f^2-12*d*e*f*g+5*d^2*g^2))*(f+g*x)^(3+n)/(g^5*(3+n))-4*c*e^2*(e*f-d*g)*(f+g*x)^(4+n)/(g^5*(4+n))+c*e^3*(f+g*x)^(5+n)/(g^5*(5+n))],
[(d+e*x)*(f+g*x)^n*(a+2*c*d*x+c*e*x^2),x,2,-(e*f-d*g)*(a*g^2+c*f*(e*f-2*d*g))*(f+g*x)^(1+n)/(g^4*(1+n))+(a*e*g^2+c*(3*e^2*f^2-6*d*e*f*g+2*d^2*g^2))*(f+g*x)^(2+n)/(g^4*(2+n))-3*c*e*(e*f-d*g)*(f+g*x)^(3+n)/(g^4*(3+n))+c*e^2*(f+g*x)^(4+n)/(g^4*(4+n))],
[(f+g*x)^n*(a+2*c*d*x+c*e*x^2),x,2,(a*g^2+c*f*(e*f-2*d*g))*(f+g*x)^(1+n)/(g^3*(1+n))-2*c*(e*f-d*g)*(f+g*x)^(2+n)/(g^3*(2+n))+c*e*(f+g*x)^(3+n)/(g^3*(3+n))],
[(f+g*x)^n*(a+2*c*d*x+c*e*x^2)/(d+e*x),x,3,-c*(e*f-d*g)*(f+g*x)^(1+n)/(e*g^2*(1+n))+c*(f+g*x)^(2+n)/(g^2*(2+n))+(c*d^2-a*e)*(f+g*x)^(1+n)*hypergeom([1,1+n],[2+n],e*(f+g*x)/(e*f-d*g))/(e*(e*f-d*g)*(1+n))],
[(f+g*x)^n*(a+2*c*d*x+c*e*x^2)/(d+e*x)^2,x,3,c*(f+g*x)^(1+n)/(e*g*(1+n))-(c*d^2-a*e)*g*(f+g*x)^(1+n)*hypergeom([2,1+n],[2+n],e*(f+g*x)/(e*f-d*g))/(e*(e*f-d*g)^2*(1+n))],
[(f+g*x)^n*(a+2*c*d*x+c*e*x^2)/(d+e*x)^3,x,3,-1/2*(a-c*d^2/e)*(f+g*x)^(1+n)/((e*f-d*g)*(d+e*x)^2)-1/2*(c*d^2-a*e)*g*(1-n)*(f+g*x)^(1+n)/(e*(e*f-d*g)^2*(d+e*x))+1/2*(a*e*g^2*(1-n)*n-c*(2*e^2*f^2-4*d*e*f*g+d^2*g^2*(2+n-n^2)))*(f+g*x)^(1+n)*hypergeom([1,1+n],[2+n],e*(f+g*x)/(e*f-d*g))/(e*(e*f-d*g)^3*(1+n))],
[(f+g*x)^n*(a+2*c*d*x+c*e*x^2)/(d+e*x)^4,x,3,-1/3*(a-c*d^2/e)*(f+g*x)^(1+n)/((e*f-d*g)*(d+e*x)^3)-1/6*(c*d^2-a*e)*g*(2-n)*(f+g*x)^(1+n)/(e*(e*f-d*g)^2*(d+e*x)^2)+1/6*g*(a*e*g^2*(2-3*n+n^2)+c*(6*e^2*f^2-12*d*e*f*g+d^2*g^2*(4+3*n-n^2)))*(f+g*x)^(1+n)*hypergeom([2,1+n],[2+n],e*(f+g*x)/(e*f-d*g))/(e*(e*f-d*g)^4*(1+n))],
[(d+e*x)^m*(f+g*x)^n*(a+2*c*d*x+c*e*x^2),x,4,-c*(e*f-d*g)*(2+m)*(d+e*x)^(1+m)*(f+g*x)^(1+n)/(e*g^2*(2+m+n)*(3+m+n))+c*(d+e*x)^(2+m)*(f+g*x)^(1+n)/(e*g*(3+m+n))+(c*(e*f-d*g)*(2+m)*(e*f*(1+m)+d*g*(1+n))+g*(2+m+n)*(a*e*g*(3+m+n)-c*d*(e*f*(2+m)+d*g*(1+n))))*(d+e*x)^(1+m)*(f+g*x)^n*hypergeom([1+m,-n],[2+m],-g*(d+e*x)/(e*f-d*g))/(e^2*g^2*(1+m)*(2+m+n)*(3+m+n)*(e*(f+g*x)/(e*f-d*g))^n)],

# Integrands of the form (d+e x)^m (f+g x)^n (a+b x+c x^2)^p

# Integrands of the form (d+e x)^m (f+g x)^n (a+b x+c x^2)^p

# p>0
[(a+b*x+c*x^2)/((d+e*x)*(f+g*x)),x,2,c*x/(e*g)+(c*d^2-b*d*e+a*e^2)*log(d+e*x)/(e^2*(e*f-d*g))-(c*f^2-b*f*g+a*g^2)*log(f+g*x)/(g^2*(e*f-d*g))],
[(a+b*x+c*x^2)^2/((d+e*x)*(f+g*x)),x,2,(b^2*e^2*g^2-2*c*e*g*(b*e*f+b*d*g-a*e*g)+c^2*(e^2*f^2+d*e*f*g+d^2*g^2))*x/(e^3*g^3)-1/2*c*(c*e*f+c*d*g-2*b*e*g)*x^2/(e^2*g^2)+1/3*c^2*x^3/(e*g)+(c*d^2-b*d*e+a*e^2)^2*log(d+e*x)/(e^4*(e*f-d*g))-(c*f^2-b*f*g+a*g^2)^2*log(f+g*x)/(g^4*(e*f-d*g))],
[(a+b*x+c*x^2)^3/((d+e*x)*(f+g*x)),x,2,-(b^2*e^3*g^3*(b*e*f+b*d*g-3*a*e*g)-c^3*(e^4*f^4+d*e^3*f^3*g+d^2*e^2*f^2*g^2+d^3*e*f*g^3+d^4*g^4)-3*c*e^2*g^2*(a^2*e^2*g^2-2*a*b*e*g*(e*f+d*g)+b^2*(e^2*f^2+d*e*f*g+d^2*g^2))-3*c^2*e*g*(a*e*g*(e^2*f^2+d*e*f*g+d^2*g^2)-b*(e^3*f^3+d*e^2*f^2*g+d^2*e*f*g^2+d^3*g^3)))*x/(e^5*g^5)+1/2*(b^3*e^3*g^3-3*b*c*e^2*g^2*(b*e*f+b*d*g-2*a*e*g)-c^3*(e^3*f^3+d*e^2*f^2*g+d^2*e*f*g^2+d^3*g^3)-3*c^2*e*g*(a*e*g*(e*f+d*g)-b*(e^2*f^2+d*e*f*g+d^2*g^2)))*x^2/(e^4*g^4)+1/3*c*(3*b^2*e^2*g^2-3*c*e*g*(b*e*f+b*d*g-a*e*g)+c^2*(e^2*f^2+d*e*f*g+d^2*g^2))*x^3/(e^3*g^3)-1/4*c^2*(c*e*f+c*d*g-3*b*e*g)*x^4/(e^2*g^2)+1/5*c^3*x^5/(e*g)+(c*d^2-b*d*e+a*e^2)^3*log(d+e*x)/(e^6*(e*f-d*g))-(c*f^2-b*f*g+a*g^2)^3*log(f+g*x)/(g^6*(e*f-d*g))],

# p<0
[1/((d+e*x)*(f+g*x)*(a+b*x+c*x^2)),x,6,e^2*log(d+e*x)/((c*d^2-b*d*e+a*e^2)*(e*f-d*g))-g^2*log(f+g*x)/((e*f-d*g)*(c*f^2-b*f*g+a*g^2))-1/2*(c*e*f+c*d*g-b*e*g)*log(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)*(c*f^2-g*(b*f-a*g)))-(2*c^2*d*f+b^2*e*g-c*(b*e*f+b*d*g+2*a*e*g))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)*(c*f^2-g*(b*f-a*g))*sqrt(b^2-4*a*c))],
[1/((d+e*x)*(f+g*x)*(a+b*x+c*x^2)^2),x,9,(-b^3*e*g+b^2*c*(e*f+d*g)-2*a*c^2*(e*f+d*g)-b*c*(c*d*f-3*a*e*g)-c*(2*c^2*d*f+b^2*e*g-c*(b*e*f+b*d*g+2*a*e*g))*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(c*f^2-g*(b*f-a*g))*(a+b*x+c*x^2))+2*c*(2*c^2*d*f+b^2*e*g-c*(b*e*f+b*d*g+2*a*e*g))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((b^2-4*a*c)^(3/2)*(c*d^2-b*d*e+a*e^2)*(c*f^2-g*(b*f-a*g)))+e^4*log(d+e*x)/((c*d^2-b*d*e+a*e^2)^2*(e*f-d*g))-g^4*log(f+g*x)/((e*f-d*g)*(c*f^2-b*f*g+a*g^2)^2)-1/2*(c*e*f+c*d*g-b*e*g)*(c*(e^2*f^2+d^2*g^2)+e*g*(2*a*e*g-b*(e*f+d*g)))*log(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)^2*(c*f^2-g*(b*f-a*g))^2)+(b^2*e^2*g^2*(b*e*f+b*d*g-2*a*e*g)-2*c^3*d*f*(e^2*f^2+d*e*f*g+d^2*g^2)+2*c*e*g*(a^2*e^2*g^2+a*b*e*g*(e*f+d*g)-b^2*(e*f+d*g)^2)-c^2*(4*a*d*e^2*f*g^2-b*(e^3*f^3+5*d*e^2*f^2*g+5*d^2*e*f*g^2+d^3*g^3)))*arctanh((b+2*c*x)/sqrt(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)^2*(c*f^2-g*(b*f-a*g))^2*sqrt(b^2-4*a*c))],

# Integrands of the form (d+e x)^m (f+g x)^(n/2) (a+b x+c x^2)^p

# p>0 & n>0

# p>0 & n<0
[(d+e*x)^3*(a+b*x+c*x^2)/sqrt(f+g*x),x,3,2/3*(e*f-d*g)^2*(c*f*(5*e*f-2*d*g)-g*(4*b*e*f-b*d*g-3*a*e*g))*(f+g*x)^(3/2)/g^6+2/5*(e*f-d*g)*(3*e*g*(2*b*e*f-b*d*g-a*e*g)-c*(10*e^2*f^2-8*d*e*f*g+d^2*g^2))*(f+g*x)^(5/2)/g^6-2/7*e*(e*g*(4*b*e*f-3*b*d*g-a*e*g)-c*(10*e^2*f^2-12*d*e*f*g+3*d^2*g^2))*(f+g*x)^(7/2)/g^6-2/9*e^2*(5*c*e*f-3*c*d*g-b*e*g)*(f+g*x)^(9/2)/g^6+2/11*c*e^3*(f+g*x)^(11/2)/g^6-2*(e*f-d*g)^3*(c*f^2-b*f*g+a*g^2)*sqrt(f+g*x)/g^6],
[(d+e*x)^2*(a+b*x+c*x^2)/sqrt(f+g*x),x,3,-2/3*(e*f-d*g)*(2*c*f*(2*e*f-d*g)-g*(3*b*e*f-b*d*g-2*a*e*g))*(f+g*x)^(3/2)/g^5-2/5*(e*g*(3*b*e*f-2*b*d*g-a*e*g)-c*(6*e^2*f^2-6*d*e*f*g+d^2*g^2))*(f+g*x)^(5/2)/g^5-2/7*e*(4*c*e*f-2*c*d*g-b*e*g)*(f+g*x)^(7/2)/g^5+2/9*c*e^2*(f+g*x)^(9/2)/g^5+2*(e*f-d*g)^2*(c*f^2-b*f*g+a*g^2)*sqrt(f+g*x)/g^5],
[(d+e*x)*(a+b*x+c*x^2)/sqrt(f+g*x),x,2,2/3*(c*f*(3*e*f-2*d*g)-g*(2*b*e*f-b*d*g-a*e*g))*(f+g*x)^(3/2)/g^4-2/5*(3*c*e*f-c*d*g-b*e*g)*(f+g*x)^(5/2)/g^4+2/7*c*e*(f+g*x)^(7/2)/g^4-2*(e*f-d*g)*(c*f^2-b*f*g+a*g^2)*sqrt(f+g*x)/g^4],
[(a+b*x+c*x^2)/sqrt(f+g*x),x,2,-2/3*(2*c*f-b*g)*(f+g*x)^(3/2)/g^3+2/5*c*(f+g*x)^(5/2)/g^3+2*(c*f^2-b*f*g+a*g^2)*sqrt(f+g*x)/g^3],
[(a+b*x+c*x^2)/((d+e*x)*sqrt(f+g*x)),x,4,2/3*c*(f+g*x)^(3/2)/(e*g^2)-2*(c*d^2-b*d*e+a*e^2)*arctanh(sqrt(e)*sqrt(f+g*x)/sqrt(e*f-d*g))/(e^(5/2)*sqrt(e*f-d*g))+2*(b*e*g-c*(e*f+d*g))*sqrt(f+g*x)/(e^2*g^2)],
[(a+b*x+c*x^2)/((d+e*x)^2*sqrt(f+g*x)),x,4,(c*d*(4*e*f-3*d*g)-e*(2*b*e*f-b*d*g-a*e*g))*arctanh(sqrt(e)*sqrt(f+g*x)/sqrt(e*f-d*g))/(e^(5/2)*(e*f-d*g)^(3/2))+2*c*sqrt(f+g*x)/(e^2*g)-(a+d*(c*d-b*e)/e^2)*sqrt(f+g*x)/((e*f-d*g)*(d+e*x))],
[(a+b*x+c*x^2)/((d+e*x)^3*sqrt(f+g*x)),x,4,1/4*(e*g*(4*b*e*f-b*d*g-3*a*e*g)-c*(8*e^2*f^2-8*d*e*f*g+3*d^2*g^2))*arctanh(sqrt(e)*sqrt(f+g*x)/sqrt(e*f-d*g))/(e^(5/2)*(e*f-d*g)^(5/2))-1/2*(a+d*(c*d-b*e)/e^2)*sqrt(f+g*x)/((e*f-d*g)*(d+e*x)^2)+1/4*(c*d*(8*e*f-5*d*g)-e*(4*b*e*f-b*d*g-3*a*e*g))*sqrt(f+g*x)/(e^2*(e*f-d*g)^2*(d+e*x))],
[(d+e*x)^3*(a+b*x+c*x^2)/(f+g*x)^(3/2),x,3,2/3*(e*f-d*g)*(3*e*g*(2*b*e*f-b*d*g-a*e*g)-c*(10*e^2*f^2-8*d*e*f*g+d^2*g^2))*(f+g*x)^(3/2)/g^6-2/5*e*(e*g*(4*b*e*f-3*b*d*g-a*e*g)-c*(10*e^2*f^2-12*d*e*f*g+3*d^2*g^2))*(f+g*x)^(5/2)/g^6-2/7*e^2*(5*c*e*f-3*c*d*g-b*e*g)*(f+g*x)^(7/2)/g^6+2/9*c*e^3*(f+g*x)^(9/2)/g^6+2*(e*f-d*g)^3*(c*f^2-b*f*g+a*g^2)/(g^6*sqrt(f+g*x))+2*(e*f-d*g)^2*(c*f*(5*e*f-2*d*g)-g*(4*b*e*f-b*d*g-3*a*e*g))*sqrt(f+g*x)/g^6],
[(d+e*x)^2*(a+b*x+c*x^2)/(f+g*x)^(3/2),x,3,-2/3*(e*g*(3*b*e*f-2*b*d*g-a*e*g)-c*(6*e^2*f^2-6*d*e*f*g+d^2*g^2))*(f+g*x)^(3/2)/g^5-2/5*e*(4*c*e*f-2*c*d*g-b*e*g)*(f+g*x)^(5/2)/g^5+2/7*c*e^2*(f+g*x)^(7/2)/g^5-2*(e*f-d*g)^2*(c*f^2-b*f*g+a*g^2)/(g^5*sqrt(f+g*x))-2*(e*f-d*g)*(2*c*f*(2*e*f-d*g)-g*(3*b*e*f-b*d*g-2*a*e*g))*sqrt(f+g*x)/g^5],
[(d+e*x)*(a+b*x+c*x^2)/(f+g*x)^(3/2),x,2,-2/3*(3*c*e*f-c*d*g-b*e*g)*(f+g*x)^(3/2)/g^4+2/5*c*e*(f+g*x)^(5/2)/g^4+2*(e*f-d*g)*(c*f^2-b*f*g+a*g^2)/(g^4*sqrt(f+g*x))+2*(c*f*(3*e*f-2*d*g)-g*(2*b*e*f-b*d*g-a*e*g))*sqrt(f+g*x)/g^4],
[(a+b*x+c*x^2)/(f+g*x)^(3/2),x,2,2/3*c*(f+g*x)^(3/2)/g^3-2*(c*f^2-b*f*g+a*g^2)/(g^3*sqrt(f+g*x))-2*(2*c*f-b*g)*sqrt(f+g*x)/g^3],
[(a+b*x+c*x^2)/((d+e*x)*(f+g*x)^(3/2)),x,4,-2*(c*d^2-b*d*e+a*e^2)*arctanh(sqrt(e)*sqrt(f+g*x)/sqrt(e*f-d*g))/(e^(3/2)*(e*f-d*g)^(3/2))+2*(c*f^2-b*f*g+a*g^2)/(g^2*(e*f-d*g)*sqrt(f+g*x))+2*c*sqrt(f+g*x)/(e*g^2)],
[(a+b*x+c*x^2)/((d+e*x)^2*(f+g*x)^(3/2)),x,4,(c*d*(4*e*f-d*g)-e*(2*b*e*f+b*d*g-3*a*e*g))*arctanh(sqrt(e)*sqrt(f+g*x)/sqrt(e*f-d*g))/(e^(3/2)*(e*f-d*g)^(5/2))-2*(c*f^2-b*f*g+a*g^2)/(g*(e*f-d*g)^2*sqrt(f+g*x))-(c*d^2-b*d*e+a*e^2)*sqrt(f+g*x)/(e*(e*f-d*g)^2*(d+e*x))],
[(a+b*x+c*x^2)/((d+e*x)^3*(f+g*x)^(3/2)),x,5,-1/4*(c*(8*e^2*f^2+8*d*e*f*g-d^2*g^2)+3*e*g*(5*a*e*g-b*(4*e*f+d*g)))*arctanh(sqrt(e)*sqrt(f+g*x)/sqrt(e*f-d*g))/(e^(3/2)*(e*f-d*g)^(7/2))+2*(c*f^2-b*f*g+a*g^2)/((e*f-d*g)^3*sqrt(f+g*x))-1/2*(c*d^2-b*d*e+a*e^2)*sqrt(f+g*x)/(e*(e*f-d*g)^2*(d+e*x)^2)+1/4*(c*d*(8*e*f-d*g)-e*(4*b*e*f+3*b*d*g-7*a*e*g))*sqrt(f+g*x)/(e*(e*f-d*g)^3*(d+e*x))],

