# Maple integration test file: "1 Algebraic functions\1.3 Miscellaneous\1.3.1 Rational functions.txt"

lst:=[

# Integrands of the form P[x]^p

# Integrands of the form P3[x]^p
[1/(-9*b*x+9*x^3+2*b^(3/2)*sqrt(3)),x,3,-1/27*log(-x*sqrt(3)+sqrt(b))/b+1/27*log(x*sqrt(3)+2*sqrt(b))/b+1/3/(sqrt(3)*sqrt(b)*(-3*x+sqrt(3)*sqrt(b)))],
[(a^3+3*a^2*b*x+3*a*b^2*x^2+b^3*x^3)^p,x,3,(a/b+x)*(b^3*(a/b+x)^3)^p/(1+3*p)],
[(a^3+3*a^2*b*x+3*a*b^2*x^2+b^3*x^3)^3,x,2,1/10*(a+b*x)^10/b],
[(a^3+3*a^2*b*x+3*a*b^2*x^2+b^3*x^3)^2,x,2,1/7*(a+b*x)^7/b],
[a^3+3*a^2*b*x+3*a*b^2*x^2+b^3*x^3,x,1,a^3*x+3/2*a^2*b*x^2+a*b^2*x^3+1/4*b^3*x^4],
[1/(a^3+3*a^2*b*x+3*a*b^2*x^2+b^3*x^3),x,2,(-1/2)/(b*(a+b*x)^2)],
[1/(a^3+3*a^2*b*x+3*a*b^2*x^2+b^3*x^3)^2,x,2,(-1/5)/(b*(a+b*x)^5)],
[1/(a^3+3*a^2*b*x+3*a*b^2*x^2+b^3*x^3)^3,x,2,(-1/8)/(b*(a+b*x)^8)],
[(3*a*b+3*b^2*x+3*b*c*x^2+c^2*x^3)^3,x,3,-b^3*(b^2-3*a*c)^3*x/c^3+3/4*b^2*(b^2-3*a*c)^2*(b+c*x)^4/c^4-3/7*b*(b^2-3*a*c)*(b+c*x)^7/c^4+1/10*(b+c*x)^10/c^4],
[(3*a*b+3*b^2*x+3*b*c*x^2+c^2*x^3)^2,x,3,b^2*(b^2-3*a*c)^2*x/c^2-1/2*b*(b^2-3*a*c)*(b+c*x)^4/c^3+1/7*(b+c*x)^7/c^3],
[3*a*b+3*b^2*x+3*b*c*x^2+c^2*x^3,x,1,3*a*b*x+3/2*b^2*x^2+b*c*x^3+1/4*c^2*x^4],
[1/(3*a*b+3*b^2*x+3*b*c*x^2+c^2*x^3),x,7,1/3*log(b-b^(1/3)*(b^2-3*a*c)^(1/3)+c*x)/(b^(2/3)*(b^2-3*a*c)^(2/3))-1/6*log(b^(2/3)*(b^2-3*a*c)^(2/3)+b^(1/3)*c*(b^2-3*a*c)^(1/3)*(b/c+x)+c^2*(b/c+x)^2)/(b^(2/3)*(b^2-3*a*c)^(2/3))-arctan((b^(1/3)+2*(b+c*x)/(b^2-3*a*c)^(1/3))/(b^(1/3)*sqrt(3)))/(b^(2/3)*(b^2-3*a*c)^(2/3)*sqrt(3))],
[1/(3*a*b+3*b^2*x+3*b*c*x^2+c^2*x^3)^2,x,8,-1/3*c*(b/c+x)/(b*(b^2-3*a*c)*(3*a*b+3*b^2*x+3*b*c*x^2+c^2*x^3))-2/9*c*log(b-b^(1/3)*(b^2-3*a*c)^(1/3)+c*x)/(b^(5/3)*(b^2-3*a*c)^(5/3))+1/9*c*log(b^(2/3)*(b^2-3*a*c)^(2/3)+b^(1/3)*c*(b^2-3*a*c)^(1/3)*(b/c+x)+c^2*(b/c+x)^2)/(b^(5/3)*(b^2-3*a*c)^(5/3))+2/3*c*arctan((b^(1/3)+2*(b+c*x)/(b^2-3*a*c)^(1/3))/(b^(1/3)*sqrt(3)))/(b^(5/3)*(b^2-3*a*c)^(5/3)*sqrt(3))],
[1/(3*a*b+3*b^2*x+3*b*c*x^2+c^2*x^3)^3,x,9,-1/6*c*(b/c+x)/(b*(b^2-3*a*c)*(3*a*b+3*b^2*x+3*b*c*x^2+c^2*x^3)^2)+5/18*c^2*(b/c+x)/(b^2*(b^2-3*a*c)^2*(3*a*b+3*b^2*x+3*b*c*x^2+c^2*x^3))+5/27*c^2*log(b-b^(1/3)*(b^2-3*a*c)^(1/3)+c*x)/(b^(8/3)*(b^2-3*a*c)^(8/3))-5/54*c^2*log(b^(2/3)*(b^2-3*a*c)^(2/3)+b^(1/3)*c*(b^2-3*a*c)^(1/3)*(b/c+x)+c^2*(b/c+x)^2)/(b^(8/3)*(b^2-3*a*c)^(8/3))-5/9*c^2*arctan((b^(1/3)+2*(b+c*x)/(b^2-3*a*c)^(1/3))/(b^(1/3)*sqrt(3)))/(b^(8/3)*(b^2-3*a*c)^(8/3)*sqrt(3))],
[(a*c*e+(b*c*e+a*d*e+a*c*f)*x+(b*d*e+b*c*f+a*d*f)*x^2+b*d*f*x^3)^3,x,3,1/4*(b*c-a*d)^3*(b*e-a*f)^3*(a+b*x)^4/b^7+3/5*(b*c-a*d)^2*(b*e-a*f)^2*(b*d*e+b*c*f-2*a*d*f)*(a+b*x)^5/b^7+1/2*(b*c-a*d)*(b*e-a*f)*(5*a^2*d^2*f^2-5*a*b*d*f*(d*e+c*f)+b^2*(d^2*e^2+3*c*d*e*f+c^2*f^2))*(a+b*x)^6/b^7+1/7*(b*d*e+b*c*f-2*a*d*f)*(10*a^2*d^2*f^2-10*a*b*d*f*(d*e+c*f)+b^2*(d^2*e^2+8*c*d*e*f+c^2*f^2))*(a+b*x)^7/b^7+3/8*d*f*(5*a^2*d^2*f^2-5*a*b*d*f*(d*e+c*f)+b^2*(d^2*e^2+3*c*d*e*f+c^2*f^2))*(a+b*x)^8/b^7+1/3*d^2*f^2*(b*d*e+b*c*f-2*a*d*f)*(a+b*x)^9/b^7+1/10*d^3*f^3*(a+b*x)^10/b^7],
[(a*c*e+(b*c*e+a*d*e+a*c*f)*x+(b*d*e+b*c*f+a*d*f)*x^2+b*d*f*x^3)^2,x,3,1/3*(b*c-a*d)^2*(b*e-a*f)^2*(a+b*x)^3/b^5+1/2*(b*c-a*d)*(b*e-a*f)*(b*d*e+b*c*f-2*a*d*f)*(a+b*x)^4/b^5+1/5*(6*a^2*d^2*f^2-6*a*b*d*f*(d*e+c*f)+b^2*(d^2*e^2+4*c*d*e*f+c^2*f^2))*(a+b*x)^5/b^5+1/3*d*f*(b*d*e+b*c*f-2*a*d*f)*(a+b*x)^6/b^5+1/7*d^2*f^2*(a+b*x)^7/b^5],
[a*c*e+(b*c*e+a*d*e+a*c*f)*x+(b*d*e+b*c*f+a*d*f)*x^2+b*d*f*x^3,x,1,a*c*e*x+1/2*(b*c*e+a*d*e+a*c*f)*x^2+1/3*(b*d*e+b*c*f+a*d*f)*x^3+1/4*b*d*f*x^4],
[1/(a*c*e+(b*c*e+a*d*e+a*c*f)*x+(b*d*e+b*c*f+a*d*f)*x^2+b*d*f*x^3),x,2,b*log(a+b*x)/((b*c-a*d)*(b*e-a*f))-d*log(c+d*x)/((b*c-a*d)*(d*e-c*f))+f*log(e+f*x)/((b*e-a*f)*(d*e-c*f))],
[1/(a*c*e+(b*c*e+a*d*e+a*c*f)*x+(b*d*e+b*c*f+a*d*f)*x^2+b*d*f*x^3)^2,x,2,-b^3/((b*c-a*d)^2*(b*e-a*f)^2*(a+b*x))-d^3/((b*c-a*d)^2*(d*e-c*f)^2*(c+d*x))-f^3/((b*e-a*f)^2*(d*e-c*f)^2*(e+f*x))-2*b^3*(b*d*e+b*c*f-2*a*d*f)*log(a+b*x)/((b*c-a*d)^3*(b*e-a*f)^3)+2*d^3*(b*d*e-2*b*c*f+a*d*f)*log(c+d*x)/((b*c-a*d)^3*(d*e-c*f)^3)+2*f^3*(2*b*d*e-b*c*f-a*d*f)*log(e+f*x)/((b*e-a*f)^3*(d*e-c*f)^3)],
[1/(a*c*e+(b*c*e+a*d*e+a*c*f)*x+(b*d*e+b*c*f+a*d*f)*x^2+b*d*f*x^3)^3,x,2,-1/2*b^5/((b*c-a*d)^3*(b*e-a*f)^3*(a+b*x)^2)+3*b^5*(b*d*e+b*c*f-2*a*d*f)/((b*c-a*d)^4*(b*e-a*f)^4*(a+b*x))+1/2*d^5/((b*c-a*d)^3*(d*e-c*f)^3*(c+d*x)^2)+3*d^5*(b*d*e-2*b*c*f+a*d*f)/((b*c-a*d)^4*(d*e-c*f)^4*(c+d*x))-1/2*f^5/((b*e-a*f)^3*(d*e-c*f)^3*(e+f*x)^2)-3*f^5*(2*b*d*e-b*c*f-a*d*f)/((b*e-a*f)^4*(d*e-c*f)^4*(e+f*x))+3*b^5*(7*a^2*d^2*f^2-7*a*b*d*f*(d*e+c*f)+b^2*(2*d^2*e^2+3*c*d*e*f+2*c^2*f^2))*log(a+b*x)/((b*c-a*d)^5*(b*e-a*f)^5)-3*d^5*(2*a^2*d^2*f^2+a*b*d*f*(3*d*e-7*c*f)+b^2*(2*d^2*e^2-7*c*d*e*f+7*c^2*f^2))*log(c+d*x)/((b*c-a*d)^5*(d*e-c*f)^5)+3*f^5*(2*a^2*d^2*f^2-a*b*d*f*(7*d*e-3*c*f)+b^2*(7*d^2*e^2-7*c*d*e*f+2*c^2*f^2))*log(e+f*x)/((b*e-a*f)^5*(d*e-c*f)^5)],
[1/(1+x+x^2+x^3),x,5,1/2*arctan(x)+1/2*log(1+x)-1/4*log(1+x^2)],
[1/(-1+4*x-4*x^2+16*x^3),x,5,-1/10*arctan(2*x)+1/5*log(1-4*x)-1/10*log(1+4*x^2)],
[1/(d*x^3),x,2,(-1/2)/(d*x^2)],
[1/(c*x^2+d*x^3),x,3,(-1)/(c*x)-d*log(x)/c^2+d*log(c+d*x)/c^2],
[1/(b*x+d*x^3),x,5,log(x)/b-1/2*log(b+d*x^2)/b],
[1/(b*x+c*x^2+d*x^3),x,7,log(x)/b-1/2*log(b+c*x+d*x^2)/b+c*arctanh((c+2*d*x)/sqrt(c^2-4*b*d))/(b*sqrt(c^2-4*b*d))],
[1/(a+d*x^3),x,6,1/3*log(a^(1/3)+d^(1/3)*x)/(a^(2/3)*d^(1/3))-1/6*log(a^(2/3)-a^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/(a^(2/3)*d^(1/3))-arctan((a^(1/3)-2*d^(1/3)*x)/(a^(1/3)*sqrt(3)))/(a^(2/3)*d^(1/3)*sqrt(3))],

#  {1/(a + 0*x + c*x^2 + d*x^3), x, 3, -((72*2^(1/3)*d*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(2/3)*Log[2^(2/3)*c^2 - c*(-4*c^3 - 6*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(1/3) + (-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(2/3) - 3*2^(1/3)*d*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(1/3)*x])/((2*(3*I + Sqrt[3])*c^2 + 2^(1/3)*(3*I - Sqrt[3])*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(2/3))*(2*(3*I - Sqrt[3])*c^2 + 2^(1/3)*(3*I + Sqrt[3])*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(2/3)))) - (12*2^(1/3)*Sqrt[3]*d*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(2/3)*Log[(2^(1/3)*c + (-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(1/3))*(2^(1/3)*(I + Sqrt[3])*c + (I - Sqrt[3])*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(1/3)) + 6*I*2^(1/3)*d*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(1/3)*x])/((2*c^2 - 2^(1/3)*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(2/3))*(2*(3*I + Sqrt[3])*c^2 + 2^(1/3)*(3*I - Sqrt[3])*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(2/3))) + (12*2^(1/3)*Sqrt[3]*d*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(2/3)*Log[(2^(1/3)*c + (-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(1/3))*(2^(1/3)*(I - Sqrt[3])*c + (I + Sqrt[3])*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(1/3)) + 6*I*2^(1/3)*d*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(1/3)*x])/((2*c^2 - 2^(1/3)*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(2/3))*(2*(3*I - Sqrt[3])*c^2 + 2^(1/3)*(3*I + Sqrt[3])*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(2/3)))} 

#  {1/(a + b*x + 0*x^2 + d*x^3), x, 3, -((6*3^(1/6)*d^(1/3)*(3*Sqrt[3]*a*d - Sqrt[d*(4*b^3 + 27*a^2*d)])*Log[2^(2/3)*3^(1/3)*b*d^(1/3) - (9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3) - 2^(1/3)*3^(2/3)*d^(2/3)*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(1/3)*x])/(18*3^(2/3)*a*b*d^(4/3) - 6*3^(1/6)*b*d^(1/3)*Sqrt[d*(4*b^3 + 27*a^2*d)] - 6*2^(2/3)*b^2*d^(2/3)*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(1/3) - 9*6^(1/3)*a*d*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3) + 2^(1/3)*3^(5/6)*Sqrt[d*(4*b^3 + 27*a^2*d)]*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3))) - (12*2^(1/3)*3^(1/6)*d^(1/3)*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3)*Log[2^(2/3)*3^(1/3)*(1 - I*Sqrt[3])*b*d^(1/3) - (1 + I*Sqrt[3])*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3) + 2*2^(1/3)*3^(2/3)*d^(2/3)*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(1/3)*x])/((6*b*d^(1/3) + 2^(1/3)*(27*a*d - 3*Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3))*(2*(3*I + Sqrt[3])*b*d^(1/3) + 2^(1/3)*3^(1/6)*(1 - I*Sqrt[3])*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3))) + (12*2^(1/3)*3^(2/3)*d^(1/3)*(3*Sqrt[3]*a*d - Sqrt[d*(4*b^3 + 27*a^2*d)])*Log[2^(2/3)*3^(1/3)*(1 + I*Sqrt[3])*b*d^(1/3) + I*(I + Sqrt[3])*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3) + 2*2^(1/3)*3^(2/3)*d^(2/3)*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(1/3)*x])/((9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(1/3)*(6*b*d^(1/3) + 2^(1/3)*(27*a*d - 3*Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3))*(2*(3*I - Sqrt[3])*b*d^(1/3) + I*2^(1/3)*3^(1/6)*(I - Sqrt[3])*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3)))} 

#  {1/(a + b*x + c*x^2 + d*x^3), x, 0, 0} 
[(d*x^3)^n,x,2,x*(d*x^3)^n/(1+3*n)],
[(c*x^2+d*x^3)^n,x,3,x*(c*x^2+d*x^3)^n*hypergeom([-n,1+2*n],[2*(1+n)],-d*x/c)/((1+2*n)*(1+d*x/c)^n)],
[(b*x+d*x^3)^n,x,3,x*(b+d*x^2)*(b*x+d*x^3)^n*hypergeom([1,3/2*(1+n)],[1/2*(3+n)],-d*x^2/b)/(b*(1+n)),x*(b*x+d*x^3)^n*hypergeom([-n,1/2*(1+n)],[1/2*(3+n)],-d*x^2/b)/((1+n)*(1+d*x^2/b)^n)],
[(b*x+c*x^2+d*x^3)^n,x,3,x*(b*x+c*x^2+d*x^3)^n*AppellF1(1+n,-n,-n,2+n,-2*d*x/(c-sqrt(c^2-4*b*d)),-2*d*x/(c+sqrt(c^2-4*b*d)))/((1+n)*(1+2*d*x/(c-sqrt(c^2-4*b*d)))^n*(1+2*d*x/(c+sqrt(c^2-4*b*d)))^n)],
[(a+d*x^3)^n,x,2,x*(a+d*x^3)^(1+n)*hypergeom([1,4/3+n],[4/3],-d*x^3/a)/a,x*(a+d*x^3)^n*hypergeom([1/3,-n],[4/3],-d*x^3/a)/(1+d*x^3/a)^n],