# p<0

# Integrands of the form (d+e x)^(m/2) (f+g x)^(n/2) (a+b x+c x^2)^p

# p>0
[sqrt(-1+x)*sqrt(1+x)/(1+x-x^2),x,-9,-arccosh(x)+arctan(sqrt(1+x)/(sqrt(-1+x)*sqrt(-2+sqrt(5))))*sqrt(2/5*(-1+sqrt(5)))+arctanh(sqrt(1+x)/(sqrt(-1+x)*sqrt(2+sqrt(5))))*sqrt(2/5*(1+sqrt(5)))],
[(a+b*x+c*x^2)/(sqrt(d+e*x)*sqrt(f+g*x)),x,5,1/4*(c*(3*e^2*f^2+2*d*e*f*g+3*d^2*g^2)+4*e*g*(2*a*e*g-b*(e*f+d*g)))*arctanh(sqrt(g)*sqrt(d+e*x)/(sqrt(e)*sqrt(f+g*x)))/(e^(5/2)*g^(5/2))+1/2*c*(d+e*x)^(3/2)*sqrt(f+g*x)/(e^2*g)-1/4*(3*c*e*f+5*c*d*g-4*b*e*g)*sqrt(d+e*x)*sqrt(f+g*x)/(e^2*g^2)],
[(d+e*x)^(3/2)*(a+b*x+c*x^2)/sqrt(f+g*x),x,7,1/64*(e*f-d*g)^2*(c*(35*e^2*f^2+10*d*e*f*g+3*d^2*g^2)+8*e*g*(6*a*e*g-b*(5*e*f+d*g)))*arctanh(sqrt(g)*sqrt(d+e*x)/(sqrt(e)*sqrt(f+g*x)))/(e^(5/2)*g^(9/2))+1/96*(c*(35*e^2*f^2+10*d*e*f*g+3*d^2*g^2)+8*e*g*(6*a*e*g-b*(5*e*f+d*g)))*(d+e*x)^(3/2)*sqrt(f+g*x)/(e^2*g^3)-1/24*(7*c*e*f+9*c*d*g-8*b*e*g)*(d+e*x)^(5/2)*sqrt(f+g*x)/(e^2*g^2)+1/4*c*(d+e*x)^(7/2)*sqrt(f+g*x)/(e^2*g)-1/64*(e*f-d*g)*(c*(35*e^2*f^2+10*d*e*f*g+3*d^2*g^2)+8*e*g*(6*a*e*g-b*(5*e*f+d*g)))*sqrt(d+e*x)*sqrt(f+g*x)/(e^2*g^4)],
[(d+e*x)^(1/2)*(a+b*x+c*x^2)/sqrt(f+g*x),x,6,-1/8*(e*f-d*g)*(c*(5*e^2*f^2+2*d*e*f*g+d^2*g^2)+2*e*g*(4*a*e*g-b*(3*e*f+d*g)))*arctanh(sqrt(g)*sqrt(d+e*x)/(sqrt(e)*sqrt(f+g*x)))/(e^(5/2)*g^(7/2))-1/12*(5*c*e*f+7*c*d*g-6*b*e*g)*(d+e*x)^(3/2)*sqrt(f+g*x)/(e^2*g^2)+1/3*c*(d+e*x)^(5/2)*sqrt(f+g*x)/(e^2*g)+1/8*(c*(5*e^2*f^2+2*d*e*f*g+d^2*g^2)+2*e*g*(4*a*e*g-b*(3*e*f+d*g)))*sqrt(d+e*x)*sqrt(f+g*x)/(e^2*g^3)],
[(a+b*x+c*x^2)/((d+e*x)^(1/2)*sqrt(f+g*x)),x,5,1/4*(c*(3*e^2*f^2+2*d*e*f*g+3*d^2*g^2)+4*e*g*(2*a*e*g-b*(e*f+d*g)))*arctanh(sqrt(g)*sqrt(d+e*x)/(sqrt(e)*sqrt(f+g*x)))/(e^(5/2)*g^(5/2))+1/2*c*(d+e*x)^(3/2)*sqrt(f+g*x)/(e^2*g)-1/4*(3*c*e*f+5*c*d*g-4*b*e*g)*sqrt(d+e*x)*sqrt(f+g*x)/(e^2*g^2)],
[(a+b*x+c*x^2)/((d+e*x)^(3/2)*sqrt(f+g*x)),x,5,-(c*e*f+3*c*d*g-2*b*e*g)*arctanh(sqrt(g)*sqrt(d+e*x)/(sqrt(e)*sqrt(f+g*x)))/(e^(5/2)*g^(3/2))-2*(a+d*(c*d-b*e)/e^2)*sqrt(f+g*x)/((e*f-d*g)*sqrt(d+e*x))+c*sqrt(d+e*x)*sqrt(f+g*x)/(e^2*g)],
[(a+b*x+c*x^2)/((d+e*x)^(5/2)*sqrt(f+g*x)),x,5,2*c*arctanh(sqrt(g)*sqrt(d+e*x)/(sqrt(e)*sqrt(f+g*x)))/(e^(5/2)*sqrt(g))-2/3*(a+d*(c*d-b*e)/e^2)*sqrt(f+g*x)/((e*f-d*g)*(d+e*x)^(3/2))+2/3*(c*(6*d*e*f-4*d^2*g)-e*(3*b*e*f-b*d*g-2*a*e*g))*sqrt(f+g*x)/(e^2*(e*f-d*g)^2*sqrt(d+e*x))],
[(a+b*x+c*x^2)/((d+e*x)^(7/2)*sqrt(f+g*x)),x,3,-2/5*(a+d*(c*d-b*e)/e^2)*sqrt(f+g*x)/((e*f-d*g)*(d+e*x)^(5/2))+2/15*(2*c*d*(5*e*f-3*d*g)-e*(5*b*e*f-b*d*g-4*a*e*g))*sqrt(f+g*x)/(e^2*(e*f-d*g)^2*(d+e*x)^(3/2))+2/15*(2*e*g*(5*b*e*f-b*d*g-4*a*e*g)-c*(15*e^2*f^2-10*d*e*f*g+3*d^2*g^2))*sqrt(f+g*x)/(e^2*(e*f-d*g)^3*sqrt(d+e*x))],
[(a+b*x+c*x^2)/((d+e*x)^(9/2)*sqrt(f+g*x)),x,4,-2/7*(a+d*(c*d-b*e)/e^2)*sqrt(f+g*x)/((e*f-d*g)*(d+e*x)^(7/2))+2/35*(2*c*d*(7*e*f-4*d*g)-e*(7*b*e*f-b*d*g-6*a*e*g))*sqrt(f+g*x)/(e^2*(e*f-d*g)^2*(d+e*x)^(5/2))+2/105*(4*e*g*(7*b*e*f-b*d*g-6*a*e*g)-c*(35*e^2*f^2-14*d*e*f*g+3*d^2*g^2))*sqrt(f+g*x)/(e^2*(e*f-d*g)^3*(d+e*x)^(3/2))-4/105*g*(4*e*g*(7*b*e*f-b*d*g-6*a*e*g)-c*(35*e^2*f^2-14*d*e*f*g+3*d^2*g^2))*sqrt(f+g*x)/(e^2*(e*f-d*g)^4*sqrt(d+e*x))],
[(d+e*x)^(1/2)*(a+b*x+c*x^2)/(e+f*x)^(3/2),x,6,-1/4*(4*e*f*(3*b*e^2-b*d*f-2*a*e*f)-c*(15*e^4-6*d*e^2*f-d^2*f^2))*arctanh(sqrt(f)*sqrt(d+e*x)/(sqrt(e)*sqrt(e+f*x)))/(e^(3/2)*f^(7/2))+2*(a+e*(c*e-b*f)/f^2)*(d+e*x)^(3/2)/((e^2-d*f)*sqrt(e+f*x))+1/2*c*(d+e*x)^(3/2)*sqrt(e+f*x)/(e*f^2)+1/4*(4*e*f*(3*b*e^2-b*d*f-2*a*e*f)-c*(15*e^4-6*d*e^2*f-d^2*f^2))*sqrt(d+e*x)*sqrt(e+f*x)/(e*f^3*(e^2-d*f))],
[(d+e*x)^(3/2)*(15*d^2+20*d*e*x+8*e^2*x^2)/sqrt(a+b*x),x,7,2*e*(a+b*x)^(3/2)*(d+e*x)^(5/2)/b^2+1/8*(b*d-a*e)^2*(73*b^2*d^2-90*a*b*d*e+35*a^2*e^2)*arctanh(sqrt(e)*sqrt(a+b*x)/(sqrt(b)*sqrt(d+e*x)))/(b^(9/2)*sqrt(e))+1/12*(73*b^2*d^2-90*a*b*d*e+35*a^2*e^2)*(d+e*x)^(3/2)*sqrt(a+b*x)/b^3+1/3*(17*b*d-13*a*e)*(d+e*x)^(5/2)*sqrt(a+b*x)/b^2+1/8*(b*d-a*e)*(73*b^2*d^2-90*a*b*d*e+35*a^2*e^2)*sqrt(a+b*x)*sqrt(d+e*x)/b^4],
[(d+e*x)^(1/2)*(15*d^2+20*d*e*x+8*e^2*x^2)/sqrt(a+b*x),x,6,8/3*e*(a+b*x)^(3/2)*(d+e*x)^(3/2)/b^2+(b*d-a*e)*(11*b^2*d^2-13*a*b*d*e+5*a^2*e^2)*arctanh(sqrt(e)*sqrt(a+b*x)/(sqrt(b)*sqrt(d+e*x)))/(b^(7/2)*sqrt(e))+2*(4*b*d-3*a*e)*(d+e*x)^(3/2)*sqrt(a+b*x)/b^2+(11*b^2*d^2-13*a*b*d*e+5*a^2*e^2)*sqrt(a+b*x)*sqrt(d+e*x)/b^3],
[(15*d^2+20*d*e*x+8*e^2*x^2)/((d+e*x)^(1/2)*sqrt(a+b*x)),x,5,2*(8*b^2*d^2-8*a*b*d*e+3*a^2*e^2)*arctanh(sqrt(e)*sqrt(a+b*x)/(sqrt(b)*sqrt(d+e*x)))/(b^(5/2)*sqrt(e))+4*e*(a+b*x)^(3/2)*sqrt(d+e*x)/b^2+2*(7*b*d-5*a*e)*sqrt(a+b*x)*sqrt(d+e*x)/b^2],
[(15*d^2+20*d*e*x+8*e^2*x^2)/((d+e*x)^(3/2)*sqrt(a+b*x)),x,5,8*(2*b*d-a*e)*arctanh(sqrt(e)*sqrt(a+b*x)/(sqrt(b)*sqrt(d+e*x)))/(b^(3/2)*sqrt(e))+6*d^2*sqrt(a+b*x)/((b*d-a*e)*sqrt(d+e*x))+8*sqrt(a+b*x)*sqrt(d+e*x)/b],
[(15*d^2+20*d*e*x+8*e^2*x^2)/((d+e*x)^(5/2)*sqrt(a+b*x)),x,5,16*arctanh(sqrt(e)*sqrt(a+b*x)/(sqrt(b)*sqrt(d+e*x)))/(sqrt(b)*sqrt(e))+2*d^2*sqrt(a+b*x)/((b*d-a*e)*(d+e*x)^(3/2))+4*d*(3*b*d-2*a*e)*sqrt(a+b*x)/((b*d-a*e)^2*sqrt(d+e*x))],
[(15*d^2+20*d*e*x+8*e^2*x^2)/((d+e*x)^(7/2)*sqrt(a+b*x)),x,3,6/5*d^2*sqrt(a+b*x)/((b*d-a*e)*(d+e*x)^(5/2))+8/15*d*(8*b*d-5*a*e)*sqrt(a+b*x)/((b*d-a*e)^2*(d+e*x)^(3/2))+16/15*(23*b^2*d^2-35*a*b*d*e+15*a^2*e^2)*sqrt(a+b*x)/((b*d-a*e)^3*sqrt(d+e*x))],
[(15*d^2+20*d*e*x+8*e^2*x^2)/((d+e*x)^(9/2)*sqrt(a+b*x)),x,4,6/7*d^2*sqrt(a+b*x)/((b*d-a*e)*(d+e*x)^(7/2))+4/35*d*(23*b*d-14*a*e)*sqrt(a+b*x)/((b*d-a*e)^2*(d+e*x)^(5/2))+16/105*(58*b^2*d^2-84*a*b*d*e+35*a^2*e^2)*sqrt(a+b*x)/((b*d-a*e)^3*(d+e*x)^(3/2))+32/105*b*(58*b^2*d^2-84*a*b*d*e+35*a^2*e^2)*sqrt(a+b*x)/((b*d-a*e)^4*sqrt(d+e*x))],

# p<0
[(d+e*x)^(3/2)/((a+b*x+c*x^2)*sqrt(f+g*x)),x,11,2*e^(3/2)*arctanh(sqrt(g)*sqrt(d+e*x)/(sqrt(e)*sqrt(f+g*x)))/(c*sqrt(g))-2*arctanh(sqrt(d+e*x)*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))/(sqrt(f+g*x)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))))*(e*(2*c*d-b*e)+(2*c^2*d^2+b^2*e^2-2*c*e*(b*d+a*e))/sqrt(b^2-4*a*c))/(c*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))))-2*arctanh(sqrt(d+e*x)*sqrt(2*c*f-g*(b+sqrt(b^2-4*a*c)))/(sqrt(f+g*x)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*(e*(2*c*d-b*e)+(-2*c^2*d^2-b^2*e^2+2*c*e*(b*d+a*e))/sqrt(b^2-4*a*c))/(c*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))*sqrt(2*c*f-g*(b+sqrt(b^2-4*a*c))))],
[(d+e*x)^(1/2)/((a+b*x+c*x^2)*sqrt(f+g*x)),x,6,-2*arctanh(sqrt(d+e*x)*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))/(sqrt(f+g*x)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))))*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(sqrt(b^2-4*a*c)*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))))+2*arctanh(sqrt(d+e*x)*sqrt(2*c*f-g*(b+sqrt(b^2-4*a*c)))/(sqrt(f+g*x)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(b^2-4*a*c)*sqrt(2*c*f-g*(b+sqrt(b^2-4*a*c))))],
[1/((d+e*x)^(1/2)*(a+b*x+c*x^2)*sqrt(f+g*x)),x,6,-4*c*arctanh(sqrt(d+e*x)*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))/(sqrt(f+g*x)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))))/(sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))))+4*c*arctanh(sqrt(d+e*x)*sqrt(2*c*f-g*(b+sqrt(b^2-4*a*c)))/(sqrt(f+g*x)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))))/(sqrt(b^2-4*a*c)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))*sqrt(2*c*f-g*(b+sqrt(b^2-4*a*c))))],
[1/((d+e*x)^(3/2)*(a+b*x+c*x^2)*sqrt(f+g*x)),x,8,4*c*e*sqrt(f+g*x)/((e*f-d*g)*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(b^2-4*a*c)*sqrt(d+e*x))-4*c*e*sqrt(f+g*x)/((e*f-d*g)*sqrt(b^2-4*a*c)*(2*c*d-e*(b+sqrt(b^2-4*a*c)))*sqrt(d+e*x))-8*c^2*arctanh(sqrt(d+e*x)*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))/(sqrt(f+g*x)*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))))/((2*c*d-e*(b-sqrt(b^2-4*a*c)))^(3/2)*sqrt(b^2-4*a*c)*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))))+8*c^2*arctanh(sqrt(d+e*x)*sqrt(2*c*f-g*(b+sqrt(b^2-4*a*c)))/(sqrt(f+g*x)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))))/(sqrt(b^2-4*a*c)*(2*c*d-e*(b+sqrt(b^2-4*a*c)))^(3/2)*sqrt(2*c*f-g*(b+sqrt(b^2-4*a*c))))],

# Integrands of the form (d+e x)^m (f+g x)^n (a+b x+c x^2)^(p/2)

# p>0
[(f+g*x)^3*sqrt(a+b*x+c*x^2)/(d+e*x),x,8,1/24*g^2*(24*c*e*f-14*c*d*g-5*b*e*g)*(a+b*x+c*x^2)^(3/2)/(c^2*e^2)+1/4*g^3*(d+e*x)*(a+b*x+c*x^2)^(3/2)/(c*e^2)-1/128*(4*c*e*(2*c*d-b*e)*(16*c^2*e^2*f^3+5*b^2*d*e*g^3-4*c*d*g^2*(6*b*e*f-2*b*d*g+a*e*g))-2*(4*c^2*d^2-1/2*b^2*e^2-2*c*e*(b*d-a*e))*g*(5*b^2*e^2*g^2-4*c*e*g*(6*b*e*f-2*b*d*g+a*e*g)+16*c^2*(3*e^2*f^2-3*d*e*f*g+d^2*g^2)))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(7/2)*e^5)+(e*f-d*g)^3*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))*sqrt(c*d^2-b*d*e+a*e^2)/e^5+1/64*(5*b^3*e^3*g^3+64*c^3*(e*f-d*g)^3-4*b*c*e^2*g^2*(6*b*e*f-2*b*d*g+a*e*g)+16*b*c^2*e*g*(3*e^2*f^2-3*d*e*f*g+d^2*g^2)+2*c*e*g*(5*b^2*e^2*g^2-4*c*e*g*(6*b*e*f-2*b*d*g+a*e*g)+16*c^2*(3*e^2*f^2-3*d*e*f*g+d^2*g^2))*x)*sqrt(a+b*x+c*x^2)/(c^3*e^4)],
[(f+g*x)^2*sqrt(a+b*x+c*x^2)/(d+e*x),x,7,1/3*g^2*(a+b*x+c*x^2)^(3/2)/(c*e)+1/16*((8*c^2*d^2-b^2*e^2-4*c*e*(b*d-a*e))*g*(4*c*e*f-2*c*d*g-b*e*g)-4*c*e*(2*c*d-b*e)*(2*c*e*f^2-b*d*g^2))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(5/2)*e^4)+(e*f-d*g)^2*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))*sqrt(c*d^2-b*d*e+a*e^2)/e^4-1/8*(b^2*e^2*g^2-8*c^2*(e*f-d*g)^2-2*b*c*e*g*(2*e*f-d*g)-2*c*e*g*(4*c*e*f-2*c*d*g-b*e*g)*x)*sqrt(a+b*x+c*x^2)/(c^2*e^3)],
[(f+g*x)*sqrt(a+b*x+c*x^2)/(d+e*x),x,6,-1/8*(b^2*e^2*g+8*c^2*d*(e*f-d*g)-4*c*e*(b*e*f-b*d*g+a*e*g))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*e^3)+(e*f-d*g)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))*sqrt(c*d^2-b*d*e+a*e^2)/e^3+1/4*(4*c*e*f-4*c*d*g+b*e*g+2*c*e*g*x)*sqrt(a+b*x+c*x^2)/(c*e^2)],
[sqrt(a+b*x+c*x^2)/(d+e*x),x,6,-1/2*(2*c*d-b*e)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(e^2*sqrt(c))+arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))*sqrt(c*d^2-b*d*e+a*e^2)/e^2+sqrt(a+b*x+c*x^2)/e],
[sqrt(a+b*x+c*x^2)/((d+e*x)*(f+g*x)),x,8,arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))*sqrt(c)/(e*g)+arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))*sqrt(c*d^2-b*d*e+a*e^2)/(e*(e*f-d*g))-arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))*sqrt(c*f^2-b*f*g+a*g^2)/(g*(e*f-d*g))],
[sqrt(a+b*x+c*x^2)/((d+e*x)*(f+g*x)^2),x,20,-1/2*(2*c*d-b*e)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)^2*sqrt(c))+1/2*e*(2*c*f-b*g)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(g*(e*f-d*g)^2*sqrt(c))-arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))*sqrt(c)/(g*(e*f-d*g))+arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))*sqrt(c*d^2-b*d*e+a*e^2)/(e*f-d*g)^2+1/2*(2*c*f-b*g)*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/(g*(e*f-d*g)*sqrt(c*f^2-b*f*g+a*g^2))-e*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))*sqrt(c*f^2-b*f*g+a*g^2)/(g*(e*f-d*g)^2)+sqrt(a+b*x+c*x^2)/((e*f-d*g)*(f+g*x))],
[sqrt(a+b*x+c*x^2)/((d+e*x)*(f+g*x)^3),x,23,1/8*(b^2-4*a*c)*g*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)*(c*f^2-b*f*g+a*g^2)^(3/2))-1/2*e*(2*c*d-b*e)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)^3*sqrt(c))+1/2*e^2*(2*c*f-b*g)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(g*(e*f-d*g)^3*sqrt(c))-e*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))*sqrt(c)/(g*(e*f-d*g)^2)+e*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))*sqrt(c*d^2-b*d*e+a*e^2)/(e*f-d*g)^3+1/2*e*(2*c*f-b*g)*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/(g*(e*f-d*g)^2*sqrt(c*f^2-b*f*g+a*g^2))-e^2*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))*sqrt(c*f^2-b*f*g+a*g^2)/(g*(e*f-d*g)^3)+e*sqrt(a+b*x+c*x^2)/((e*f-d*g)^2*(f+g*x))-1/4*g*(b*f-2*a*g+(2*c*f-b*g)*x)*sqrt(a+b*x+c*x^2)/((e*f-d*g)*(c*f^2-b*f*g+a*g^2)*(f+g*x)^2)],
[sqrt(a+b*x+c*x^2)/((d+e*x)*(f+g*x)^4),x,27,1/3*g^2*(a+b*x+c*x^2)^(3/2)/((e*f-d*g)*(c*f^2-b*f*g+a*g^2)*(f+g*x)^3)+1/16*(b^2-4*a*c)*g*(2*c*f-b*g)*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)*(c*f^2-b*f*g+a*g^2)^(5/2))+1/8*(b^2-4*a*c)*e*g*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)^2*(c*f^2-b*f*g+a*g^2)^(3/2))-1/2*e^2*(2*c*d-b*e)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)^4*sqrt(c))+1/2*e^3*(2*c*f-b*g)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(g*(e*f-d*g)^4*sqrt(c))-e^2*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))*sqrt(c)/(g*(e*f-d*g)^3)+e^2*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))*sqrt(c*d^2-b*d*e+a*e^2)/(e*f-d*g)^4+1/2*e^2*(2*c*f-b*g)*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/(g*(e*f-d*g)^3*sqrt(c*f^2-b*f*g+a*g^2))-e^3*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))*sqrt(c*f^2-b*f*g+a*g^2)/(g*(e*f-d*g)^4)+e^2*sqrt(a+b*x+c*x^2)/((e*f-d*g)^3*(f+g*x))-1/8*g*(2*c*f-b*g)*(b*f-2*a*g+(2*c*f-b*g)*x)*sqrt(a+b*x+c*x^2)/((e*f-d*g)*(c*f^2-b*f*g+a*g^2)^2*(f+g*x)^2)-1/4*e*g*(b*f-2*a*g+(2*c*f-b*g)*x)*sqrt(a+b*x+c*x^2)/((e*f-d*g)^2*(c*f^2-b*f*g+a*g^2)*(f+g*x)^2)],
[(f+g*x)^3*(a+b*x+c*x^2)^(3/2)/(d+e*x),x,9,1/192*(7*b^3*e^3*g^3+64*c^3*(e*f-d*g)^3-4*b*c*e^2*g^2*(9*b*e*f-3*b*d*g+a*e*g)+24*b*c^2*e*g*(3*e^2*f^2-3*d*e*f*g+d^2*g^2)+2*c*e*g*(7*b^2*e^2*g^2-4*c*e*g*(9*b*e*f-3*b*d*g+a*e*g)+24*c^2*(3*e^2*f^2-3*d*e*f*g+d^2*g^2))*x)*(a+b*x+c*x^2)^(3/2)/(c^3*e^4)+1/60*g^2*(36*c*e*f-22*c*d*g-7*b*e*g)*(a+b*x+c*x^2)^(5/2)/(c^2*e^2)+1/6*g^3*(d+e*x)*(a+b*x+c*x^2)^(5/2)/(c*e^2)+1/3072*(4*c*e*(2*c*d-b*e)*(8*c*e*(b*d-2*a*e)*(24*c^2*e^2*f^3+7*b^2*d*e*g^3-4*c*d*g^2*(9*b*e*f-3*b*d*g+a*e*g))-d*(8*b*c*d-3*b^2*e-4*a*c*e)*g*(7*b^2*e^2*g^2-4*c*e*g*(9*b*e*f-3*b*d*g+a*e*g)+24*c^2*(3*e^2*f^2-3*d*e*f*g+d^2*g^2)))-2*(4*c^2*d^2-1/2*b^2*e^2-2*c*e*(b*d-a*e))*(8*c*e*(2*c*d-b*e)*(24*c^2*e^2*f^3+7*b^2*d*e*g^3-4*c*d*g^2*(9*b*e*f-3*b*d*g+a*e*g))-2*(8*c^2*d^2-4*b*c*d*e-3/2*b^2*e^2+6*a*c*e^2)*g*(7*b^2*e^2*g^2-4*c*e*g*(9*b*e*f-3*b*d*g+a*e*g)+24*c^2*(3*e^2*f^2-3*d*e*f*g+d^2*g^2))))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(9/2)*e^7)+(c*d^2-b*d*e+a*e^2)^(3/2)*(e*f-d*g)^3*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/e^7-1/1536*(3*(7*b^5*e^5*g^3-512*c^5*d^2*(e*f-d*g)^3+128*c^4*e*(5*b*d-4*a*e)*(e*f-d*g)^3-4*b^3*c*e^4*g^2*(9*b*e*f-3*b*d*g+8*a*e*g)+8*b*c^2*e^3*g*(2*a^2*e^2*g^2+6*a*b*e*g*(3*e*f-d*g)+3*b^2*(3*e^2*f^2-3*d*e*f*g+d^2*g^2))-32*b*c^3*e^2*(2*b*(e*f-d*g)^3+3*a*e*g*(3*e^2*f^2-3*d*e*f*g+d^2*g^2)))+2*c*e*(8*c*e*(2*c*d-b*e)*(24*c^2*e^2*f^3+7*b^2*d*e*g^3-4*c*d*g^2*(9*b*e*f-3*b*d*g+a*e*g))-2*(8*c^2*d^2-4*b*c*d*e-3/2*b^2*e^2+6*a*c*e^2)*g*(7*b^2*e^2*g^2-4*c*e*g*(9*b*e*f-3*b*d*g+a*e*g)+24*c^2*(3*e^2*f^2-3*d*e*f*g+d^2*g^2)))*x)*sqrt(a+b*x+c*x^2)/(c^4*e^6)],
[(f+g*x)^2*(a+b*x+c*x^2)^(3/2)/(d+e*x),x,8,-1/48*(3*b^2*e^2*g^2-16*c^2*(e*f-d*g)^2-6*b*c*e*g*(2*e*f-d*g)-6*c*e*g*(4*c*e*f-2*c*d*g-b*e*g)*x)*(a+b*x+c*x^2)^(3/2)/(c^2*e^3)+1/5*g^2*(a+b*x+c*x^2)^(5/2)/(c*e)-1/256*(3*b^5*e^5*g^2+256*c^5*d^3*(e*f-d*g)^2-384*c^4*d*e*(b*d-a*e)*(e*f-d*g)^2-6*b^3*c*e^4*g*(2*b*e*f-b*d*g+4*a*e*g)+16*b*c^2*e^3*(3*a^2*e^2*g^2+b^2*(e*f-d*g)^2+3*a*b*e*g*(2*e*f-d*g))+96*c^3*e^2*(b^2*d*(e*f-d*g)^2-2*a*b*e*(e*f-d*g)^2-a^2*e^2*g*(2*e*f-d*g)))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(7/2)*e^6)+(c*d^2-b*d*e+a*e^2)^(3/2)*(e*f-d*g)^2*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/e^6+1/128*(3*b^4*e^4*g^2+128*c^4*d^2*(e*f-d*g)^2-32*c^3*e*(5*b*d-4*a*e)*(e*f-d*g)^2-6*b^2*c*e^3*g*(2*b*e*f-b*d*g+2*a*e*g)+8*b*c^2*e^2*(2*b*(e*f-d*g)^2+3*a*e*g*(2*e*f-d*g))+2*c*e*((16*c^2*d^2-3*b^2*e^2-4*c*e*(2*b*d-3*a*e))*g*(4*c*e*f-2*c*d*g-b*e*g)-8*c*e*(2*c*d-b*e)*(2*c*e*f^2-b*d*g^2))*x)*sqrt(a+b*x+c*x^2)/(c^3*e^5)],
[(f+g*x)*(a+b*x+c*x^2)^(3/2)/(d+e*x),x,7,1/24*(8*c*e*f-8*c*d*g+3*b*e*g+6*c*e*g*x)*(a+b*x+c*x^2)^(3/2)/(c*e^2)+1/128*(3*b^4*e^4*g-128*c^4*d^3*(e*f-d*g)+192*c^3*d*e*(b*d-a*e)*(e*f-d*g)-8*b^2*c*e^3*(b*e*f-b*d*g+3*a*e*g)+48*c^2*e^2*(a^2*e^2*g-b^2*d*(e*f-d*g)+2*a*b*e*(e*f-d*g)))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(5/2)*e^5)+(c*d^2-b*d*e+a*e^2)^(3/2)*(e*f-d*g)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/e^5-1/64*(3*b^3*e^3*g-64*c^3*d^2*(e*f-d*g)+16*c^2*e*(5*b*d-4*a*e)*(e*f-d*g)-4*b*c*e^2*(2*b*e*f-2*b*d*g+3*a*e*g)+2*c*e*(3*b^2*e^2*g+16*c^2*d*(e*f-d*g)-4*c*e*(2*b*e*f-2*b*d*g+3*a*e*g))*x)*sqrt(a+b*x+c*x^2)/(c^2*e^4)],
[(a+b*x+c*x^2)^(3/2)/(d+e*x),x,7,1/3*(a+b*x+c*x^2)^(3/2)/e-1/16*(2*c*d-b*e)*(8*c^2*d^2-b^2*e^2-4*c*e*(2*b*d-3*a*e))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*e^4)+(c*d^2-b*d*e+a*e^2)^(3/2)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/e^4+1/8*(8*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-4*a*e)-2*c*e*(2*c*d-b*e)*x)*sqrt(a+b*x+c*x^2)/(c*e^3)],
[(a+b*x+c*x^2)^(3/2)/((d+e*x)*(f+g*x)),x,13,(c*d^2-b*d*e+a*e^2)^(3/2)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(e^3*(e*f-d*g))-(c*f^2-b*f*g+a*g^2)^(3/2)*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/(g^3*(e*f-d*g))-1/2*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(e^3*(e*f-d*g)*sqrt(c))+1/8*(8*c^2*e*f^3+b*g^2*(3*b*e*f+b*d*g-4*a*e*g)-4*c*g*(3*b*e*f^2-a*g*(3*e*f-d*g)))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(e*g^3*(e*f-d*g)*sqrt(c))+(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)/(e^2*(e*f-d*g))-1/4*(4*c*e*f^2-g*(5*b*e*f-b*d*g-4*a*e*g)-2*c*g*(e*f-d*g)*x)*sqrt(a+b*x+c*x^2)/(e*g^2*(e*f-d*g))],
[(a+b*x+c*x^2)^(3/2)/((d+e*x)*(f+g*x)^2),x,23,(a+b*x+c*x^2)^(3/2)/((e*f-d*g)*(f+g*x))-1/16*(2*c*d-b*e)*(8*c^2*d^2-b^2*e^2-4*c*e*(2*b*d-3*a*e))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*e^2*(e*f-d*g)^2)+1/16*e*(2*c*f-b*g)*(8*c^2*f^2-b^2*g^2-4*c*g*(2*b*f-3*a*g))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*g^3*(e*f-d*g)^2)+(c*d^2-b*d*e+a*e^2)^(3/2)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(e^2*(e*f-d*g)^2)-e*(c*f^2-b*f*g+a*g^2)^(3/2)*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/(g^3*(e*f-d*g)^2)-3/8*(8*c^2*f^2+b^2*g^2-4*c*g*(2*b*f-a*g))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(g^3*(e*f-d*g)*sqrt(c))+3/2*(2*c*f-b*g)*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))*sqrt(c*f^2-b*f*g+a*g^2)/(g^3*(e*f-d*g))+1/8*(8*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-4*a*e)-2*c*e*(2*c*d-b*e)*x)*sqrt(a+b*x+c*x^2)/(c*e*(e*f-d*g)^2)+3/4*(4*c*f-3*b*g-2*c*g*x)*sqrt(a+b*x+c*x^2)/(g^2*(e*f-d*g))-1/8*e*(8*c^2*f^2+b^2*g^2-2*c*g*(5*b*f-4*a*g)-2*c*g*(2*c*f-b*g)*x)*sqrt(a+b*x+c*x^2)/(c*g^2*(e*f-d*g)^2)],
[(a+b*x+c*x^2)^(3/2)/((d+e*x)*(f+g*x)^3),x,30,1/2*(a+b*x+c*x^2)^(3/2)/((e*f-d*g)*(f+g*x)^2)+e*(a+b*x+c*x^2)^(3/2)/((e*f-d*g)^2*(f+g*x))-1/16*(2*c*d-b*e)*(8*c^2*d^2-b^2*e^2-4*c*e*(2*b*d-3*a*e))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*e*(e*f-d*g)^3)+1/16*e^2*(2*c*f-b*g)*(8*c^2*f^2-b^2*g^2-4*c*g*(2*b*f-3*a*g))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*g^3*(e*f-d*g)^3)+(c*d^2-b*d*e+a*e^2)^(3/2)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(e*(e*f-d*g)^3)-e^2*(c*f^2-b*f*g+a*g^2)^(3/2)*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/(g^3*(e*f-d*g)^3)-3/8*e*(8*c^2*f^2+b^2*g^2-4*c*g*(2*b*f-a*g))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(g^3*(e*f-d*g)^2*sqrt(c))+3/2*(2*c*f-b*g)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))*sqrt(c)/(g^3*(e*f-d*g))-3/8*(8*c^2*f^2+b^2*g^2-4*c*g*(2*b*f-a*g))*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/(g^3*(e*f-d*g)*sqrt(c*f^2-b*f*g+a*g^2))+3/2*e*(2*c*f-b*g)*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))*sqrt(c*f^2-b*f*g+a*g^2)/(g^3*(e*f-d*g)^2)+1/8*(8*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-4*a*e)-2*c*e*(2*c*d-b*e)*x)*sqrt(a+b*x+c*x^2)/(c*(e*f-d*g)^3)+3/4*e*(4*c*f-3*b*g-2*c*g*x)*sqrt(a+b*x+c*x^2)/(g^2*(e*f-d*g)^2)-3/4*(4*c*f-b*g+2*c*g*x)*sqrt(a+b*x+c*x^2)/(g^2*(e*f-d*g)*(f+g*x))-1/8*e^2*(8*c^2*f^2+b^2*g^2-2*c*g*(5*b*f-4*a*g)-2*c*g*(2*c*f-b*g)*x)*sqrt(a+b*x+c*x^2)/(c*g^2*(e*f-d*g)^3)],
[(a+b*x+c*x^2)^(5/2)/((d+e*x)*(f+g*x)),x,15,1/3*(c*d^2-b*d*e+a*e^2)*(a+b*x+c*x^2)^(3/2)/(e^2*(e*f-d*g))-1/24*(8*c*e*f^2-g*(11*b*e*f-3*b*d*g-8*a*e*g)-6*c*g*(e*f-d*g)*x)*(a+b*x+c*x^2)^(3/2)/(e*g^2*(e*f-d*g))-1/16*(2*c*d-b*e)*(c*d^2-b*d*e+a*e^2)*(8*c^2*d^2-b^2*e^2-4*c*e*(2*b*d-3*a*e))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*e^5*(e*f-d*g))+1/128*(128*c^4*e*f^5-320*c^3*e*f^3*g*(b*f-a*g)-b^3*g^4*(5*b*e*f+3*b*d*g-8*a*e*g)+48*c^2*g^2*(5*b^2*e*f^3-10*a*b*e*f^2*g+a^2*g^2*(5*e*f-d*g))-8*b*c*g^3*(5*b^2*e*f^2+12*a^2*e*g^2-3*a*b*g*(5*e*f+d*g)))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*e*g^5*(e*f-d*g))+(c*d^2-b*d*e+a*e^2)^(5/2)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(e^5*(e*f-d*g))-(c*f^2-b*f*g+a*g^2)^(5/2)*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/(g^5*(e*f-d*g))+1/8*(c*d^2-b*d*e+a*e^2)*(8*c^2*d^2+b^2*e^2-2*c*e*(5*b*d-4*a*e)-2*c*e*(2*c*d-b*e)*x)*sqrt(a+b*x+c*x^2)/(c*e^4*(e*f-d*g))-1/64*(64*c^3*e*f^4-16*c^2*e*f^2*g*(9*b*f-8*a*g)-b^2*g^3*(5*b*e*f+3*b*d*g-8*a*e*g)+4*c*g^2*(22*b^2*e*f^2+16*a^2*e*g^2-3*a*b*g*(13*e*f-d*g))-2*c*g*(16*c^2*e*f^3+b*g^2*(5*b*e*f+3*b*d*g-8*a*e*g)-4*c*g*(6*b*e*f^2-a*g*(7*e*f-3*d*g)))*x)*sqrt(a+b*x+c*x^2)/(c*e*g^4*(e*f-d*g))],