#  {(a + 0*x + c*x^2 + d*x^3)^n, x, 2, (1/(3*d*(1 + n)))*((2^(-1 + (2*n)/3)*(2*c - (2*2^(1/3)*c^2)/(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(1/3) - 2^(2/3)*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(1/3) + 6*d*x)*(a + c*x^2 + d*x^3)^n*AppellF1[1 + n, -n, -n, 2 + n, -((3*d*(c - (2*c^2 + 2^(1/3)*(-2*c^3 - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[a*(4*c^3 + 27*a*d^2)])^(2/3))/(2^(2/3)*(-2*c^3 - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[a*(4*c^3 + 27*a*d^2)])^(1/3)) + 3*d*x))/(-3*d*(c - (2*c^2 + 2^(1/3)*(-2*c^3 - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[a*(4*c^3 + 27*a*d^2)])^(2/3))/(2^(2/3)*(-2*c^3 - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[a*(4*c^3 + 27*a*d^2)])^(1/3))) + 3*d*(c + (2*(1 + I*Sqrt[3])*c^2 + 2^(1/3)*(1 - I*Sqrt[3])*(-2*c^3 - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[a*(4*c^3 + 27*a*d^2)])^(2/3))/(2*2^(2/3)*(-2*c^3 - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[a*(4*c^3 + 27*a*d^2)])^(1/3))))), -((3*d*(c - (2*c^2 + 2^(1/3)*(-2*c^3 - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[a*(4*c^3 + 27*a*d^2)])^(2/3))/(2^(2/3)*(-2*c^3 - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[a*(4*c^3 + 27*a*d^2)])^(1/3)) + 3*d*x))/(-3*d*(c - (2*c^2 + 2^(1/3)*(-2*c^3 - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[a*(4*c^3 + 27*a*d^2)])^(2/3))/(2^(2/3)*(-2*c^3 - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[a*(4*c^3 + 27*a*d^2)])^(1/3))) + 3*d*(c + (2*(1 - I*Sqrt[3])*c^2 + 2^(1/3)*(1 + I*Sqrt[3])*(-2*c^3 - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[a*(4*c^3 + 27*a*d^2)])^(2/3))/(2*2^(2/3)*(-2*c^3 - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[a*(4*c^3 + 27*a*d^2)])^(1/3)))))])/(((I*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(1/3)*(4*c + (2^(1/3)*((2 + 2*I*Sqrt[3])*c^2 + 2^(1/3)*(1 - I*Sqrt[3])*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(2/3)))/(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(1/3) + 12*d*x))/(2*(3*I - Sqrt[3])*c^2 + 2^(1/3)*(3*I + Sqrt[3])*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(2/3)))^n*((I*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(1/3)*(4*c + (2^(1/3)*((2 - 2*I*Sqrt[3])*c^2 + 2^(1/3)*(1 + I*Sqrt[3])*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(2/3)))/(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(1/3) + 12*d*x))/(2*(3*I + Sqrt[3])*c^2 + 2^(1/3)*(3*I - Sqrt[3])*(-2*c^3 - 3*d*(9*a*d - Sqrt[3]*Sqrt[a*(4*c^3 + 27*a*d^2)]))^(2/3)))^n))} 

#  {(a + b*x + 0*x^2 + d*x^3)^n, x, 2, -((1/(d^(2/3)*(1 + n)))*((2^(-2 + (2*n)/3)*3^(-2 + (17*n)/6)*((2^(1/3)*3^(2/3)*(6*I*(I - Sqrt[3])*b*d^(1/3) - 2^(1/3)*3^(1/6)*(3*I - Sqrt[3])*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3)))/(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(1/3) - 36*d^(2/3)*x)*(a + b*x + d*x^3)^n*AppellF1[1 + n, -n, -n, 2 + n, (3*d*(-((6*(1 + I*Sqrt[3])*b*d - 2^(1/3)*3^(2/3)*(1 - I*Sqrt[3])*d^(2/3)*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3))/(2*2^(2/3)*3^(1/3)*d^(1/3)*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(1/3))) - 3*d*x))/(-(((3/2)^(2/3)*d^(2/3)*(6*(1 + I*Sqrt[3])*b*d - 2^(1/3)*3^(2/3)*(1 - I*Sqrt[3])*d^(2/3)*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3)))/(2*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(1/3))) + ((3/2)^(2/3)*d^(2/3)*(6*(1 - I*Sqrt[3])*b*d - 2^(1/3)*3^(2/3)*(1 + I*Sqrt[3])*d^(2/3)*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3)))/(2*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(1/3))), -((3*d*(-((6*(1 + I*Sqrt[3])*b*d - 2^(1/3)*3^(2/3)*(1 - I*Sqrt[3])*d^(2/3)*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3))/(2*2^(2/3)*3^(1/3)*d^(1/3)*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(1/3))) - 3*d*x))/(((3/2)^(2/3)*d^(2/3)*(6*b*d - 2^(1/3)*3^(2/3)*d^(2/3)*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3)))/(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(1/3) + ((3/2)^(2/3)*d^(2/3)*(6*(1 + I*Sqrt[3])*b*d - 2^(1/3)*3^(2/3)*(1 - I*Sqrt[3])*d^(2/3)*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3)))/(2*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(1/3))))])/((-((I*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(1/3)*((2^(1/3)*3^(2/3)*(6*I*(I + Sqrt[3])*b*d^(1/3) + 2^(1/3)*3^(1/6)*(3*I + Sqrt[3])*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3)))/(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(1/3) - 36*d^(2/3)*x))/(6*b*d^(1/3) + 2^(1/3)*(27*a*d - 3*Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3))))^n*(-(((9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(1/3)*((2^(1/3)*3^(2/3)*(6*b*d^(1/3) - 2^(1/3)*(27*a*d - 3*Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3)))/(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(1/3) - 18*d^(2/3)*x))/(2*I*(3*I - Sqrt[3])*b*d^(1/3) - 2^(1/3)*3^(1/6)*(I - Sqrt[3])*(9*a*d - Sqrt[3]*Sqrt[d*(4*b^3 + 27*a^2*d)])^(2/3))))^n)))} 

#  {(a + b*x + c*x^2 + d*x^3)^n, x, 2, (1/(3*d*(1 + n)))*((2^(-1 + (2*n)/3)*(2*c - (2^(1/3)*(2*c^2 - 6*b*d + 2^(1/3)*(-2*c^3 + 9*b*c*d - 3*d*(9*a*d - Sqrt[3]*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2]))^(2/3)))/(-2*c^3 + 9*b*c*d - 3*d*(9*a*d - Sqrt[3]*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2]))^(1/3) + 6*d*x)*(a + b*x + c*x^2 + d*x^3)^n*AppellF1[1 + n, -n, -n, 2 + n, -((3*d*(c - (2*c^2 - 6*b*d + 2^(1/3)*(-2*c^3 + 9*b*c*d - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2])^(2/3))/(2^(2/3)*(-2*c^3 + 9*b*c*d - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2])^(1/3)) + 3*d*x))/(-3*d*(c - (2*c^2 - 6*b*d + 2^(1/3)*(-2*c^3 + 9*b*c*d - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2])^(2/3))/(2^(2/3)*(-2*c^3 + 9*b*c*d - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2])^(1/3))) + 3*d*(c + (2*(1 - I*Sqrt[3])*c^2 - 6*(1 - I*Sqrt[3])*b*d + I*2^(1/3)*(-I + Sqrt[3])*(-2*c^3 + 9*b*c*d - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2])^(2/3))/(2*2^(2/3)*(-2*c^3 + 9*b*c*d - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2])^(1/3))))), -((3*d*(c - (2*c^2 - 6*b*d + 2^(1/3)*(-2*c^3 + 9*b*c*d - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2])^(2/3))/(2^(2/3)*(-2*c^3 + 9*b*c*d - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2])^(1/3)) + 3*d*x))/(-3*d*(c - (2*c^2 - 6*b*d + 2^(1/3)*(-2*c^3 + 9*b*c*d - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2])^(2/3))/(2^(2/3)*(-2*c^3 + 9*b*c*d - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2])^(1/3))) + 3*d*(c + (2*(1 + I*Sqrt[3])*c^2 - 6*(1 + I*Sqrt[3])*b*d - I*2^(1/3)*(I + Sqrt[3])*(-2*c^3 + 9*b*c*d - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2])^(2/3))/(2*2^(2/3)*(-2*c^3 + 9*b*c*d - 27*a*d^2 + 3*Sqrt[3]*d*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2])^(1/3)))))])/(((I*(-2*c^3 + 9*b*c*d - 3*d*(9*a*d - Sqrt[3]*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2]))^(1/3)*(4*c + (2^(1/3)*((2 + 2*I*Sqrt[3])*c^2 + 6*I*(I - Sqrt[3])*b*d + 2^(1/3)*(1 - I*Sqrt[3])*(-2*c^3 + 9*b*c*d - 3*d*(9*a*d - Sqrt[3]*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2]))^(2/3)))/(-2*c^3 + 9*b*c*d - 3*d*(9*a*d - Sqrt[3]*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2]))^(1/3) + 12*d*x))/(2*(3*I - Sqrt[3])*c^2 - 6*(3*I - Sqrt[3])*b*d + 2^(1/3)*(3*I + Sqrt[3])*(-2*c^3 + 9*b*c*d - 3*d*(9*a*d - Sqrt[3]*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2]))^(2/3)))^n*((I*(-2*c^3 + 9*b*c*d - 3*d*(9*a*d - Sqrt[3]*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2]))^(1/3)*(4*c + (2^(1/3)*((2 - 2*I*Sqrt[3])*c^2 + 6*I*(I + Sqrt[3])*b*d + 2^(1/3)*(1 + I*Sqrt[3])*(-2*c^3 + 9*b*c*d - 3*d*(9*a*d - Sqrt[3]*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2]))^(2/3)))/(-2*c^3 + 9*b*c*d - 3*d*(9*a*d - Sqrt[3]*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2]))^(1/3) + 12*d*x))/(2*(3*I + Sqrt[3])*c^2 - 6*(3*I + Sqrt[3])*b*d + 2^(1/3)*(3*I - Sqrt[3])*(-2*c^3 + 9*b*c*d - 3*d*(9*a*d - Sqrt[3]*Sqrt[(-b^2)*c^2 + 4*a*c^3 + 4*b^3*d - 18*a*b*c*d + 27*a^2*d^2]))^(2/3)))^n))} 