# p<0
[(f+g*x)^4/((d+e*x)*sqrt(a+b*x+c*x^2)),x,8,-1/16*g*(5*b^3*e^3*g^3-6*b*c*e^2*g^2*(4*b*e*f-b*d*g+2*a*e*g)-16*c^3*(4*e^3*f^3-6*d*e^2*f^2*g+4*d^2*e*f*g^2-d^3*g^3)+8*c^2*e*g*(a*e*g*(4*e*f-d*g)+b*(6*e^2*f^2-4*d*e*f*g+d^2*g^2)))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(7/2)*e^4)+(e*f-d*g)^4*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(e^4*sqrt(c*d^2-b*d*e+a*e^2))+1/24*g^2*(15*b^2*e^2*g^2-4*c*e*g*(18*b*e*f-7*b*d*g+4*a*e*g)+4*c^2*(36*e^2*f^2-36*d*e*f*g+11*d^2*g^2))*sqrt(a+b*x+c*x^2)/(c^3*e^3)+1/12*g^3*(24*c*e*f-14*c*d*g-5*b*e*g)*(d+e*x)*sqrt(a+b*x+c*x^2)/(c^2*e^3)+1/3*g^4*(d+e*x)^2*sqrt(a+b*x+c*x^2)/(c*e^3)],
[(f+g*x)^3/((d+e*x)*sqrt(a+b*x+c*x^2)),x,7,1/8*g*(3*b^2*e^2*g^2-4*c*e*g*(3*b*e*f-b*d*g+a*e*g)+8*c^2*(3*e^2*f^2-3*d*e*f*g+d^2*g^2))*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(5/2)*e^3)+(e*f-d*g)^3*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(e^3*sqrt(c*d^2-b*d*e+a*e^2))+3/4*g^2*(4*c*e*f-2*c*d*g-b*e*g)*sqrt(a+b*x+c*x^2)/(c^2*e^2)+1/2*g^3*(d+e*x)*sqrt(a+b*x+c*x^2)/(c*e^2)],
[(f+g*x)^2/((d+e*x)*sqrt(a+b*x+c*x^2)),x,6,1/2*g*(4*c*e*f-2*c*d*g-b*e*g)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*e^2)+(e*f-d*g)^2*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(e^2*sqrt(c*d^2-b*d*e+a*e^2))+g^2*sqrt(a+b*x+c*x^2)/(c*e)],
[(f+g*x)/((d+e*x)*sqrt(a+b*x+c*x^2)),x,5,g*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(e*sqrt(c))+(e*f-d*g)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(e*sqrt(c*d^2-b*d*e+a*e^2))],
[1/((d+e*x)*sqrt(a+b*x+c*x^2)),x,2,arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/sqrt(c*d^2-b*d*e+a*e^2)],
[1/((d+e*x)*(f+g*x)*sqrt(a+b*x+c*x^2)),x,6,e*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)*sqrt(c*d^2-b*d*e+a*e^2))-g*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)*sqrt(c*f^2-b*f*g+a*g^2))],
[1/((d+e*x)*(f+g*x)^2*sqrt(a+b*x+c*x^2)),x,9,-1/2*g*(2*c*f-b*g)*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)*(c*f^2-b*f*g+a*g^2)^(3/2))+e^2*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)^2*sqrt(c*d^2-b*d*e+a*e^2))-e*g*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)^2*sqrt(c*f^2-b*f*g+a*g^2))+g^2*sqrt(a+b*x+c*x^2)/((e*f-d*g)*(c*f^2-b*f*g+a*g^2)*(f+g*x))],
[1/((d+e*x)*(f+g*x)^3*sqrt(a+b*x+c*x^2)),x,13,-1/2*e*g*(2*c*f-b*g)*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)^2*(c*f^2-b*f*g+a*g^2)^(3/2))-1/8*g*(8*c^2*f^2+3*b^2*g^2-4*c*g*(2*b*f+a*g))*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)*(c*f^2-b*f*g+a*g^2)^(5/2))+e^3*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)^3*sqrt(c*d^2-b*d*e+a*e^2))-e^2*g*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)^3*sqrt(c*f^2-b*f*g+a*g^2))+1/2*g^2*sqrt(a+b*x+c*x^2)/((e*f-d*g)*(c*f^2-b*f*g+a*g^2)*(f+g*x)^2)+3/4*g^2*(2*c*f-b*g)*sqrt(a+b*x+c*x^2)/((e*f-d*g)*(c*f^2-b*f*g+a*g^2)^2*(f+g*x))+e*g^2*sqrt(a+b*x+c*x^2)/((e*f-d*g)^2*(c*f^2-b*f*g+a*g^2)*(f+g*x))],
[(f+g*x)^4/((d+e*x)*(a+b*x+c*x^2)^(3/2)),x,7,1/2*g^3*(8*c*e*f-2*c*d*g-3*b*e*g)*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(5/2)*e^2)+(e*f-d*g)^4*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(e^2*(c*d^2-b*d*e+a*e^2)^(3/2))-2*(a*b^3*d*g^4-b^2*(c^2*e*f^4+4*a*c*d*f*g^3+a^2*e*g^4)+2*a*c*(a^2*e*g^4+c^2*f^3*(e*f-4*d*g)-2*a*c*f*g^2*(3*e*f-2*d*g))+b*c*(c^2*d*f^4+a^2*g^3*(4*e*f-3*d*g)+2*a*c*f^2*g*(2*e*f+3*d*g))+(2*c^4*d*f^4+b^3*(b*d-a*e)*g^4-b*c*g^3*(4*b^2*d*f-3*a^2*e*g-4*a*b*(e*f-d*g))+2*c^2*g^2*(3*b^2*d*f^2-3*a*b*f*(e*f-2*d*g)-a^2*g*(4*e*f-d*g))+c^3*f^2*(4*a*g*(2*e*f-3*d*g)-b*f*(e*f+4*d*g)))*x)/(c^2*(b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2))+g^4*sqrt(a+b*x+c*x^2)/(c^2*e)],
[(f+g*x)^3/((d+e*x)*(a+b*x+c*x^2)^(3/2)),x,6,g^3*arctanh(1/2*(b+2*c*x)/(sqrt(c)*sqrt(a+b*x+c*x^2)))/(c^(3/2)*e)+(e*f-d*g)^3*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(e*(c*d^2-b*d*e+a*e^2)^(3/2))+2*(b^2*(c*e*f^3+a*d*g^3)-2*a*c*(c*f^2*(e*f-3*d*g)-a*g^2*(3*e*f-d*g))-b*(c^2*d*f^3+a^2*e*g^3+3*a*c*f*g*(e*f+d*g))-(2*c^3*d*f^3-b^2*(b*d-a*e)*g^3+c*g^2*(3*b^2*d*f-3*a*b*e*f+3*a*b*d*g-2*a^2*e*g)+c^2*f*(6*a*g*(e*f-d*g)-b*f*(e*f+3*d*g)))*x)/(c*(b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2))],
[(f+g*x)^2/((d+e*x)*(a+b*x+c*x^2)^(3/2)),x,4,(e*f-d*g)^2*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(3/2)+2*(b^2*e*f^2+2*a*(a*e*g^2-c*f*(e*f-2*d*g))-b*(c*d*f^2+a*g*(2*e*f+d*g))-(2*c^2*d*f^2+b*(b*d-a*e)*g^2+c*(2*a*g*(2*e*f-d*g)-b*f*(e*f+2*d*g)))*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2))],
[(f+g*x)/((d+e*x)*(a+b*x+c*x^2)^(3/2)),x,4,e*(e*f-d*g)*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(3/2)-2*(b*c*d*f-b^2*e*f+2*a*c*e*f-2*a*c*d*g+a*b*e*g+c*(2*c*d*f+2*a*e*g-b*(e*f+d*g))*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2))],
[1/((d+e*x)*(a+b*x+c*x^2)^(3/2)),x,4,e^2*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/(c*d^2-b*d*e+a*e^2)^(3/2)-2*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2))],
[1/((d+e*x)*(f+g*x)*(a+b*x+c*x^2)^(3/2)),x,10,e^3*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/((c*d^2-b*d*e+a*e^2)^(3/2)*(e*f-d*g))-g^3*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)*(c*f^2-b*f*g+a*g^2)^(3/2))-2*e*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(e*f-d*g)*sqrt(a+b*x+c*x^2))+2*g*(b*c*f-b^2*g+2*a*c*g+c*(2*c*f-b*g)*x)/((b^2-4*a*c)*(e*f-d*g)*(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2))],
[1/((d+e*x)*(f+g*x)^2*(a+b*x+c*x^2)^(3/2)),x,14,e^4*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/((c*d^2-b*d*e+a*e^2)^(3/2)*(e*f-d*g)^2)-3/2*g^3*(2*c*f-b*g)*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)*(c*f^2-b*f*g+a*g^2)^(5/2))-e*g^3*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)^2*(c*f^2-b*f*g+a*g^2)^(3/2))-2*e^2*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(e*f-d*g)^2*sqrt(a+b*x+c*x^2))+2*e*g*(b*c*f-b^2*g+2*a*c*g+c*(2*c*f-b*g)*x)/((b^2-4*a*c)*(e*f-d*g)^2*(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2))+2*g*(b*c*f-b^2*g+2*a*c*g+c*(2*c*f-b*g)*x)/((b^2-4*a*c)*(e*f-d*g)*(c*f^2-b*f*g+a*g^2)*(f+g*x)*sqrt(a+b*x+c*x^2))+g^2*(4*c^2*f^2+3*b^2*g^2-4*c*g*(b*f+2*a*g))*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)*(e*f-d*g)*(c*f^2-b*f*g+a*g^2)^2*(f+g*x))],
[1/((d+e*x)*(f+g*x)^3*(a+b*x+c*x^2)^(3/2)),x,19,e^5*arctanh(1/2*(b*d-2*a*e+(2*c*d-b*e)*x)/(sqrt(c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2)))/((c*d^2-b*d*e+a*e^2)^(3/2)*(e*f-d*g)^3)-3/2*e*g^3*(2*c*f-b*g)*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)^2*(c*f^2-b*f*g+a*g^2)^(5/2))-e^2*g^3*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)^3*(c*f^2-b*f*g+a*g^2)^(3/2))-3/8*g^3*(16*c^2*f^2+5*b^2*g^2-4*c*g*(4*b*f+a*g))*arctanh(1/2*(b*f-2*a*g+(2*c*f-b*g)*x)/(sqrt(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)))/((e*f-d*g)*(c*f^2-b*f*g+a*g^2)^(7/2))-2*e^3*(b*c*d-b^2*e+2*a*c*e+c*(2*c*d-b*e)*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*(e*f-d*g)^3*sqrt(a+b*x+c*x^2))+2*e^2*g*(b*c*f-b^2*g+2*a*c*g+c*(2*c*f-b*g)*x)/((b^2-4*a*c)*(e*f-d*g)^3*(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2))+2*g*(b*c*f-b^2*g+2*a*c*g+c*(2*c*f-b*g)*x)/((b^2-4*a*c)*(e*f-d*g)*(c*f^2-b*f*g+a*g^2)*(f+g*x)^2*sqrt(a+b*x+c*x^2))+2*e*g*(b*c*f-b^2*g+2*a*c*g+c*(2*c*f-b*g)*x)/((b^2-4*a*c)*(e*f-d*g)^2*(c*f^2-b*f*g+a*g^2)*(f+g*x)*sqrt(a+b*x+c*x^2))+1/2*g^2*(8*c^2*f^2+5*b^2*g^2-4*c*g*(2*b*f+3*a*g))*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)*(e*f-d*g)*(c*f^2-b*f*g+a*g^2)^2*(f+g*x)^2)+e*g^2*(4*c^2*f^2+3*b^2*g^2-4*c*g*(b*f+2*a*g))*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)*(e*f-d*g)^2*(c*f^2-b*f*g+a*g^2)^2*(f+g*x))+1/4*g^2*(2*c*f-b*g)*(8*c^2*f^2+15*b^2*g^2-4*c*g*(2*b*f+13*a*g))*sqrt(a+b*x+c*x^2)/((b^2-4*a*c)*(e*f-d*g)*(c*f^2-b*f*g+a*g^2)^3*(f+g*x))],

# Integrands of the form (d+e x)^m (f+g x)^(n/2) (a+b x+c x^2)^(p/2)

# p>0 & n>0
[(d+e*x)^3*sqrt(f+g*x)*sqrt(a+b*x+c*x^2),x,10,2/3465*(48*b^3*e^3*g^3+b*c*e^2*g^2*(67*b*e*f-198*b*d*g-157*a*e*g)+c^3*(233*e^3*f^3-843*d*e^2*f^2*g+1107*d^2*e*f*g^2-567*d^3*g^3)-c^2*e*g*(2*a*e*g*(74*e*f-231*d*g)-3*b*(24*e^2*f^2-88*d*e*f*g+99*d^2*g^2)))*(f+g*x)^(3/2)*sqrt(a+b*x+c*x^2)/(c^3*g^4)-2/693*e*(8*b^2*e^2*g^2+c*e*g*(19*b*e*f-33*b*d*g-18*a*e*g)+c^2*(29*e^2*f^2-96*d*e*f*g+81*d^2*g^2))*(f+g*x)^(5/2)*sqrt(a+b*x+c*x^2)/(c^2*g^4)+2/99*e^2*(c*e*f-3*c*d*g+b*e*g)*(f+g*x)^(7/2)*sqrt(a+b*x+c*x^2)/(c*g^4)-2/3465*(64*b^4*e^4*g^4+4*b^2*c*e^3*g^3*(7*b*e*f-66*b*d*g-69*a*e*g)+c^4*(187*e^4*f^4-732*d*e^3*f^3*g+1098*d^2*e^2*f^2*g^2-798*d^3*e*f*g^3+315*d^4*g^4)+3*c^2*e^2*g^2*(50*a^2*e^2*g^2-a*b*e*g*(29*e*f-297*d*g)+3*b^2*(e^2*f^2-11*d*e*f*g+44*d^2*g^2))-c^3*e*g*(6*a*e*g*(2*e^2*f^2-33*d*e*f*g+165*d^2*g^2)+b*(8*e^3*f^3-99*d^2*e*f*g^2+231*d^3*g^3)))*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(c^4*e*g^4)+2/11*(d+e*x)^4*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/e+1/3465*(128*b^5*e^3*g^5-8*b^3*c*e^2*g^4*(7*b*e*f+66*b*d*g+87*a*e*g)+2*c^5*f^2*(64*e^3*f^3-264*d*e^2*f^2*g+396*d^2*e*f*g^2-231*d^3*g^3)+b*c^2*e*g^3*(771*a^2*e^2*g^2+6*a*b*e*g*(43*e*f+396*d*g)-b^2*(37*e^2*f^2-264*d*e*f*g-792*d^2*g^2))-c^4*g*(b*f*(56*e^3*f^3-264*d*e^2*f^2*g+495*d^2*e*f*g^2-462*d^3*g^3)-18*a*g*(6*e^3*f^3-33*d*e^2*f^2*g+88*d^2*e*f*g^2+77*d^3*g^3))-c^3*g^2*(6*a^2*e^2*g^2*(26*e*f+231*d*g)-9*a*b*e*g*(15*e^2*f^2-110*d*e*f*g-319*d^2*g^2)+b^2*(37*e^3*f^3-198*d*e^2*f^2*g+495*d^2*e*f*g^2+462*d^3*g^3)))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^5*g^5*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))+2/3465*(c*f^2-b*f*g+a*g^2)*(64*b^4*e^3*g^4+4*b^2*c*e^2*g^3*(7*b*e*f-66*b*d*g-69*a*e*g)-2*c^4*f*(64*e^3*f^3-264*d*e^2*f^2*g+396*d^2*e*f*g^2-231*d^3*g^3)+3*c^2*e*g^2*(50*a^2*e^2*g^2-a*b*e*g*(29*e*f-297*d*g)+3*b^2*(e^2*f^2-11*d*e*f*g+44*d^2*g^2))-c^3*g*(6*a*e*g*(2*e^2*f^2-33*d*e*f*g+165*d^2*g^2)+b*(8*e^3*f^3-99*d^2*e*f*g^2+231*d^3*g^3)))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c^5*g^5*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^2*sqrt(f+g*x)*sqrt(a+b*x+c*x^2),x,9,-4/315*(3*b^2*e^2*g^2+c*e*g*(4*b*e*f-9*b*d*g-7*a*e*g)+c^2*(8*e^2*f^2-24*d*e*f*g+21*d^2*g^2))*(f+g*x)^(3/2)*sqrt(a+b*x+c*x^2)/(c^2*g^3)+2/63*e*(c*e*f-3*c*d*g+b*e*g)*(f+g*x)^(5/2)*sqrt(a+b*x+c*x^2)/(c*g^3)+2/315*(8*b^3*e^3*g^3+3*b*c*e^2*g^2*(b*e*f-8*b*d*g-9*a*e*g)+c^3*(19*e^3*f^3-57*d*e^2*f^2*g+63*d^2*e*f*g^2-35*d^3*g^3)-3*c^2*e*g^2*(2*a*e*(e*f-10*d*g)+b*d*(2*e*f-7*d*g)))*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(c^3*e*g^3)+2/9*(d+e*x)^3*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/e-2/315*(8*b^4*e^2*g^4-4*b^2*c*e*g^3*(b*e*f+6*b*d*g+9*a*e*g)+c^4*f^2*(8*e^2*f^2-24*d*e*f*g+21*d^2*g^2)+3*c^2*g^2*(7*a^2*e^2*g^2+a*b*e*g*(5*e*f+29*d*g)-b^2*(e^2*f^2-5*d*e*f*g-7*d^2*g^2))+c^3*g*(3*a*g*(3*e^2*f^2-16*d*e*f*g-21*d^2*g^2)-b*f*(4*e^2*f^2-15*d*e*f*g+21*d^2*g^2)))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*g^4*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))-2/315*(c*f^2-b*f*g+a*g^2)*(8*b^3*e^2*g^3+3*b*c*e*g^2*(b*e*f-8*b*d*g-9*a*e*g)-2*c^3*f*(8*e^2*f^2-24*d*e*f*g+21*d^2*g^2)-3*c^2*g^2*(2*a*e*(e*f-10*d*g)+b*d*(2*e*f-7*d*g)))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c^4*g^4*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2),x,7,2/7*e*(a+b*x+c*x^2)^(3/2)*sqrt(f+g*x)/c-2/105*(4*b^2*e*g^2+c^2*f*(4*e*f-7*d*g)-c*g*(2*b*e*f+7*b*d*g-5*a*e*g)-3*c*g*(c*e*f+7*c*d*g-4*b*e*g)*x)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(c^2*g^2)+1/105*((c*e*f+7*c*d*g-4*b*e*g)*(8*c^2*f^2-2*b^2*g^2-3*c*g*(b*f-2*a*g))-5*c*g*(2*c*f-b*g)*(7*c*d*f-e*(3*b*f+a*g)))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*g^3*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))+2/105*(c*f^2-b*f*g+a*g^2)*(4*b^2*e*g^2-2*c^2*f*(4*e*f-7*d*g)+c*g*(b*e*f-7*b*d*g-10*a*e*g))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c^3*g^3*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))],
[sqrt(f+g*x)*sqrt(a+b*x+c*x^2),x,7,2/5*(f+g*x)^(3/2)*sqrt(a+b*x+c*x^2)/g-2/15*(2*c*f-b*g)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(c*g)-2/15*(c^2*f^2+b^2*g^2-c*g*(b*f+3*a*g))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*g^2*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))+2/15*(2*c*f-b*g)*(c*f^2-b*f*g+a*g^2)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c^2*g^2*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))],
[sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(d+e*x),x,15,2/3*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/e+1/3*(c*e*f-3*c*d*g+b*e*g)*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*e^2*g*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))+2/3*(e*g*(b*e*f-3*b*d*g+2*a*e*g)+c*(-e^2*f^2+3*d^2*g^2))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(2*g*sqrt(b^2-4*a*c)/(-2*c*f+g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(c*(a+x*(b+c*x))/(-b^2+4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c*e^3*g*sqrt(f+g*x)*sqrt(a+x*(b+c*x)))-(c*d^2-b*d*e+a*e^2)*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^3*sqrt(c)*sqrt(a+b*x+c*x^2)),2/3*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/e+1/3*(c*e*f-3*c*d*g+b*e*g)*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*e^2*g*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))-2/3*f*(c*e*f-3*c*d*g+b*e*g)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c*e^2*g*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))-2/3*(3*c*d*(e*f-d*g)-e*(2*b*e*f-3*b*d*g+2*a*e*g))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c*e^3*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))-(c*d^2-b*d*e+a*e^2)*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^3*sqrt(c)*sqrt(a+b*x+c*x^2))],
[sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(d+e*x)^2,x,15,-sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(e*(d+e*x))+3*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(e^2*sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))+(2*b*e*g-c*(e*f+3*d*g))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(2*g*sqrt(b^2-4*a*c)/(-2*c*f+g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(c*(a+x*(b+c*x))/(-b^2+4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c*e^3*sqrt(f+g*x)*sqrt(a+x*(b+c*x)))+(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^3*(e*f-d*g)*sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2)),-sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(e*(d+e*x))+3*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(e^2*sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))-3*f*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^2*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))+(2*c*e*f-3*c*d*g+2*b*e*g)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c*e^3*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))+(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^3*(e*f-d*g)*sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2))],
[sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(d+e*x)^3,x,25,-1/2*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(e*(d+e*x)^2)+1/4*(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(e*(c*d^2-b*d*e+a*e^2)*(e*f-d*g)*(d+e*x))-1/4*(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(e^2*(c*d^2-b*d*e+a*e^2)*(e*f-d*g)*sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))-1/2*(-c*d*(2*e*f+3*d*g)+e*(b*e*f+4*b*d*g-5*a*e*g))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(2*g*sqrt(b^2-4*a*c)/(-2*c*f+g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(c*(a+x*(b+c*x))/(-b^2+4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^3*(c*d^2+e*(-b*d+a*e))*sqrt(2)*sqrt(f+g*x)*sqrt(a+x*(b+c*x)))+1/4*(b^2*e^4*f^2+a^2*e^4*g^2+c^2*d^3*g*(4*e*f-3*d*g)-2*a*c*e^2*(2*e^2*f^2-6*d*e*f*g+3*d^2*g^2)-2*b*e*g*(a*e^3*f+c*d^2*(3*e*f-2*d*g)))*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-b*g+g*sqrt(b^2-4*a*c)),(2*c*e*f-b*e*g+e*g*sqrt(b^2-4*a*c))/(2*c*e*f-2*c*d*g),sqrt((2*c*f+g*(-b+sqrt(b^2-4*a*c)))/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2*c*f-b*g+g*sqrt(b^2-4*a*c))*sqrt(g*(-b-2*c*x+sqrt(b^2-4*a*c))/(2*c*f+g*(-b+sqrt(b^2-4*a*c))))*sqrt(g*(b+2*c*x+sqrt(b^2-4*a*c))/(-2*c*f+g*(b+sqrt(b^2-4*a*c))))/(e^3*(c*d^2+e*(-b*d+a*e))*(e*f-d*g)^2*sqrt(2)*sqrt(c)*sqrt(a+x*(b+c*x))),-1/2*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(e*(d+e*x)^2)+1/4*(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(e*(c*d^2-b*d*e+a*e^2)*(e*f-d*g)*(d+e*x))-1/4*(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(e^2*(c*d^2-b*d*e+a*e^2)*(e*f-d*g)*sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))+3*g*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^3*sqrt(2)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))+1/2*f*(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^2*(c*d^2-b*d*e+a*e^2)*(e*f-d*g)*sqrt(2)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))-1/2*d*g*(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^3*(c*d^2-b*d*e+a*e^2)*(e*f-d*g)*sqrt(2)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))-(c*e*f-3*c*d*g+b*e*g)*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^3*(e*f-d*g)*sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2))+1/4*(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))^2*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^3*(c*d^2-b*d*e+a*e^2)*(e*f-d*g)^2*sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2))],