# Integrands of the form P4[x]^p

# Integrands of the form (a + 0 x + c x^2 + d x^3 + e x^4)^p when d^3 - 4 c d e=0
[(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)^4,x,3,c^4*(c^3+4*a*d^2)^4*x/d^8-8/3*c^5*(c^3+4*a*d^2)^3*(c/d+x)^3/d^6+4/5*c^3*(c^3+4*a*d^2)^2*(7*c^3+4*a*d^2)*(c/d+x)^5/d^4-8/7*c^4*(c^3+4*a*d^2)*(7*c^3+12*a*d^2)*(c/d+x)^7/d^2+2/9*c^2*(35*c^6+120*a*c^3*d^2+48*a^2*d^4)*(c/d+x)^9-8/11*c^3*d^2*(7*c^3+12*a*d^2)*(c/d+x)^11+4/13*c*d^4*(7*c^3+4*a*d^2)*(c/d+x)^13-8/15*c^2*d^6*(c/d+x)^15+1/17*d^8*(c/d+x)^17],
[(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)^3,x,2,64*a^3*c^3*x+64*a^2*c^4*x^3+48*a^2*c^3*d*x^4+48/5*a*c^2*(4*c^3+a*d^2)*x^5+64*a*c^4*d*x^6+32/7*c^3*(2*c^3+9*a*d^2)*x^7+12*c^2*d*(2*c^3+a*d^2)*x^8+4/3*c*d^2*(20*c^3+a*d^2)*x^9+16*c^3*d^3*x^10+60/11*c^2*d^4*x^11+c*d^5*x^12+1/13*d^6*x^13],
[(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)^2,x,2,16*a^2*c^2*x+32/3*a*c^3*x^3+8*a*c^2*d*x^4+8/5*c*(2*c^3+a*d^2)*x^5+16/3*c^3*d*x^6+24/7*c^2*d^2*x^7+c*d^3*x^8+1/9*d^4*x^9],
[4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4,x,1,4*a*c*x+4/3*c^2*x^3+c*d*x^4+1/5*d^2*x^5],
[1/(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4),x,10,-1/2*d*arctanh((c*sqrt(2)+d*x*sqrt(2)+c^(1/4)*sqrt(c^(3/2)+sqrt(c^3+4*a*d^2)))/(c^(1/4)*sqrt(c^(3/2)-sqrt(c^3+4*a*d^2))))/(c^(3/4)*sqrt(2)*sqrt(c^3+4*a*d^2)*sqrt(c^(3/2)-sqrt(c^3+4*a*d^2)))+1/2*d*arctanh((-(c+d*x)*sqrt(2)+c^(1/4)*sqrt(c^(3/2)+sqrt(c^3+4*a*d^2)))/(c^(1/4)*sqrt(c^(3/2)-sqrt(c^3+4*a*d^2))))/(c^(3/4)*sqrt(2)*sqrt(c^3+4*a*d^2)*sqrt(c^(3/2)-sqrt(c^3+4*a*d^2)))-1/4*d*log(d^2*(c/d+x)^2+sqrt(c)*sqrt(c^3+4*a*d^2)-c^(1/4)*d*(c/d+x)*sqrt(2)*sqrt(c^(3/2)+sqrt(c^3+4*a*d^2)))/(c^(3/4)*sqrt(2)*sqrt(c^3+4*a*d^2)*sqrt(c^(3/2)+sqrt(c^3+4*a*d^2)))+1/4*d*log(d^2*(c/d+x)^2+sqrt(c)*sqrt(c^3+4*a*d^2)+c^(1/4)*d*(c/d+x)*sqrt(2)*sqrt(c^(3/2)+sqrt(c^3+4*a*d^2)))/(c^(3/4)*sqrt(2)*sqrt(c^3+4*a*d^2)*sqrt(c^(3/2)+sqrt(c^3+4*a*d^2)))],
[1/(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)^2,x,11,-1/16*(c/d+x)*(c^3-4*a*d^2-c*d^2*(c/d+x)^2)/(a*c*(c^3+4*a*d^2)*(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4))-1/32*d*arctanh((c*sqrt(2)+d*x*sqrt(2)+c^(1/4)*sqrt(c^(3/2)+sqrt(c^3+4*a*d^2)))/(c^(1/4)*sqrt(c^(3/2)-sqrt(c^3+4*a*d^2))))*(c^3+12*a*d^2+c^(3/2)*sqrt(c^3+4*a*d^2))/(a*c^(7/4)*(c^3+4*a*d^2)^(3/2)*sqrt(2)*sqrt(c^(3/2)-sqrt(c^3+4*a*d^2)))+1/32*d*arctanh((-(c+d*x)*sqrt(2)+c^(1/4)*sqrt(c^(3/2)+sqrt(c^3+4*a*d^2)))/(c^(1/4)*sqrt(c^(3/2)-sqrt(c^3+4*a*d^2))))*(c^3+12*a*d^2+c^(3/2)*sqrt(c^3+4*a*d^2))/(a*c^(7/4)*(c^3+4*a*d^2)^(3/2)*sqrt(2)*sqrt(c^(3/2)-sqrt(c^3+4*a*d^2)))-1/64*d*log(d^2*(c/d+x)^2+sqrt(c)*sqrt(c^3+4*a*d^2)-c^(1/4)*d*(c/d+x)*sqrt(2)*sqrt(c^(3/2)+sqrt(c^3+4*a*d^2)))*(c^3+12*a*d^2-c^(3/2)*sqrt(c^3+4*a*d^2))/(a*c^(7/4)*(c^3+4*a*d^2)^(3/2)*sqrt(2)*sqrt(c^(3/2)+sqrt(c^3+4*a*d^2)))+1/64*d*log(d^2*(c/d+x)^2+sqrt(c)*sqrt(c^3+4*a*d^2)+c^(1/4)*d*(c/d+x)*sqrt(2)*sqrt(c^(3/2)+sqrt(c^3+4*a*d^2)))*(c^3+12*a*d^2-c^(3/2)*sqrt(c^3+4*a*d^2))/(a*c^(7/4)*(c^3+4*a*d^2)^(3/2)*sqrt(2)*sqrt(c^(3/2)+sqrt(c^3+4*a*d^2)))],