# p>0 & n<0
[(d+e*x)^3*sqrt(a+b*x+c*x^2)/sqrt(f+g*x),x,9,-2/315*e*(6*b^2*e^2*g^2+c*e*g*(17*b*e*f-27*b*d*g-14*a*e*g)-2*c^2*(64*e^2*f^2-111*d*e*f*g+42*d^2*g^2))*(f+g*x)^(3/2)*sqrt(a+b*x+c*x^2)/(c^2*g^4)-2/63*e^2*(8*c*e*f-6*c*d*g-b*e*g)*(f+g*x)^(5/2)*sqrt(a+b*x+c*x^2)/(c*g^4)+2/315*(8*b^3*e^3*g^3+3*b*c*e^2*g^2*(5*b*e*f-12*b*d*g-9*a*e*g)-c^3*(152*e^3*f^3-408*d*e^2*f^2*g+336*d^2*e*f*g^2-70*d^3*g^3)-3*c^2*e*g*(6*a*e*g*(2*e*f-5*d*g)-b*(8*e^2*f^2-24*d*e*f*g+21*d^2*g^2)))*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(c^3*g^4)+2/9*(d+e*x)^3*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/g-1/315*(16*b^4*e^3*g^4+8*b^2*c*e^2*g^3*(2*b*e*f-9*b*d*g-9*a*e*g)-2*c^4*f*(64*e^3*f^3-216*d*e^2*f^2*g+252*d^2*e*f*g^2-105*d^3*g^3)+3*c^2*e*g^2*(14*a^2*e^2*g^2-a*b*e*g*(19*e*f-87*d*g)+b^2*(7*e^2*f^2-27*d*e*f*g+42*d^2*g^2))-c^3*g*(6*a*e*g*(10*e^2*f^2-39*d*e*f*g+63*d^2*g^2)-b*(40*e^3*f^3-144*d*e^2*f^2*g+189*d^2*e*f*g^2-105*d^3*g^3)))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*g^5*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))-2/315*(c*f^2-b*f*g+a*g^2)*(8*b^3*e^3*g^3+3*b*c*e^2*g^2*(5*b*e*f-12*b*d*g-9*a*e*g)+2*c^3*(64*e^3*f^3-216*d*e^2*f^2*g+252*d^2*e*f*g^2-105*d^3*g^3)-3*c^2*e*g*(6*a*e*g*(2*e*f-5*d*g)-b*(8*e^2*f^2-24*d*e*f*g+21*d^2*g^2)))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c^4*g^5*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^2*sqrt(a+b*x+c*x^2)/sqrt(f+g*x),x,8,-2/35*e*(6*c*e*f-4*c*d*g-b*e*g)*(f+g*x)^(3/2)*sqrt(a+b*x+c*x^2)/(c*g^3)-4/105*(2*b^2*e^2*g^2+c*e*g*(4*b*e*f-7*b*d*g-5*a*e*g)-c^2*(21*e^2*f^2-34*d*e*f*g+10*d^2*g^2))*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(c^2*g^3)+2/7*(d+e*x)^2*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/g+1/105*(8*b^3*e^2*g^3+b*c*e*g^2*(9*b*e*f-28*b*d*g-29*a*e*g)-2*c^3*f*(24*e^2*f^2-56*d*e*f*g+35*d^2*g^2)-c^2*g*(2*a*e*g*(13*e*f-42*d*g)-b*(16*e^2*f^2-42*d*e*f*g+35*d^2*g^2)))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*g^4*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))+4/105*(c*f^2-b*f*g+a*g^2)*(2*b^2*e^2*g^2+c*e*g*(4*b*e*f-7*b*d*g-5*a*e*g)+c^2*(24*e^2*f^2-56*d*e*f*g+35*d^2*g^2))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c^3*g^4*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)*sqrt(a+b*x+c*x^2)/sqrt(f+g*x),x,6,-2/15*(4*c*e*f-5*c*d*g-b*e*g-3*c*e*g*x)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(c*g^2)-1/15*(2*b^2*e*g^2-2*c^2*f*(4*e*f-5*d*g)+c*g*(3*b*e*f-5*b*d*g-6*a*e*g))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*g^3*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))-2/15*(8*c*e*f-10*c*d*g+b*e*g)*(c*f^2-b*f*g+a*g^2)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c^2*g^3*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))],
[sqrt(a+b*x+c*x^2)/sqrt(f+g*x),x,6,2/3*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/g-1/3*(2*c*f-b*g)*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*g^2*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))+4/3*(c*f^2-b*f*g+a*g^2)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c*g^2*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))],
[sqrt(a+b*x+c*x^2)/((d+e*x)*sqrt(f+g*x)),x,11,EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(e*g*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))-2*(c*e*f+c*d*g-b*e*g)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c*e^2*g*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))-(c*d^2-b*d*e+a*e^2)*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^2*(e*f-d*g)*sqrt(c)*sqrt(a+b*x+c*x^2))],
[sqrt(a+b*x+c*x^2)/((d+e*x)^2*sqrt(f+g*x)),x,15,-sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/((e*f-d*g)*(d+e*x))+EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(e*(e*f-d*g)*sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))+EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(2*g*sqrt(b^2-4*a*c)/(-2*c*f+g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(c*(a+x*(b+c*x))/(-b^2+4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^2*sqrt(f+g*x)*sqrt(a+x*(b+c*x)))-(e^2*(b*f-a*g)-c*d*(2*e*f-d*g))*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^2*(e*f-d*g)^2*sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2)),-sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/((e*f-d*g)*(d+e*x))+EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(e*(e*f-d*g)*sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))-f*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e*(e*f-d*g)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))+(2*e*f-d*g)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^2*(e*f-d*g)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))-(e^2*(b*f-a*g)-c*d*(2*e*f-d*g))*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^2*(e*f-d*g)^2*sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2))],
[sqrt(a+b*x+c*x^2)/((d+e*x)^3*sqrt(f+g*x)),x,25,-1/2*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/((e*f-d*g)*(d+e*x)^2)+1/4*(c*d*(2*e*f+d*g)-e*(b*e*f+2*b*d*g-3*a*e*g))*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)*(e*f-d*g)^2*(d+e*x))-1/4*(c*d*(2*e*f+d*g)-e*(b*e*f+2*b*d*g-3*a*e*g))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(e*(c*d^2-b*d*e+a*e^2)*(e*f-d*g)^2*sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))-1/2*(e^2*(b*f-a*g)+c*d*(-2*e*f+d*g))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(2*g*sqrt(b^2-4*a*c)/(-2*c*f+g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(c*(a+x*(b+c*x))/(-b^2+4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^2*(c*d^2+e*(-b*d+a*e))*(e*f-d*g)*sqrt(2)*sqrt(f+g*x)*sqrt(a+x*(b+c*x)))-1/4*(3*a^2*e^4*g^2+c^2*d^3*g*(4*e*f-d*g)+b^2*e^3*f*(-e*f+4*d*g)+2*a*c*e^2*(2*e^2*f^2-2*d*e*f*g+3*d^2*g^2)-2*b*e^2*g*(3*c*d^2*f+a*e*(e*f+2*d*g)))*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-b*g+g*sqrt(b^2-4*a*c)),(2*c*e*f-b*e*g+e*g*sqrt(b^2-4*a*c))/(2*c*e*f-2*c*d*g),sqrt((2*c*f+g*(-b+sqrt(b^2-4*a*c)))/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2*c*f-b*g+g*sqrt(b^2-4*a*c))*sqrt(g*(-b-2*c*x+sqrt(b^2-4*a*c))/(2*c*f+g*(-b+sqrt(b^2-4*a*c))))*sqrt(g*(b+2*c*x+sqrt(b^2-4*a*c))/(-2*c*f+g*(b+sqrt(b^2-4*a*c))))/(e^2*(c*d^2+e*(-b*d+a*e))*(e*f-d*g)^3*sqrt(2)*sqrt(c)*sqrt(a+x*(b+c*x))),-1/2*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/((e*f-d*g)*(d+e*x)^2)+1/4*(c*d*(2*e*f+d*g)-e*(b*e*f+2*b*d*g-3*a*e*g))*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)*(e*f-d*g)^2*(d+e*x))-1/4*(c*d*(2*e*f+d*g)-e*(b*e*f+2*b*d*g-3*a*e*g))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(e*(c*d^2-b*d*e+a*e^2)*(e*f-d*g)^2*sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))-g*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^2*(e*f-d*g)*sqrt(2)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))+1/2*f*(c*d*(2*e*f+d*g)-e*(b*e*f+2*b*d*g-3*a*e*g))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e*(c*d^2-b*d*e+a*e^2)*(e*f-d*g)^2*sqrt(2)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))-1/2*d*g*(c*d*(2*e*f+d*g)-e*(b*e*f+2*b*d*g-3*a*e*g))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^2*(c*d^2-b*d*e+a*e^2)*(e*f-d*g)^2*sqrt(2)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))-(c*e*f+c*d*g-b*e*g)*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^2*(e*f-d*g)^2*sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2))+1/4*(c*d*(2*e*f+d*g)-e*(b*e*f+2*b*d*g-3*a*e*g))*(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^2*(c*d^2-b*d*e+a*e^2)*(e*f-d*g)^3*sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2))],

# p<0 & n>0
[(d+e*x)^3*sqrt(f+g*x)/sqrt(a+b*x+c*x^2),x,8,2/35*e^2*(c*e*f+11*c*d*g-6*b*e*g)*(f+g*x)^(3/2)*sqrt(a+b*x+c*x^2)/(c^2*g^2)+2/105*e*(24*b^2*e^2*g^2+c*e*g*(13*b*e*f-84*b*d*g-25*a*e*g)-c^2*(7*e^2*f^2+12*d*e*f*g-90*d^2*g^2))*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(c^3*g^2)+2/7*e*(d+e*x)^2*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/c-1/105*(48*b^3*e^3*g^3-8*b*c*e^2*g^2*(2*b*e*f+21*b*d*g+13*a*e*g)-c^3*(8*e^3*f^3-42*d*e^2*f^2*g+105*d^2*e*f*g^2+105*d^3*g^3)+c^2*e*g*(a*e*g*(19*e*f+189*d*g)-b*(9*e^2*f^2-63*d*e*f*g-210*d^2*g^2)))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^4*g^3*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))-2/105*e*(c*f^2-b*f*g+a*g^2)*(24*b^2*e^2*g^2+c*e*g*(13*b*e*f-84*b*d*g-25*a*e*g)+c^2*(8*e^2*f^2-42*d*e*f*g+105*d^2*g^2))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c^4*g^3*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^2*sqrt(f+g*x)/sqrt(a+b*x+c*x^2),x,7,2/15*e*(c*e*f+7*c*d*g-4*b*e*g)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(c^2*g)+2/5*e*(d+e*x)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/c+1/15*(8*b^2*e^2*g^2-c*e*g*(3*b*e*f+20*b*d*g+9*a*e*g)-c^2*(2*e^2*f^2-10*d*e*f*g-15*d^2*g^2))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*g^2*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))+4/15*e*(c*e*f-5*c*d*g+2*b*e*g)*(c*f^2-b*f*g+a*g^2)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c^3*g^2*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)*sqrt(f+g*x)/sqrt(a+b*x+c*x^2),x,6,2/3*e*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/c+1/3*(c*e*f+3*c*d*g-2*b*e*g)*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*g*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))-2/3*e*(c*f^2-b*f*g+a*g^2)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c^2*g*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))],
[sqrt(f+g*x)/sqrt(a+b*x+c*x^2),x,2,EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))],
[sqrt(f+g*x)/((d+e*x)*sqrt(a+b*x+c*x^2)),x,8,2*g*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c*e*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))-EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e*sqrt(c)*sqrt(a+b*x+c*x^2))],
[sqrt(f+g*x)/((d+e*x)^2*sqrt(a+b*x+c*x^2)),x,15,-e*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)*(d+e*x))+EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)*sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))-f*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/((c*d^2-b*d*e+a*e^2)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))+d*g*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e*(c*d^2-b*d*e+a*e^2)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))+(e^2*(b*f-a*g)-c*d*(2*e*f-d*g))*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e*(c*d^2-b*d*e+a*e^2)*(e*f-d*g)*sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2))],
[sqrt(f+g*x)/((d+e*x)^3*sqrt(a+b*x+c*x^2)),x,25,-1/2*e*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)*(d+e*x)^2)-1/4*e*(c*d*(6*e*f-5*d*g)-e*(3*b*e*f-2*b*d*g-a*e*g))*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)^2*(e*f-d*g)*(d+e*x))+1/4*(c*d*(6*e*f-5*d*g)-e*(3*b*e*f-2*b*d*g-a*e*g))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)^2*(e*f-d*g)*sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))-g*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e*(c*d^2-b*d*e+a*e^2)*sqrt(2)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))-1/2*f*(c*d*(6*e*f-5*d*g)-e*(3*b*e*f-2*b*d*g-a*e*g))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/((c*d^2-b*d*e+a*e^2)^2*(e*f-d*g)*sqrt(2)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))+1/2*d*g*(c*d*(6*e*f-5*d*g)-e*(3*b*e*f-2*b*d*g-a*e*g))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e*(c*d^2-b*d*e+a*e^2)^2*(e*f-d*g)*sqrt(2)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))+(c*e*f-3*c*d*g+b*e*g)*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e*(c*d^2-b*d*e+a*e^2)*(e*f-d*g)*sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2))-1/4*(c*d*(6*e*f-5*d*g)-e*(3*b*e*f-2*b*d*g-a*e*g))*(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e*(c*d^2-b*d*e+a*e^2)^2*(e*f-d*g)^2*sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2))],
[(f+g*x)^(3/2)/((d+e*x)*sqrt(a+b*x+c*x^2)),x,11,g*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*e*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))+2*g*(e*f-d*g)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c*e^2*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))-(e*f-d*g)*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^2*sqrt(c)*sqrt(a+b*x+c*x^2))],
[(f+g*x)^(5/2)/((d+e*x)*sqrt(a+b*x+c*x^2)),x,17,2/3*g^2*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(c*e)+2/3*g*(2*c*f-b*g)*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*e*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))+g*(e*f-d*g)*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*e^2*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))+2*g*(e*f-d*g)^2*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c*e^3*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))-2/3*g*(c*f^2-b*f*g+a*g^2)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c^2*e*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))-(e*f-d*g)^2*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(e^3*sqrt(c)*sqrt(a+b*x+c*x^2))],

# p<0 & n<0
[(d+e*x)^3/(sqrt(f+g*x)*sqrt(a+b*x+c*x^2)),x,7,-8/15*e^2*(c*e*f-3*c*d*g+b*e*g)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(c^2*g^2)+2/5*e^2*(d+e*x)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(c*g)+1/15*e*(8*b^2*e^2*g^2+c*e*g*(7*b*e*f-30*b*d*g-9*a*e*g)+c^2*(8*e^2*f^2-30*d*e*f*g+45*d^2*g^2))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^3*g^3*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))-2/15*(4*b*e^3*g^2*(b*f-a*g)+c^2*(8*e^3*f^3-30*d*e^2*f^2*g+45*d^2*e*f*g^2-15*d^3*g^3)-c*e^2*g*(a*g*(7*e*f-15*d*g)-3*b*f*(e*f-5*d*g)))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c^3*g^3*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^2/(sqrt(f+g*x)*sqrt(a+b*x+c*x^2)),x,7,2/3*e^2*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/(c*g)-2/3*e*(c*e*f-3*c*d*g+b*e*g)*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c^2*g^2*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))+2/3*(e^2*g*(b*f-a*g)+c*(2*e^2*f^2-6*d*e*f*g+3*d^2*g^2))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c^2*g^2*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))],
[(d+e*x)/(sqrt(f+g*x)*sqrt(a+b*x+c*x^2)),x,5,e*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/(c*g*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))-2*(e*f-d*g)*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c*g*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))],
[1/(sqrt(f+g*x)*sqrt(a+b*x+c*x^2)),x,2,2*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/(c*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))],
[1/((d+e*x)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)),x,5,-EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/((e*f-d*g)*sqrt(c)*sqrt(a+b*x+c*x^2))],
[1/((d+e*x)^2*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)),x,15,-e^2*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)*(e*f-d*g)*(d+e*x))+e*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)*(e*f-d*g)*sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))-e*f*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/((c*d^2-b*d*e+a*e^2)*(e*f-d*g)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))+d*g*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/((c*d^2-b*d*e+a*e^2)*(e*f-d*g)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))-(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/((c*d^2-b*d*e+a*e^2)*(e*f-d*g)^2*sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2))],
[1/((d+e*x)^3*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)),x,25,-1/2*e^2*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)*(e*f-d*g)*(d+e*x)^2)-3/4*e^2*(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)^2*(e*f-d*g)^2*(d+e*x))+3/4*e*(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)^2*(e*f-d*g)^2*sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))+1/2*(c*d*(-6*e*f+7*d*g)+e*(3*b*e*f-4*b*d*g+a*e*g))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(2*g*sqrt(b^2-4*a*c)/(-2*c*f+g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(c*(a+x*(b+c*x))/(-b^2+4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/((c*d^2+e*(-b*d+a*e))^2*(e*f-d*g)*sqrt(2)*sqrt(f+g*x)*sqrt(a+x*(b+c*x)))+1/4*(c^2*d^2*(8*e^2*f^2-20*d*e*f*g+15*d^2*g^2)+2*c*e*(b*d*(-4*e^2*f^2+11*d*e*f*g-10*d^2*g^2)+a*e*(-2*e^2*f^2+2*d*e*f*g+3*d^2*g^2))+e^2*(3*a^2*e^2*g^2+2*a*b*e*g*(e*f-4*d*g)+b^2*(3*e^2*f^2-8*d*e*f*g+8*d^2*g^2)))*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-b*g+g*sqrt(b^2-4*a*c)),(2*c*e*f-b*e*g+e*g*sqrt(b^2-4*a*c))/(2*c*e*f-2*c*d*g),sqrt((2*c*f+g*(-b+sqrt(b^2-4*a*c)))/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2*c*f-b*g+g*sqrt(b^2-4*a*c))*sqrt(g*(-b-2*c*x+sqrt(b^2-4*a*c))/(2*c*f+g*(-b+sqrt(b^2-4*a*c))))*sqrt(g*(b+2*c*x+sqrt(b^2-4*a*c))/(-2*c*f+g*(b+sqrt(b^2-4*a*c))))/((c*d^2+e*(-b*d+a*e))^2*(-e*f+d*g)^3*sqrt(2)*sqrt(c)*sqrt(a+x*(b+c*x))),-1/2*e^2*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)*(e*f-d*g)*(d+e*x)^2)-3/4*e^2*(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)^2*(e*f-d*g)^2*(d+e*x))+3/4*e*(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((c*d^2-b*d*e+a*e^2)^2*(e*f-d*g)^2*sqrt(2)*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))-g*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/((c*d^2-b*d*e+a*e^2)*(e*f-d*g)*sqrt(2)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))-3/2*e*f*(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/((c*d^2-b*d*e+a*e^2)^2*(e*f-d*g)^2*sqrt(2)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))+3/2*d*g*(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/((c*d^2-b*d*e+a*e^2)^2*(e*f-d*g)^2*sqrt(2)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))+(c*e*f-3*c*d*g+b*e*g)*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/((c*d^2-b*d*e+a*e^2)*(e*f-d*g)^2*sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2))-3/4*(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))^2*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/((c*d^2-b*d*e+a*e^2)^2*(e*f-d*g)^3*sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2))],
[1/((d+e*x)*(f+g*x)^(3/2)*sqrt(a+b*x+c*x^2)),x,11,2*g^2*sqrt(a+b*x+c*x^2)/((e*f-d*g)*(c*f^2-b*f*g+a*g^2)*sqrt(f+g*x))-g*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((e*f-d*g)*(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))-e*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/((e*f-d*g)^2*sqrt(c)*sqrt(a+b*x+c*x^2))],
[1/((d+e*x)*(f+g*x)^(5/2)*sqrt(a+b*x+c*x^2)),x,18,2/3*g^2*sqrt(a+b*x+c*x^2)/((e*f-d*g)*(c*f^2-b*f*g+a*g^2)*(f+g*x)^(3/2))+4/3*g^2*(2*c*f-b*g)*sqrt(a+b*x+c*x^2)/((e*f-d*g)*(c*f^2-b*f*g+a*g^2)^2*sqrt(f+g*x))+2*e*g^2*sqrt(a+b*x+c*x^2)/((e*f-d*g)^2*(c*f^2-b*f*g+a*g^2)*sqrt(f+g*x))-2/3*g*(2*c*f-b*g)*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((e*f-d*g)*(c*f^2-b*f*g+a*g^2)^2*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))-e*g*EllipticE(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(f+g*x)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))/((e*f-d*g)^2*(c*f^2-b*f*g+a*g^2)*sqrt(a+b*x+c*x^2)*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))+2/3*g*EllipticF(sqrt((b+2*c*x+sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c))/sqrt(2),sqrt(-2*g*sqrt(b^2-4*a*c)/(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-c*(a+b*x+c*x^2)/(b^2-4*a*c))*sqrt(c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/((e*f-d*g)*(c*f^2-b*f*g+a*g^2)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2))-e^2*EllipticPi(sqrt(2)*sqrt(c)*sqrt(f+g*x)/sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c))),1/2*e*(2*c*f-b*g+g*sqrt(b^2-4*a*c))/(c*(e*f-d*g)),sqrt((b-2*c*f/g-sqrt(b^2-4*a*c))/(b-2*c*f/g+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b-sqrt(b^2-4*a*c))))*sqrt(2*c*f-g*(b-sqrt(b^2-4*a*c)))*sqrt(1-2*c*(f+g*x)/(2*c*f-g*(b+sqrt(b^2-4*a*c))))/((e*f-d*g)^3*sqrt(c)*sqrt(a+b*x+c*x^2))],

# Integrands of the form (d+e x)^(m/2) (f+g x)^(n/2) (a+b x+c x^2)^(p/2)

# p>0 & n>0

# p>0 & n<0

# p<0 & n>0

# p<0 & n<0
#  {(d + e*x)^(5/2)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x, 0, 0}

# {(d + e*x)^(3/2)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x, 0, 0} 
[(d+e*x)^(1/2)/(sqrt(f+g*x)*sqrt(a+b*x+c*x^2)),x,1,(d+e*x)*EllipticPi(sqrt(f+g*x)*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(sqrt(d+e*x)*sqrt(2*c*f-g*(b+sqrt(b^2-4*a*c)))),e*(2*c*f-g*(b+sqrt(b^2-4*a*c)))/(g*(2*c*d-e*(b+sqrt(b^2-4*a*c)))),sqrt((b*d-2*a*e+d*sqrt(b^2-4*a*c))*(2*c*f-g*(b+sqrt(b^2-4*a*c)))/((b*f-2*a*g+f*sqrt(b^2-4*a*c))*(2*c*d-e*(b+sqrt(b^2-4*a*c))))))*sqrt(2)*sqrt(b+2*c*x-sqrt(b^2-4*a*c))*sqrt((e*f-d*g)*(b+2*c*x+sqrt(b^2-4*a*c))/((d+e*x)*(2*c*f-g*(b+sqrt(b^2-4*a*c)))))*sqrt(2*c*f-g*(b+sqrt(b^2-4*a*c)))*sqrt((e*f-d*g)*(2*a+x*(b+sqrt(b^2-4*a*c)))/((d+e*x)*(b*f-2*a*g+f*sqrt(b^2-4*a*c))))/(g*sqrt(a+b*x+c*x^2)*sqrt(c*x+2*a*c/(b+sqrt(b^2-4*a*c)))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c))))],
[1/((d+e*x)^(1/2)*sqrt(f+g*x)*sqrt(a+b*x+c*x^2)),x,2,-(c*f^2-g*(b*f-a*g))^(1/4)*(d+e*x)*sqrt(cos(2*arctan((c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(f+g*x)/((c*f^2-b*f*g+a*g^2)^(1/4)*sqrt(d+e*x))))^2)/cos(2*arctan((c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(f+g*x)/((c*f^2-b*f*g+a*g^2)^(1/4)*sqrt(d+e*x))))*EllipticF(sin(2*arctan((c*d^2-b*d*e+a*e^2)^(1/4)*sqrt(f+g*x)/((c*f^2-b*f*g+a*g^2)^(1/4)*sqrt(d+e*x)))),sqrt(1/4*(2+(2*c*d*f+2*a*e*g-b*(e*f+d*g))/(sqrt(c*d^2-e*(b*d-a*e))*sqrt(c*f^2-g*(b*f-a*g))))))*(1+(f+g*x)*sqrt(c*d^2-b*d*e+a*e^2)/((d+e*x)*sqrt(c*f^2-g*(b*f-a*g))))*sqrt((e*f-d*g)^2*(a+b*x+c*x^2)/((c*f^2-b*f*g+a*g^2)*(d+e*x)^2))*sqrt((1-(2*c*d*f+2*a*e*g-b*(e*f+d*g))*(f+g*x)/((c*f^2-b*f*g+a*g^2)*(d+e*x))+(c*d^2-b*d*e+a*e^2)*(f+g*x)^2/((c*f^2-g*(b*f-a*g))*(d+e*x)^2))/(1+(f+g*x)*sqrt(c*d^2-b*d*e+a*e^2)/((d+e*x)*sqrt(c*f^2-g*(b*f-a*g))))^2)/((c*d^2-b*d*e+a*e^2)^(1/4)*(e*f-d*g)*sqrt(a+b*x+c*x^2)*sqrt(1-(2*c*d*f+2*a*e*g-b*(e*f+d*g))*(f+g*x)/((c*f^2-b*f*g+a*g^2)*(d+e*x))+(c*d^2-b*d*e+a*e^2)*(f+g*x)^2/((c*f^2-g*(b*f-a*g))*(d+e*x)^2)))],
#  {1/((d + e*x)^(3/2)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x, 0, 0}

# {1/((d + e*x)^(5/2)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x, 0, 0} 

# Integrands of the form (d+e x)^m (f+g x)^n (a+b x+c x^2)^p when m symbolic

# Integrands of the form (d+e x)^m (f+g x)^n (a+b x+c x^2)^p when m symbolic

# p>0
[(d+e*x)^m*(f+g*x)^2*(a+b*x+c*x^2),x,2,(c*d^2-b*d*e+a*e^2)*(e*f-d*g)^2*(d+e*x)^(1+m)/(e^5*(1+m))-(e*f-d*g)*(2*c*d*(e*f-2*d*g)-e*(b*e*f-3*b*d*g+2*a*e*g))*(d+e*x)^(2+m)/(e^5*(2+m))+(e*g*(2*b*e*f-3*b*d*g+a*e*g)+c*(e^2*f^2-6*d*e*f*g+6*d^2*g^2))*(d+e*x)^(3+m)/(e^5*(3+m))+g*(2*c*e*f-4*c*d*g+b*e*g)*(d+e*x)^(4+m)/(e^5*(4+m))+c*g^2*(d+e*x)^(5+m)/(e^5*(5+m))],
[(d+e*x)^m*(f+g*x)*(a+b*x+c*x^2),x,2,(c*d^2-b*d*e+a*e^2)*(e*f-d*g)*(d+e*x)^(1+m)/(e^4*(1+m))-(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))*(d+e*x)^(2+m)/(e^4*(2+m))+(c*e*f-3*c*d*g+b*e*g)*(d+e*x)^(3+m)/(e^4*(3+m))+c*g*(d+e*x)^(4+m)/(e^4*(4+m))],
[(d+e*x)^m*(a+b*x+c*x^2)/(f+g*x),x,3,-(c*e*f+c*d*g-b*e*g)*(d+e*x)^(1+m)/(e^2*g^2*(1+m))+c*(d+e*x)^(2+m)/(e^2*g*(2+m))+(c*f^2-b*f*g+a*g^2)*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],-g*(d+e*x)/(e*f-d*g))/(g^2*(e*f-d*g)*(1+m))],
[(d+e*x)^m*(a+b*x+c*x^2)/(f+g*x)^2,x,3,c*(d+e*x)^(1+m)/(e*g^2*(1+m))+(a+f*(c*f-b*g)/g^2)*(d+e*x)^(1+m)/((e*f-d*g)*(f+g*x))+(c*f*(2*d*g-e*f*(2+m))-g*(a*e*g*m+b*(d*g-e*f*(1+m))))*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],-g*(d+e*x)/(e*f-d*g))/(g^2*(e*f-d*g)^2*(1+m))],
[(d+e*x)^m*(a+b*x+c*x^2)/(f+g*x)^3,x,3,1/2*(a+f*(c*f-b*g)/g^2)*(d+e*x)^(1+m)/((e*f-d*g)*(f+g*x)^2)+1/2*(c*f*(4*d*g-e*f*(3+m))+g*(a*e*g*(1-m)-b*(2*d*g-e*f*(1+m))))*(d+e*x)^(1+m)/(g^2*(e*f-d*g)^2*(f+g*x))+1/2*(c*(2*d^2*g^2-4*d*e*f*g*(1+m)+e^2*f^2*(2+3*m+m^2))-e*g*m*(a*e*g*(1-m)-b*(2*d*g-e*f*(1+m))))*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],-g*(d+e*x)/(e*f-d*g))/(g^2*(e*f-d*g)^3*(1+m))],
[(d+e*x)^m*(f+g*x)^2*(a+b*x+c*x^2)^2,x,2,(c*d^2-b*d*e+a*e^2)^2*(e*f-d*g)^2*(d+e*x)^(1+m)/(e^7*(1+m))-2*(c*d^2-b*d*e+a*e^2)*(e*f-d*g)*(c*d*(2*e*f-3*d*g)-e*(b*e*f-2*b*d*g+a*e*g))*(d+e*x)^(2+m)/(e^7*(2+m))+(c^2*d^2*(6*e^2*f^2-20*d*e*f*g+15*d^2*g^2)+e^2*(a^2*e^2*g^2+2*a*b*e*g*(2*e*f-3*d*g)+b^2*(e^2*f^2-6*d*e*f*g+6*d^2*g^2))+2*c*e*(a*e*(e^2*f^2-6*d*e*f*g+6*d^2*g^2)-b*d*(3*e^2*f^2-12*d*e*f*g+10*d^2*g^2)))*(d+e*x)^(3+m)/(e^7*(3+m))+2*(b*e^2*g*(b*e*f-2*b*d*g+a*e*g)-2*c^2*d*(e^2*f^2-5*d*e*f*g+5*d^2*g^2)+c*e*(2*a*e*g*(e*f-2*d*g)+b*(e^2*f^2-8*d*e*f*g+10*d^2*g^2)))*(d+e*x)^(4+m)/(e^7*(4+m))+(b^2*e^2*g^2+2*c*e*g*(2*b*e*f-5*b*d*g+a*e*g)+c^2*(e^2*f^2-10*d*e*f*g+15*d^2*g^2))*(d+e*x)^(5+m)/(e^7*(5+m))+2*c*g*(c*e*f-3*c*d*g+b*e*g)*(d+e*x)^(6+m)/(e^7*(6+m))+c^2*g^2*(d+e*x)^(7+m)/(e^7*(7+m))],
[(d+e*x)^m*(f+g*x)*(a+b*x+c*x^2)^2,x,2,(c*d^2-b*d*e+a*e^2)^2*(e*f-d*g)*(d+e*x)^(1+m)/(e^6*(1+m))-(c*d^2-b*d*e+a*e^2)*(c*d*(4*e*f-5*d*g)-e*(2*b*e*f-3*b*d*g+a*e*g))*(d+e*x)^(2+m)/(e^6*(2+m))+(2*c^2*d^2*(3*e*f-5*d*g)+b*e^2*(b*e*f-3*b*d*g+2*a*e*g)+2*c*e*(a*e*(e*f-3*d*g)-3*b*d*(e*f-2*d*g)))*(d+e*x)^(3+m)/(e^6*(3+m))+(b^2*e^2*g-2*c^2*d*(2*e*f-5*d*g)+2*c*e*(b*e*f-4*b*d*g+a*e*g))*(d+e*x)^(4+m)/(e^6*(4+m))+c*(c*e*f-5*c*d*g+2*b*e*g)*(d+e*x)^(5+m)/(e^6*(5+m))+c^2*g*(d+e*x)^(6+m)/(e^6*(6+m))],
[(d+e*x)^m*(a+b*x+c*x^2)^2/(f+g*x),x,4,(b*e*g-c*(e*f+d*g))*(c*(e^2*f^2+d^2*g^2)+e*g*(2*a*e*g-b*(e*f+d*g)))*(d+e*x)^(1+m)/(e^4*g^4*(1+m))+(b^2*e^2*g^2+c^2*(e^2*f^2+2*d*e*f*g+3*d^2*g^2)+2*c*e*g*(a*e*g-b*(e*f+2*d*g)))*(d+e*x)^(2+m)/(e^4*g^3*(2+m))-c*(c*e*f+3*c*d*g-2*b*e*g)*(d+e*x)^(3+m)/(e^4*g^2*(3+m))+c^2*(d+e*x)^(4+m)/(e^4*g*(4+m))+(c*f^2-b*f*g+a*g^2)^2*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],-g*(d+e*x)/(e*f-d*g))/(g^4*(e*f-d*g)*(1+m))],
[(d+e*x)^m*(a+b*x+c*x^2)^2/(f+g*x)^2,x,4,(b^2*e^2*g^2+c^2*(3*e^2*f^2+2*d*e*f*g+d^2*g^2)+2*c*e*g*(a*e*g-b*(2*e*f+d*g)))*(d+e*x)^(1+m)/(e^3*g^4*(1+m))-2*c*(c*e*f+c*d*g-b*e*g)*(d+e*x)^(2+m)/(e^3*g^3*(2+m))+c^2*(d+e*x)^(3+m)/(e^3*g^2*(3+m))+(c*f^2-b*f*g+a*g^2)^2*(d+e*x)^(1+m)/(g^4*(e*f-d*g)*(f+g*x))+(c*f^2-b*f*g+a*g^2)*(c*f*(4*d*g-e*f*(4+m))-g*(a*e*g*m+b*(2*d*g-e*f*(2+m))))*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],-g*(d+e*x)/(e*f-d*g))/(g^4*(e*f-d*g)^2*(1+m))],
[(d+e*x)^m*(a+b*x+c*x^2)^2/(f+g*x)^3,x,5,-c*(3*c*e*f+c*d*g-2*b*e*g)*(d+e*x)^(1+m)/(e^2*g^4*(1+m))+c^2*(d+e*x)^(2+m)/(e^2*g^3*(2+m))+1/2*(c*f^2-b*f*g+a*g^2)^2*(d+e*x)^(1+m)/(g^4*(e*f-d*g)*(f+g*x)^2)+1/2*(c*f^2-b*f*g+a*g^2)*(c*f*(8*d*g-e*f*(7+m))+g*(a*e*g*(1-m)-b*(4*d*g-e*f*(3+m))))*(d+e*x)^(1+m)/(g^4*(e*f-d*g)^2*(f+g*x))+1/2*(c^2*f^2*(12*d^2*g^2-8*d*e*f*g*(3+m)+e^2*f^2*(12+7*m+m^2))-g^2*(a^2*e^2*g^2*(1-m)*m-2*a*b*e*g*m*(2*d*g-e*f*(1+m))-b^2*(2*d^2*g^2-4*d*e*f*g*(1+m)+e^2*f^2*(2+3*m+m^2)))+2*c*g*(a*g*(2*d^2*g^2-4*d*e*f*g*(1+m)+e^2*f^2*(2+3*m+m^2))-b*f*(6*d^2*g^2-6*d*e*f*g*(2+m)+e^2*f^2*(6+5*m+m^2))))*(d+e*x)^(1+m)*hypergeom([1,1+m],[2+m],-g*(d+e*x)/(e*f-d*g))/(g^4*(e*f-d*g)^3*(1+m))],