# Integrands of the form (a + b x + 0 x^2 + d x^3 + e x^4)^p when d^3 + 8 b e^2=0
[(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4)^4,x,3,1/1048576*(5*d^4+256*a*e^3)^4*x/e^4-1/8192*d^2*(5*d^4+256*a*e^3)^3*(1/4*d/e+x)^3/e^2+1/5120*(5*d^4+256*a*e^3)^2*(59*d^4+256*a*e^3)*(1/4*d/e+x)^5-9/224*d^2*e^2*(5*d^4+256*a*e^3)*(17*d^4+256*a*e^3)*(1/4*d/e+x)^7+1/24*e^4*(601*d^8+20992*a*d^4*e^3+65536*a^2*e^6)*(1/4*d/e+x)^9-72/11*d^2*e^6*(17*d^4+256*a*e^3)*(1/4*d/e+x)^11+64/13*e^8*(59*d^4+256*a*e^3)*(1/4*d/e+x)^13-2048/5*d^2*e^10*(1/4*d/e+x)^15+4096/17*e^12*(1/4*d/e+x)^17],
[(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4)^3,x,2,512*a^3*e^6*x-96*a^2*d^3*e^4*x^2+8*a*d^6*e^2*x^3-1/4*d*(d^8-1536*a^2*e^6)*x^4-384/5*a*e^4*(d^4-4*a*e^3)*x^5+4*d^3*e^2*(d^4-16*a*e^3)*x^6+24/7*d^2*e^3*(d^4+64*a*e^3)*x^7-24*d*e^4*(d^4-16*a*e^3)*x^8-128/3*e^5*(d^4-4*a*e^3)*x^9+32*d^3*e^6*x^10+1536/11*d^2*e^7*x^11+128*d*e^8*x^12+512/13*e^9*x^13],
[(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4)^2,x,2,64*a^2*e^4*x-8*a*d^3*e^2*x^2+1/3*d^6*x^3+32*a*d*e^4*x^4-16/5*e^2*(d^4-8*a*e^3)*x^5-8/3*d^3*e^3*x^6+64/7*d^2*e^4*x^7+16*d*e^5*x^8+64/9*e^6*x^9],
[8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4,x,1,8*a*e^2*x-1/2*d^3*x^2+2*d*e^2*x^4+8/5*e^3*x^5],
[1/(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4),x,4,2*arctanh((d+4*e*x)/sqrt(3*d^2-2*sqrt(d^4-64*a*e^3)))/(sqrt(d^4-64*a*e^3)*sqrt(3*d^2-2*sqrt(d^4-64*a*e^3)))-2*arctanh((d+4*e*x)/sqrt(3*d^2+2*sqrt(d^4-64*a*e^3)))/(sqrt(d^4-64*a*e^3)*sqrt(3*d^2+2*sqrt(d^4-64*a*e^3)))],
[1/(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4)^2,x,5,2*e*(1/4*d/e+x)*(13*d^4-256*a*e^3-48*d^2*e^2*(1/4*d/e+x)^2)/((5*d^8-64*a*d^4*e^3-16384*a^2*e^6)*(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4))-24*e*arctanh((d+4*e*x)/sqrt(3*d^2-2*sqrt(d^4-64*a*e^3)))*(d^4+128*a*e^3-d^2*sqrt(d^4-64*a*e^3))/((d^4-64*a*e^3)^(3/2)*(5*d^4+256*a*e^3)*sqrt(3*d^2-2*sqrt(d^4-64*a*e^3)))+24*e*arctanh((d+4*e*x)/sqrt(3*d^2+2*sqrt(d^4-64*a*e^3)))*(d^4+128*a*e^3+d^2*sqrt(d^4-64*a*e^3))/((d^4-64*a*e^3)^(3/2)*(5*d^4+256*a*e^3)*sqrt(3*d^2+2*sqrt(d^4-64*a*e^3)))],

# Integrands of the form (a + b x + 0 x^2 + d x^3 + e x^4)^p when b^3 + 8 a^2 d=0
[(8+8*x-x^3+8*x^4)^4,x,2,4096*x+8192*x^2+8192*x^3+3584*x^4+14336/5*x^5+7168*x^6+6784*x^7+1376*x^8+1408*x^9+21488/5*x^10+25312/11*x^11-448*x^12+10241/13*x^13+1168*x^14+128/5*x^15-128*x^16+4096/17*x^17],
[(8+8*x-x^3+8*x^4)^3,x,2,512*x+768*x^2+512*x^3+80*x^4+1152/5*x^5+480*x^6+1560/7*x^7-45*x^8+128*x^9+307/2*x^10+24/11*x^11-16*x^12+512/13*x^13],
[(8+8*x-x^3+8*x^4)^2,x,2,64*x+64*x^2+64/3*x^3-4*x^4+112/5*x^5+64/3*x^6+1/7*x^7-2*x^8+64/9*x^9],
[8+8*x-x^3+8*x^4,x,1,8*x+4*x^2-1/4*x^4+8/5*x^5],
[1/(8+8*x-x^3+8*x^4),x,16,-1/12*arctan(1/6*(3-(1+4/x)^2)/sqrt(7))/sqrt(7)-1/24*log((1+4/x)^2+3*sqrt(29)-(1+4/x)*sqrt(6*(1+sqrt(29))))*sqrt(1/1218*(-109+67*sqrt(29)))+1/24*log((1+4/x)^2+3*sqrt(29)+(1+4/x)*sqrt(6*(1+sqrt(29))))*sqrt(1/1218*(-109+67*sqrt(29)))-1/12*arctan((2+8/x-sqrt(6*(1+sqrt(29))))/sqrt(6*(-1+sqrt(29))))*sqrt(1/1218*(109+67*sqrt(29)))-1/12*arctan((2+8/x+sqrt(6*(1+sqrt(29))))/sqrt(6*(-1+sqrt(29))))*sqrt(1/1218*(109+67*sqrt(29)))],
[1/(8+8*x-x^3+8*x^4)^2,x,18,1/336*(-207-29*(1+4/x)^2)/(261-6*(1+4/x)^2+(1+4/x)^4)+5/87696*(5157+199*(1+4/x)^2)*(1+4/x)/(261-6*(1+4/x)^2+(1+4/x)^4)-17/1008*arctan(1/6*(3-(1+4/x)^2)/sqrt(7))/sqrt(7)-1/175392*log((1+4/x)^2+3*sqrt(29)-(1+4/x)*sqrt(6*(1+sqrt(29))))*sqrt(1/1218*(-180983329+45923327*sqrt(29)))+1/175392*log((1+4/x)^2+3*sqrt(29)+(1+4/x)*sqrt(6*(1+sqrt(29))))*sqrt(1/1218*(-180983329+45923327*sqrt(29)))-1/87696*arctan((2+8/x-sqrt(6*(1+sqrt(29))))/sqrt(6*(-1+sqrt(29))))*sqrt(1/1218*(180983329+45923327*sqrt(29)))-1/87696*arctan((2+8/x+sqrt(6*(1+sqrt(29))))/sqrt(6*(-1+sqrt(29))))*sqrt(1/1218*(180983329+45923327*sqrt(29)))],

# Integrands of the form (a + b x + c x^2 + 0 x^3 + e x^4)^p when b^2 - 4 a c=0
[(1+4*x+4*x^2+4*x^4)^4,x,2,x+8*x^2+112/3*x^3+112*x^4+1136/5*x^5+992/3*x^6+2752/7*x^7+448*x^8+4192/9*x^9+384*x^10+3328/11*x^11+256*x^12+1792/13*x^13+512/7*x^14+1024/15*x^15+256/17*x^17],
[(1+4*x+4*x^2+4*x^4)^3,x,2,x+6*x^2+20*x^3+40*x^4+252/5*x^5+48*x^6+352/7*x^7+48*x^8+80/3*x^9+96/5*x^10+192/11*x^11+64/13*x^13],
[(1+4*x+4*x^2+4*x^4)^2,x,2,x+4*x^2+8*x^3+8*x^4+24/5*x^5+16/3*x^6+32/7*x^7+16/9*x^9],
[1+4*x+4*x^2+4*x^4,x,1,x+2*x^2+4/3*x^3+4/5*x^5],
[1/(1+4*x+4*x^2+4*x^4),x,15,1/2*arctan(1/2*(-1+(1+1/x)^2))-1/4*log((1+1/x)^2+sqrt(5)-(1+1/x)*sqrt(2*(1+sqrt(5))))*sqrt(1/5*(-2+sqrt(5)))+1/4*log((1+1/x)^2+sqrt(5)+(1+1/x)*sqrt(2*(1+sqrt(5))))*sqrt(1/5*(-2+sqrt(5)))-1/2*arctan((2+2/x-sqrt(2*(1+sqrt(5))))/sqrt(2*(-1+sqrt(5))))*sqrt(1/5*(2+sqrt(5)))-1/2*arctan((2+2/x+sqrt(2*(1+sqrt(5))))/sqrt(2*(-1+sqrt(5))))*sqrt(1/5*(2+sqrt(5)))],
[1/(1+4*x+4*x^2+4*x^4)^2,x,17,1/2*(-17+(1+1/x)^2)/(5-2*(1+1/x)^2+(1+1/x)^4)+1/10*(59-17*(1+1/x)^2)*(1+1/x)/(5-2*(1+1/x)^2+(1+1/x)^4)+7/4*arctan(1/2*(-1+(1+1/x)^2))+1/40*log((1+1/x)^2+sqrt(5)-(1+1/x)*sqrt(2*(1+sqrt(5))))*sqrt(1/10*(-5959+2665*sqrt(5)))-1/40*log((1+1/x)^2+sqrt(5)+(1+1/x)*sqrt(2*(1+sqrt(5))))*sqrt(1/10*(-5959+2665*sqrt(5)))-1/20*arctan((2+2/x-sqrt(2*(1+sqrt(5))))/sqrt(2*(-1+sqrt(5))))*sqrt(1/10*(5959+2665*sqrt(5)))-1/20*arctan((2+2/x+sqrt(2*(1+sqrt(5))))/sqrt(2*(-1+sqrt(5))))*sqrt(1/10*(5959+2665*sqrt(5)))],

# Integrands of the form (a + b x + c x^2 + d x^3 + e x^4)^p when b^3 - 4 a b c + 8 a^2 d=0
[(8+24*x+8*x^2-15*x^3+8*x^4)^4,x,2,4096*x+24576*x^2+237568/3*x^3+139776*x^4+538624/5*x^5-30720*x^6-566912/7*x^7+36384*x^8+641152/9*x^9-169584/5*x^10-331040/11*x^11+31128*x^12-12095/13*x^13-75504/7*x^14+102784/15*x^15-1920*x^16+4096/17*x^17],
[(8+24*x+8*x^2-15*x^3+8*x^4)^3,x,2,512*x+2304*x^2+5120*x^3+5040*x^4-384/5*x^5-2976*x^6+5528/7*x^7+2097*x^8-2936/3*x^9-4527/10*x^10+6936/11*x^11-240*x^12+512/13*x^13],
[(8+24*x+8*x^2-15*x^3+8*x^4)^2,x,2,64*x+192*x^2+704/3*x^3+36*x^4-528/5*x^5+24*x^6+353/7*x^7-30*x^8+64/9*x^9],
[8+24*x+8*x^2-15*x^3+8*x^4,x,1,8*x+12*x^2+8/3*x^3-15/4*x^4+8/5*x^5],
[1/(8+24*x+8*x^2-15*x^3+8*x^4),x,16,1/4*arctan((8+12*x-5*x^2)/(x^2*sqrt(39)))*sqrt(3/13)-1/8*log((3+4/x)^2+sqrt(517)-(3+4/x)*sqrt(2*(19+sqrt(517))))*sqrt(1/40326*(-5167+235*sqrt(517)))+1/8*log((3+4/x)^2+sqrt(517)+(3+4/x)*sqrt(2*(19+sqrt(517))))*sqrt(1/40326*(-5167+235*sqrt(517)))-1/4*arctan((6+8/x-sqrt(2*(19+sqrt(517))))/sqrt(2*(-19+sqrt(517))))*sqrt(1/40326*(5167+235*sqrt(517)))-1/4*arctan((6+8/x+sqrt(2*(19+sqrt(517))))/sqrt(2*(-19+sqrt(517))))*sqrt(1/40326*(5167+235*sqrt(517)))],
[1/(8+24*x+8*x^2-15*x^3+8*x^4)^2,x,18,-3/208*(3359-107*(3+4/x)^2)/(517-38*(3+4/x)^2+(3+4/x)^4)+1/322608*(3327931-129631*(3+4/x)^2)*(3+4/x)/(517-38*(3+4/x)^2+(3+4/x)^4)+73/208*arctan((8+12*x-5*x^2)/(x^2*sqrt(39)))*sqrt(3/13)-1/645216*arctan((6+8/x-sqrt(2*(19+sqrt(517))))/sqrt(2*(-19+sqrt(517))))*(1678181+74897*sqrt(517))*sqrt(1/40326*(19+sqrt(517)))-1/645216*arctan((6+8/x+sqrt(2*(19+sqrt(517))))/sqrt(2*(-19+sqrt(517))))*(1678181+74897*sqrt(517))*sqrt(1/40326*(19+sqrt(517)))-1/645216*log((3+4/x)^2+sqrt(517)-(3+4/x)*sqrt(2*(19+sqrt(517))))*sqrt(1/40326*(-59644114671451+2623170438295*sqrt(517)))+1/645216*log((3+4/x)^2+sqrt(517)+(3+4/x)*sqrt(2*(19+sqrt(517))))*sqrt(1/40326*(-59644114671451+2623170438295*sqrt(517)))],

# Integrands of the form P5[x]^p
[(a^5+5*a^4*b*x+10*a^3*b^2*x^2+10*a^2*b^3*x^3+5*a*b^4*x^4+b^5*x^5)^3,x,2,1/16*(a+b*x)^16/b],
[(a^5+5*a^4*b*x+10*a^3*b^2*x^2+10*a^2*b^3*x^3+5*a*b^4*x^4+b^5*x^5)^2,x,2,1/11*(a+b*x)^11/b],
[a^5+5*a^4*b*x+10*a^3*b^2*x^2+10*a^2*b^3*x^3+5*a*b^4*x^4+b^5*x^5,x,1,1/6*(a+b*x)^6/b,a^5*x+5/2*a^4*b*x^2+10/3*a^3*b^2*x^3+5/2*a^2*b^3*x^4+a*b^4*x^5+1/6*b^5*x^6],
[1/(a^5+5*a^4*b*x+10*a^3*b^2*x^2+10*a^2*b^3*x^3+5*a*b^4*x^4+b^5*x^5),x,2,(-1/4)/(b*(a+b*x)^4)],
[1/(a^5+5*a^4*b*x+10*a^3*b^2*x^2+10*a^2*b^3*x^3+5*a*b^4*x^4+b^5*x^5)^2,x,2,(-1/9)/(b*(a+b*x)^9)],
[1/(a^5+5*a^4*b*x+10*a^3*b^2*x^2+10*a^2*b^3*x^3+5*a*b^4*x^4+b^5*x^5)^3,x,2,(-1/14)/(b*(a+b*x)^14)],
[1/(1+x^2+x^3+x^5),x,6,1/2*arctan(x)+1/6*log(1+x)+1/4*log(1+x^2)-1/3*log(1-x+x^2)],

# Integrands of the form P6[x]^p

# Integrands of the form P3[x^2]^p
[(3-19*x^2+32*x^4-16*x^6)^4,x,5,81*x-684*x^3+4590*x^5-149700/7*x^7+634321/9*x^9-1841600/11*x^11+3764416/13*x^13-1094656/3*x^15+5633536/17*x^17-4014080/19*x^19+1884160/21*x^21-524288/23*x^23+65536/25*x^25],
[(3-19*x^2+32*x^4-16*x^6)^3,x,5,27*x-171*x^3+4113/5*x^5-2605*x^7+16448/3*x^9-84912/11*x^11+93440/13*x^13-21248/5*x^15+24576/17*x^17-4096/19*x^19],
[(3-19*x^2+32*x^4-16*x^6)^2,x,5,9*x-38*x^3+553/5*x^5-1312/7*x^7+544/3*x^9-1024/11*x^11+256/13*x^13],
[3-19*x^2+32*x^4-16*x^6,x,1,3*x-19/3*x^3+32/5*x^5-16/7*x^7],
[1/(3-19*x^2+32*x^4-16*x^6),x,5,1/3*arctanh(x)+1/3*arctanh(2*x)-arctanh(2*x/sqrt(3))/sqrt(3)],
[1/(3-19*x^2+32*x^4-16*x^6)^2,x,7,1/18/(1-2*x)+1/36/(1-x)+(-1/36)/(1+x)+(-1/18)/(1+2*x)+2/3*x/(3-4*x^2)+67/54*arctanh(x)-7/27*arctanh(2*x)-5/3*arctanh(2*x/sqrt(3))/sqrt(3)],
[1/(3-19*x^2+32*x^4-16*x^6)^3,x,10,1/108/(1-2*x)^2+(-7/108)/(1-2*x)+1/432/(1-x)^2+67/432/(1-x)+(-1/432)/(1+x)^2+(-67/432)/(1+x)+(-1/108)/(1+2*x)^2+7/108/(1+2*x)-2/3*x/(3-4*x^2)^2+5/3*x/(3-4*x^2)+3913/648*arctanh(x)+67/162*arctanh(2*x)+5/6*arctanh(2*x/sqrt(3))/sqrt(3)-4*arctanh(2*x/sqrt(3))*sqrt(3)],
[1/(-1+7*x^2-7*x^4+x^6)^2,x,15,1/32*x/(1-x^2)+1/128*x*(99-17*x^2)/(1-6*x^2+x^4)+5/32*arctanh(x)+1/512*arctanh(x*(-1+sqrt(2)))*(-4+3*sqrt(2))+1/512*arctanh(x*(1+sqrt(2)))*(4+3*sqrt(2)),1/64/(1-x)+(-1/64)/(1+x)+1/256*(41+17*x)/(1-2*x-x^2)+1/256*(-41+17*x)/(1+2*x-x^2)+5/32*arctanh(x)+1/512*log(1-x-sqrt(2))*(2-7*sqrt(2))-1/512*log(1+x-sqrt(2))*(2-7*sqrt(2))-17/256*arctanh((1-x)/sqrt(2))/sqrt(2)+17/256*arctanh((1+x)/sqrt(2))/sqrt(2)+1/512*log(1-x+sqrt(2))*(2+7*sqrt(2))-1/512*log(1+x+sqrt(2))*(2+7*sqrt(2))],

# Integrands of the form x^m P[x]^p

# Integrands of the form x^m P2[x]^p

# Integrands of the form x^m (a+b (c+d x)^2)^p
[x^3/(c+(a+b*x)^2),x,6,-3*a*x/b^3+1/2*(a+b*x)^2/b^4+1/2*(3*a^2-c)*log(c+(a+b*x)^2)/b^4-a*(a^2-3*c)*arctan((a+b*x)/sqrt(c))/(b^4*sqrt(c))],
[x^2/(c+(a+b*x)^2),x,6,x/b^2-a*log(c+(a+b*x)^2)/b^3+(a^2-c)*arctan((a+b*x)/sqrt(c))/(b^3*sqrt(c))],
[x/(c+(a+b*x)^2),x,4,1/2*log(c+(a+b*x)^2)/b^2-a*arctan((a+b*x)/sqrt(c))/(b^2*sqrt(c))],
[1/(c+(a+b*x)^2),x,2,arctan((a+b*x)/sqrt(c))/(b*sqrt(c))],
[1/(x*(c+(a+b*x)^2)),x,6,log(x)/(a^2+c)-1/2*log(c+(a+b*x)^2)/(a^2+c)-a*arctan((a+b*x)/sqrt(c))/((a^2+c)*sqrt(c))],
[1/(x^2*(c+(a+b*x)^2)),x,7,(-1)/((a^2+c)*x)-2*a*b*log(x)/(a^2+c)^2+a*b*log(c+(a+b*x)^2)/(a^2+c)^2+b*(a^2-c)*arctan((a+b*x)/sqrt(c))/((a^2+c)^2*sqrt(c))],
[1/(x^3*(c+(a+b*x)^2)),x,7,(-1/2)/((a^2+c)*x^2)+2*a*b/((a^2+c)^2*x)+b^2*(3*a^2-c)*log(x)/(a^2+c)^3-1/2*b^2*(3*a^2-c)*log(c+(a+b*x)^2)/(a^2+c)^3-a*b^2*(a^2-3*c)*arctan((a+b*x)/sqrt(c))/((a^2+c)^3*sqrt(c))],
[1/(a+b*(c+d*x)^2),x,2,arctan((c+d*x)*sqrt(b)/sqrt(a))/(d*sqrt(a)*sqrt(b))],
[1/(a+b*(c+d*x)^2)^2,x,3,1/2*(c+d*x)/(a*d*(a+b*(c+d*x)^2))+1/2*arctan((c+d*x)*sqrt(b)/sqrt(a))/(a^(3/2)*d*sqrt(b))],
[1/(a+b*(c+d*x)^2)^3,x,4,1/4*(c+d*x)/(a*d*(a+b*(c+d*x)^2)^2)+3/8*(c+d*x)/(a^2*d*(a+b*(c+d*x)^2))+3/8*arctan((c+d*x)*sqrt(b)/sqrt(a))/(a^(5/2)*d*sqrt(b))],
[1/(b*(c+d*x)^2+sqrt(-a)),x,2,arctan((c+d*x)*sqrt(b)/(-a)^(1/4))/((-a)^(1/4)*d*sqrt(b))],
[1/(1+(c+d*x)^2),x,2,arctan(c+d*x)/d],
[1/(1+(c+d*x)^2)^2,x,3,1/2*(c+d*x)/(d*(1+(c+d*x)^2))+1/2*arctan(c+d*x)/d],
[1/(1+(c+d*x)^2)^3,x,4,1/4*(c+d*x)/(d*(1+(c+d*x)^2)^2)+3/8*(c+d*x)/(d*(1+(c+d*x)^2))+3/8*arctan(c+d*x)/d],
[1/(1-(c+d*x)^2),x,2,arctanh(c+d*x)/d],
[1/(1-(c+d*x)^2)^2,x,3,1/2*(c+d*x)/(d*(1-(c+d*x)^2))+1/2*arctanh(c+d*x)/d],
[1/(1-(c+d*x)^2)^3,x,4,1/4*(c+d*x)/(d*(1-(c+d*x)^2)^2)+3/8*(c+d*x)/(d*(1-(c+d*x)^2))+3/8*arctanh(c+d*x)/d],
[1/(1-(1+x)^2),x,2,arctanh(1+x)],
[1/(1-(1+x)^2)^2,x,3,1/2*(1+x)/(1-(1+x)^2)+1/2*arctanh(1+x)],
[1/(1-(1+x)^2)^3,x,4,1/4*(1+x)/(1-(1+x)^2)^2+3/8*(1+x)/(1-(1+x)^2)+3/8*arctanh(1+x)],
[(1+(a+b*x)^2)^2/x,x,3,a*(2+a^2)*b*x+1/2*(2+a^2)*(a+b*x)^2+1/3*a*(a+b*x)^3+1/4*(a+b*x)^4+(1+a^2)^2*log(x)],
[x^2/(1+(-1+x)^2),x,4,x+log(1+(-1+x)^2)],

# Integrands of the form x^m (a+b (c+d x)^2)^(p/2)
[x^2/sqrt(1-(1+x)^2),x,4,3/2*arcsin(1+x)+3/2*sqrt(1-(1+x)^2)-1/2*x*sqrt(1-(1+x)^2)],
[x^2/sqrt(1-(a+b*x)^2),x,4,1/2*(1+2*a^2)*arcsin(a+b*x)/b^3+3/2*a*sqrt(1-(a+b*x)^2)/b^3-1/2*x*sqrt(1-(a+b*x)^2)/b^2],
[x^2/sqrt(1+(a+b*x)^2),x,4,-1/2*(1-2*a^2)*arcsinh(a+b*x)/b^3-3/2*a*sqrt(1+(a+b*x)^2)/b^3+1/2*x*sqrt(1+(a+b*x)^2)/b^2],

# Integrands of the form x^m P3[x]^p

# Integrands of the form x^m (a+b (c+d x)^3)^p
[x^3/(a+b*(c+d*x)^3),x,11,x/(b*d^3)-1/3*(a+3*a^(1/3)*b^(2/3)*c^2+b*c^3)*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(2/3)*b^(4/3)*d^4)+1/6*(a+3*a^(1/3)*b^(2/3)*c^2+b*c^3)*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(2/3)*b^(4/3)*d^4)-c*log(a+b*(c+d*x)^3)/(b*d^4)+(a-3*a^(1/3)*b^(2/3)*c^2+b*c^3)*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(4/3)*d^4*sqrt(3))],
[x^2/(a+b*(c+d*x)^3),x,9,1/3*c*(2*a^(1/3)+b^(1/3)*c)*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(2/3)*b^(2/3)*d^3)-1/6*c*(2*a^(1/3)+b^(1/3)*c)*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(2/3)*b^(2/3)*d^3)+1/3*log(a+b*(c+d*x)^3)/(b*d^3)+c*(2*a^(1/3)-b^(1/3)*c)*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(2/3)*d^3*sqrt(3))],
[x/(a+b*(c+d*x)^3),x,7,-1/3*(a^(1/3)+b^(1/3)*c)*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(2/3)*b^(2/3)*d^2)+1/6*(a^(1/3)+b^(1/3)*c)*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(2/3)*b^(2/3)*d^2)-(a^(1/3)-b^(1/3)*c)*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(2/3)*d^2*sqrt(3))],
[1/(a+b*(c+d*x)^3),x,7,1/3*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(2/3)*b^(1/3)*d)-1/6*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(2/3)*b^(1/3)*d)-arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(2/3)*b^(1/3)*d*sqrt(3))],
[1/(x*(a+b*(c+d*x)^3)),x,11,log(x)/(a+b*c^3)-1/3*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(2/3)*(a^(1/3)+b^(1/3)*c))-1/6*(2*a^(1/3)-b^(1/3)*c)*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(2/3)*(a^(2/3)-a^(1/3)*b^(1/3)*c+b^(2/3)*c^2))+b^(1/3)*c*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(2/3)*(a^(2/3)-a^(1/3)*b^(1/3)*c+b^(2/3)*c^2)*sqrt(3)),log(x)/(a+b*c^3)+1/3*b^(1/3)*c*(a^(1/3)-b^(1/3)*c)*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(2/3)*(a+b*c^3))-1/6*b^(1/3)*c*(a^(1/3)-b^(1/3)*c)*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(2/3)*(a+b*c^3))-1/3*log(a+b*(c+d*x)^3)/(a+b*c^3)+b^(1/3)*c*(a^(1/3)+b^(1/3)*c)*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(2/3)*(a+b*c^3)*sqrt(3))],
[1/(x^2*(a+b*(c+d*x)^3)),x,11,(-1)/((a+b*c^3)*x)-3*b*c^2*d*log(x)/(a+b*c^3)^2+1/3*b^(1/3)*(a^(1/3)*(a-2*b*c^3)-b^(1/3)*c*(2*a-b*c^3))*d*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(2/3)*(a+b*c^3)^2)-1/6*b^(1/3)*(a^(1/3)*(a-2*b*c^3)-b^(1/3)*c*(2*a-b*c^3))*d*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(2/3)*(a+b*c^3)^2)+b*c^2*d*log(a+b*(c+d*x)^3)/(a+b*c^3)^2+b^(1/3)*(a^(1/3)-b^(1/3)*c)*(a^(1/3)+b^(1/3)*c)^3*d*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(2/3)*(a+b*c^3)^2*sqrt(3))],
[1/(x^3*(a+b*(c+d*x)^3)),x,11,(-1/2)/((a+b*c^3)*x^2)+3*b*c^2*d/((a+b*c^3)^2*x)-3*b*c*(a-2*b*c^3)*d^2*log(x)/(a+b*c^3)^3-1/3*b^(2/3)*(a^2+6*a^(4/3)*b^(2/3)*c^2-7*a*b*c^3-3*a^(1/3)*b^(5/3)*c^5+b^2*c^6)*d^2*log(a^(1/3)+b^(1/3)*(c+d*x))/(a^(2/3)*(a+b*c^3)^3)+1/6*b^(2/3)*(a^2+6*a^(4/3)*b^(2/3)*c^2-7*a*b*c^3-3*a^(1/3)*b^(5/3)*c^5+b^2*c^6)*d^2*log(a^(2/3)-a^(1/3)*b^(1/3)*(c+d*x)+b^(2/3)*(c+d*x)^2)/(a^(2/3)*(a+b*c^3)^3)+b*c*(a-2*b*c^3)*d^2*log(a+b*(c+d*x)^3)/(a+b*c^3)^3+b^(2/3)*(a^(1/3)+b^(1/3)*c)^3*(a-3*a^(2/3)*b^(1/3)*c+b*c^3)*d^2*arctan((a^(1/3)-2*b^(1/3)*(c+d*x))/(a^(1/3)*sqrt(3)))/(a^(2/3)*(a+b*c^3)^3*sqrt(3))],

# Integrands of the form x^m P4[x]^p

# Integrands of the form x^m (a+b (c+d x)^4)^p
[x^3/(a+b*(c+d*x)^4),x,16,1/4*log(a+b*(c+d*x)^4)/(b*d^4)+3/2*c^2*arctan((c+d*x)^2*sqrt(b)/sqrt(a))/(d^4*sqrt(a)*sqrt(b))-1/4*c*log(-a^(1/4)*b^(1/4)*(c+d*x)*sqrt(2)+sqrt(a)+(c+d*x)^2*sqrt(b))*(3*sqrt(a)-c^2*sqrt(b))/(a^(3/4)*b^(3/4)*d^4*sqrt(2))+1/4*c*log(a^(1/4)*b^(1/4)*(c+d*x)*sqrt(2)+sqrt(a)+(c+d*x)^2*sqrt(b))*(3*sqrt(a)-c^2*sqrt(b))/(a^(3/4)*b^(3/4)*d^4*sqrt(2))+1/2*c*arctan(1-b^(1/4)*(c+d*x)*sqrt(2)/a^(1/4))*(3*sqrt(a)+c^2*sqrt(b))/(a^(3/4)*b^(3/4)*d^4*sqrt(2))-1/2*c*arctan(1+b^(1/4)*(c+d*x)*sqrt(2)/a^(1/4))*(3*sqrt(a)+c^2*sqrt(b))/(a^(3/4)*b^(3/4)*d^4*sqrt(2))],
[x^2/(a+b*(c+d*x)^4),x,14,-c*arctan((c+d*x)^2*sqrt(b)/sqrt(a))/(d^3*sqrt(a)*sqrt(b))+1/4*log(-a^(1/4)*b^(1/4)*(c+d*x)*sqrt(2)+sqrt(a)+(c+d*x)^2*sqrt(b))*(sqrt(a)-c^2*sqrt(b))/(a^(3/4)*b^(3/4)*d^3*sqrt(2))-1/4*log(a^(1/4)*b^(1/4)*(c+d*x)*sqrt(2)+sqrt(a)+(c+d*x)^2*sqrt(b))*(sqrt(a)-c^2*sqrt(b))/(a^(3/4)*b^(3/4)*d^3*sqrt(2))-1/2*arctan(1-b^(1/4)*(c+d*x)*sqrt(2)/a^(1/4))*(sqrt(a)+c^2*sqrt(b))/(a^(3/4)*b^(3/4)*d^3*sqrt(2))+1/2*arctan(1+b^(1/4)*(c+d*x)*sqrt(2)/a^(1/4))*(sqrt(a)+c^2*sqrt(b))/(a^(3/4)*b^(3/4)*d^3*sqrt(2))],
[x/(a+b*(c+d*x)^4),x,14,1/2*c*arctan(1-b^(1/4)*(c+d*x)*sqrt(2)/a^(1/4))/(a^(3/4)*b^(1/4)*d^2*sqrt(2))-1/2*c*arctan(1+b^(1/4)*(c+d*x)*sqrt(2)/a^(1/4))/(a^(3/4)*b^(1/4)*d^2*sqrt(2))+1/4*c*log(-a^(1/4)*b^(1/4)*(c+d*x)*sqrt(2)+sqrt(a)+(c+d*x)^2*sqrt(b))/(a^(3/4)*b^(1/4)*d^2*sqrt(2))-1/4*c*log(a^(1/4)*b^(1/4)*(c+d*x)*sqrt(2)+sqrt(a)+(c+d*x)^2*sqrt(b))/(a^(3/4)*b^(1/4)*d^2*sqrt(2))+1/2*arctan((c+d*x)^2*sqrt(b)/sqrt(a))/(d^2*sqrt(a)*sqrt(b))],
[1/(a+b*(c+d*x)^4),x,10,-1/2*arctan(1-b^(1/4)*(c+d*x)*sqrt(2)/a^(1/4))/(a^(3/4)*b^(1/4)*d*sqrt(2))+1/2*arctan(1+b^(1/4)*(c+d*x)*sqrt(2)/a^(1/4))/(a^(3/4)*b^(1/4)*d*sqrt(2))-1/4*log(-a^(1/4)*b^(1/4)*(c+d*x)*sqrt(2)+sqrt(a)+(c+d*x)^2*sqrt(b))/(a^(3/4)*b^(1/4)*d*sqrt(2))+1/4*log(a^(1/4)*b^(1/4)*(c+d*x)*sqrt(2)+sqrt(a)+(c+d*x)^2*sqrt(b))/(a^(3/4)*b^(1/4)*d*sqrt(2))],
[1/(x*(a+b*(c+d*x)^4)),x,18,log(x)/(a+b*c^4)-1/4*log(a+b*(c+d*x)^4)/(a+b*c^4)-1/2*c^2*arctan((c+d*x)^2*sqrt(b)/sqrt(a))*sqrt(b)/((a+b*c^4)*sqrt(a))-1/4*b^(1/4)*c*log(-a^(1/4)*b^(1/4)*(c+d*x)*sqrt(2)+sqrt(a)+(c+d*x)^2*sqrt(b))*(sqrt(a)-c^2*sqrt(b))/(a^(3/4)*(a+b*c^4)*sqrt(2))+1/4*b^(1/4)*c*log(a^(1/4)*b^(1/4)*(c+d*x)*sqrt(2)+sqrt(a)+(c+d*x)^2*sqrt(b))*(sqrt(a)-c^2*sqrt(b))/(a^(3/4)*(a+b*c^4)*sqrt(2))+1/2*b^(1/4)*c*arctan(1-b^(1/4)*(c+d*x)*sqrt(2)/a^(1/4))*(sqrt(a)+c^2*sqrt(b))/(a^(3/4)*(a+b*c^4)*sqrt(2))-1/2*b^(1/4)*c*arctan(1+b^(1/4)*(c+d*x)*sqrt(2)/a^(1/4))*(sqrt(a)+c^2*sqrt(b))/(a^(3/4)*(a+b*c^4)*sqrt(2))],
[1/(x^2*(a+b*(c+d*x)^4)),x,18,(-1)/((a+b*c^4)*x)-4*b*c^3*d*log(x)/(a+b*c^4)^2+b*c^3*d*log(a+b*(c+d*x)^4)/(a+b*c^4)^2-c*(a-b*c^4)*d*arctan((c+d*x)^2*sqrt(b)/sqrt(a))*sqrt(b)/((a+b*c^4)^2*sqrt(a))-1/4*b^(1/4)*d*log(-a^(1/4)*b^(1/4)*(c+d*x)*sqrt(2)+sqrt(a)+(c+d*x)^2*sqrt(b))*((a-3*b*c^4)*sqrt(a)-c^2*(3*a-b*c^4)*sqrt(b))/(a^(3/4)*(a+b*c^4)^2*sqrt(2))+1/4*b^(1/4)*d*log(a^(1/4)*b^(1/4)*(c+d*x)*sqrt(2)+sqrt(a)+(c+d*x)^2*sqrt(b))*((a-3*b*c^4)*sqrt(a)-c^2*(3*a-b*c^4)*sqrt(b))/(a^(3/4)*(a+b*c^4)^2*sqrt(2))+1/2*b^(1/4)*d*arctan(1-b^(1/4)*(c+d*x)*sqrt(2)/a^(1/4))*((a-3*b*