# p<0
[(2+3*x)^4*(1+4*x)^m/(1-5*x+3*x^2),x,4,3687/64*(1+4*x)^(1+m)/(1+m)+207/32*(1+4*x)^(2+m)/(2+m)+27/64*(1+4*x)^(3+m)/(3+m)-3/26*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13-2*sqrt(13)))*(5499-1631*sqrt(13))/((1+m)*(13-2*sqrt(13)))-3/26*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13+2*sqrt(13)))*(5499+1631*sqrt(13))/((1+m)*(13+2*sqrt(13)))],
[(2+3*x)^3*(1+4*x)^m/(1-5*x+3*x^2),x,4,123/16*(1+4*x)^(1+m)/(1+m)+9/16*(1+4*x)^(2+m)/(2+m)-3/13*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13-2*sqrt(13)))*(416-135*sqrt(13))/((1+m)*(13-2*sqrt(13)))-3/13*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13+2*sqrt(13)))*(416+135*sqrt(13))/((1+m)*(13+2*sqrt(13)))],
[(2+3*x)^2*(1+4*x)^m/(1-5*x+3*x^2),x,4,3/4*(1+4*x)^(1+m)/(1+m)-3/26*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13-2*sqrt(13)))*(117-47*sqrt(13))/((1+m)*(13-2*sqrt(13)))-3/26*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13+2*sqrt(13)))*(117+47*sqrt(13))/((1+m)*(13+2*sqrt(13)))],
[(2+3*x)*(1+4*x)^m/(1-5*x+3*x^2),x,4,-3/26*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13-2*sqrt(13)))*(13-9*sqrt(13))/((1+m)*(13-2*sqrt(13)))-3/26*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13+2*sqrt(13)))*(13+9*sqrt(13))/((1+m)*(13+2*sqrt(13)))],
[(1+4*x)^m/(1-5*x+3*x^2),x,4,3*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13-2*sqrt(13)))/((1+m)*(13-2*sqrt(13))*sqrt(13))-3*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13+2*sqrt(13)))/((1+m)*sqrt(13)*(13+2*sqrt(13)))],
[(1+4*x)^m/((2+3*x)*(1-5*x+3*x^2)),x,7,3/85*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],-3/5*(1+4*x))/(1+m)+3/442*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13+2*sqrt(13)))*(13-9*sqrt(13))/((1+m)*(13+2*sqrt(13)))+3/442*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13-2*sqrt(13)))*(13+9*sqrt(13))/((1+m)*(13-2*sqrt(13)))],
[(1+4*x)^m/((2+3*x)^2*(1-5*x+3*x^2)),x,8,27/1445*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],-3/5*(1+4*x))/(1+m)+12/425*(1+4*x)^(1+m)*hypergeom([2,1+m],[2+m],-3/5*(1+4*x))/(1+m)+3/7514*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13+2*sqrt(13)))*(117-47*sqrt(13))/((1+m)*(13+2*sqrt(13)))+3/7514*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13-2*sqrt(13)))*(117+47*sqrt(13))/((1+m)*(13-2*sqrt(13)))],
[(2+3*x)^4*(1+4*x)^m/(1-5*x+3*x^2)^2,x,5,9/4*(1+4*x)^(1+m)/(1+m)+1/39*(844-2355*x)*(1+4*x)^(1+m)/(1-5*x+3*x^2)-1/169*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13-2*sqrt(13)))*(13689-sqrt(13)*(297+4474*m-1570*m*sqrt(13)))/((1+m)*(13-2*sqrt(13)))-1/169*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13+2*sqrt(13)))*(13689+sqrt(13)*(297+4474*m+1570*m*sqrt(13)))/((1+m)*(13+2*sqrt(13)))],
[(2+3*x)^3*(1+4*x)^m/(1-5*x+3*x^2)^2,x,5,1/39*(209-426*x)*(1+4*x)^(1+m)/(1-5*x+3*x^2)+1/338*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13+2*sqrt(13)))*(-13*(117+568*m)+(1701-1168*m)*sqrt(13))/((1+m)*(13+2*sqrt(13)))-1/338*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13-2*sqrt(13)))*(1521+sqrt(13)*(1701-1168*m+568*m*sqrt(13)))/((1+m)*(13-2*sqrt(13)))],
[(2+3*x)^2*(1+4*x)^m/(1-5*x+3*x^2)^2,x,5,1/39*(61-87*x)*(1+4*x)^(1+m)/(1-5*x+3*x^2)-2/13*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13-2*sqrt(13)))*(153-m*(23-29*sqrt(13)))/((1+m)*(13-2*sqrt(13))*sqrt(13))+2/13*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13+2*sqrt(13)))*(153-m*(23+29*sqrt(13)))/((1+m)*sqrt(13)*(13+2*sqrt(13)))],
[(2+3*x)*(1+4*x)^m/(1-5*x+3*x^2)^2,x,5,1/39*(20-21*x)*(1+4*x)^(1+m)/(1-5*x+3*x^2)+1/13*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13+2*sqrt(13)))*(81+2*m*(5-7*sqrt(13)))/((1+m)*sqrt(13)*(13+2*sqrt(13)))-1/13*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13-2*sqrt(13)))*(81+2*m*(5+7*sqrt(13)))/((1+m)*(13-2*sqrt(13))*sqrt(13))],
[(1+4*x)^m/(1-5*x+3*x^2)^2,x,5,1/39*(7-6*x)*(1+4*x)^(1+m)/(1-5*x+3*x^2)+2/13*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13+2*sqrt(13)))*(9+2*m*(2-sqrt(13)))/((1+m)*sqrt(13)*(13+2*sqrt(13)))-2/13*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13-2*sqrt(13)))*(9+2*m*(2+sqrt(13)))/((1+m)*(13-2*sqrt(13))*sqrt(13))],
[(1+4*x)^m/((2+3*x)*(1-5*x+3*x^2)^2),x,12,1/663*(43-33*x)*(1+4*x)^(1+m)/(1-5*x+3*x^2)+9/1445*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],-3/5*(1+4*x))/(1+m)+9/7514*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13+2*sqrt(13)))*(13-9*sqrt(13))/((1+m)*(13+2*sqrt(13)))+1/221*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13+2*sqrt(13)))*(81+m*(62-22*sqrt(13)))/((1+m)*sqrt(13)*(13+2*sqrt(13)))+9/7514*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13-2*sqrt(13)))*(13+9*sqrt(13))/((1+m)*(13-2*sqrt(13)))-1/221*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13-2*sqrt(13)))*(81+m*(62+22*sqrt(13)))/((1+m)*(13-2*sqrt(13))*sqrt(13))],
[(1+4*x)^m/((2+3*x)^2*(1-5*x+3*x^2)^2),x,13,1/11271*(268-195*x)*(1+4*x)^(1+m)/(1-5*x+3*x^2)+162/24565*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],-3/5*(1+4*x))/(1+m)+36/7225*(1+4*x)^(1+m)*hypergeom([2,1+m],[2+m],-3/5*(1+4*x))/(1+m)+9/63869*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13+2*sqrt(13)))*(117-64*sqrt(13))/((1+m)*(13+2*sqrt(13)))+1/3757*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13+2*sqrt(13)))*(423+m*(422-130*sqrt(13)))/((1+m)*sqrt(13)*(13+2*sqrt(13)))+9/63869*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13-2*sqrt(13)))*(117+64*sqrt(13))/((1+m)*(13-2*sqrt(13)))-1/3757*(1+4*x)^(1+m)*hypergeom([1,1+m],[2+m],3*(1+4*x)/(13-2*sqrt(13)))*(423+2*m*(211+65*sqrt(13)))/((1+m)*(13-2*sqrt(13))*sqrt(13))],

# Integrands of the form (d+e x)^m (f+g x)^(n/2) (a+b x+c x^2)^p when m symbolic

# p>0
[(d+e*x)^m*(a+b*x+c*x^2)/(e+f*x)^(3/2),x,4,2*(a+e*(c*e-b*f)/f^2)*(d+e*x)^(1+m)/((e^2-d*f)*sqrt(e+f*x))+2*c*(d+e*x)^(1+m)*sqrt(e+f*x)/(e*f^2*(3+2*m))+2*(c*(d^2*f^2+4*d*e^2*f*(1+m)-4*e^4*(2+3*m+m^2))-e*f*(3+2*m)*(a*e*f*(1+2*m)+b*(d*f-2*e^2*(1+m))))*(d+e*x)^m*hypergeom([1/2,-m],[3/2],e*(e+f*x)/(e^2-d*f))*sqrt(e+f*x)/(e*f^3*(e^2-d*f)*(3+2*m)*(-f*(d+e*x)/(e^2-d*f))^m)],

# p<0

# Integrands of the form (d+e x)^m (f+g x)^n (a+b x+c x^2)^(p/2) when m symbolic

# p>0
[(d+e*x)^m*(f+g*x)^2*sqrt(a+b*x+c*x^2),x,6,g^2*(d+e*x)^(1+m)*(a+b*x+c*x^2)^(3/2)/(c*e*(4+m))+(e*(b*d-a*e)*g^2*(1+m)+c*(3*d^2*g^2+e^2*f^2*(4+m)-2*d*e*f*g*(4+m)))*(d+e*x)^(1+m)*AppellF1(1+m,-1/2,-1/2,2+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(a+b*x+c*x^2)/(c*e^3*(1+m)*(4+m)*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))-1/2*g*(b*e*g*(5+2*m)+2*c*(3*d*g-2*e*f*(4+m)))*(d+e*x)^(2+m)*AppellF1(2+m,-1/2,-1/2,3+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(a+b*x+c*x^2)/(c*e^3*(2+m)*(4+m)*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))],
[(d+e*x)^m*(f+g*x)*sqrt(a+b*x+c*x^2),x,5,(e*f-d*g)*(d+e*x)^(1+m)*AppellF1(1+m,-1/2,-1/2,2+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(a+b*x+c*x^2)/(e^2*(1+m)*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))+g*(d+e*x)^(2+m)*AppellF1(2+m,-1/2,-1/2,3+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(a+b*x+c*x^2)/(e^2*(2+m)*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))],
[(d+e*x)^m*sqrt(a+b*x+c*x^2),x,2,(d+e*x)^(1+m)*AppellF1(1+m,-1/2,-1/2,2+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(a+b*x+c*x^2)/(e*(1+m)*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c)))))],
[(d+e*x)^m*sqrt(a+b*x+c*x^2)/(f+g*x),x,0,Unintegrable((d+e*x)^m*sqrt(a+b*x+c*x^2)/(f+g*x),x)],

# p<0
[(d+e*x)^m*(f+g*x)^2/sqrt(a+b*x+c*x^2),x,6,g^2*(d+e*x)^(1+m)*sqrt(a+b*x+c*x^2)/(c*e*(2+m))+(e*(b*d-a*e)*g^2*(1+m)+c*(d^2*g^2+e^2*f^2*(2+m)-2*d*e*f*g*(2+m)))*(d+e*x)^(1+m)*AppellF1(1+m,1/2,1/2,2+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c*e^3*(1+m)*(2+m)*sqrt(a+b*x+c*x^2))-1/2*g*(b*e*g*(3+2*m)+c*(2*d*g-4*e*f*(2+m)))*(d+e*x)^(2+m)*AppellF1(2+m,1/2,1/2,3+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c*e^3*(2+m)^2*sqrt(a+b*x+c*x^2))],
[(d+e*x)^m*(f+g*x)/sqrt(a+b*x+c*x^2),x,5,(e*f-d*g)*(d+e*x)^(1+m)*AppellF1(1+m,1/2,1/2,2+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e^2*(1+m)*sqrt(a+b*x+c*x^2))+g*(d+e*x)^(2+m)*AppellF1(2+m,1/2,1/2,3+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e^2*(2+m)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^m/sqrt(a+b*x+c*x^2),x,2,(d+e*x)^(1+m)*AppellF1(1+m,1/2,1/2,2+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e*(1+m)*sqrt(a+b*x+c*x^2))],
[(d+e*x)^m/((f+g*x)*sqrt(a+b*x+c*x^2)),x,0,Unintegrable((d+e*x)^m/((f+g*x)*sqrt(a+b*x+c*x^2)),x)],

# Integrands of the form (d+e x)^m (f+g x)^n (a+b x+c x^2)^p when m and n symbolic
[(d+e*x)^m*(f+g*x)^n*(a+b*x+c*x^2),x,4,(b*e*g*(3+m+n)-c*(e*f*(2+m)+d*g*(4+m+2*n)))*(d+e*x)^(1+m)*(f+g*x)^(1+n)/(e^2*g^2*(2+m+n)*(3+m+n))+c*(d+e*x)^(2+m)*(f+g*x)^(1+n)/(e^2*g*(3+m+n))+(g*(2+m+n)*(a*e^2*g*(3+m+n)-c*d*(e*f*(2+m)+d*g*(1+n)))-(e*f*(1+m)+d*g*(1+n))*(b*e*g*(3+m+n)-c*(e*f*(2+m)+d*g*(4+m+2*n))))*(d+e*x)^(1+m)*(f+g*x)^n*hypergeom([1+m,-n],[2+m],-g*(d+e*x)/(e*f-d*g))/(e^3*g^2*(1+m)*(2+m+n)*(3+m+n)*(e*(f+g*x)/(e*f-d*g))^n)],

# Integrands of the form (d+e x)^m (f+g x)^n (a+b x+c x^2)^p when m and p symbolic
[(d+e*x)^m*(f+g*x)^2*(a+b*x+c*x^2)^p,x,6,g^2*(d+e*x)^(1+m)*(a+b*x+c*x^2)^(1+p)/(c*e*(3+m+2*p))+(e*(b*d-a*e)*g^2*(1+m)+c*(2*d^2*g^2*(1+p)+e^2*f^2*(3+m+2*p)-2*d*e*f*g*(3+m+2*p)))*(d+e*x)^(1+m)*(a+b*x+c*x^2)^p*AppellF1(1+m,-p,-p,2+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c*e^3*(1+m)*(3+m+2*p)*(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))^p*(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))^p)-g*(b*e*g*(2+m+p)+2*c*(d*g*(1+p)-e*f*(3+m+2*p)))*(d+e*x)^(2+m)*(a+b*x+c*x^2)^p*AppellF1(2+m,-p,-p,3+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c*e^3*(2+m)*(3+m+2*p)*(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))^p*(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))^p)],
[(d+e*x)^m*(f+g*x)*(a+b*x+c*x^2)^p,x,5,(e*f-d*g)*(d+e*x)^(1+m)*(a+b*x+c*x^2)^p*AppellF1(1+m,-p,-p,2+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e^2*(1+m)*(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))^p*(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))^p)+g*(d+e*x)^(2+m)*(a+b*x+c*x^2)^p*AppellF1(2+m,-p,-p,3+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e^2*(2+m)*(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))^p*(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))^p)],
[(d+e*x)^m*(a+b*x+c*x^2)^p,x,2,(d+e*x)^(1+m)*(a+b*x+c*x^2)^p*AppellF1(1+m,-p,-p,2+m,2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))),2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e*(1+m)*(1-2*c*(d+e*x)/(2*c*d-e*(b-sqrt(b^2-4*a*c))))^p*(1-2*c*(d+e*x)/(2*c*d-e*(b+sqrt(b^2-4*a*c))))^p)],
[(d+e*x)^m*(a+b*x+c*x^2)^p/(f+g*x),x,0,Unintegrable((d+e*x)^m*(a+b*x+c*x^2)^p/(f+g*x),x)],

# Integrands of the form (g x)^m (d+e x)^n (a+b/x+c/x^2)^p

# Integrands of the form (g x)^m (d+e x)^n (a+b/x+c/x^2)^p with b=0

# Integrands of the form x^m (d+e x)^(n/2) (a+c/x^2)^p
[1/(x^2*sqrt(1+(-1)/(c^2*x^2))*sqrt(d+e*x)),x,5,-2*EllipticPi(sqrt(1-c*x)/sqrt(2),2,sqrt(2*e/(c*d+e)))*sqrt(c*(d+e*x)/(c*d+e))*sqrt(1-c^2*x^2)/(x*sqrt(1+(-1)/(c^2*x^2))*sqrt(d+e*x))]]:
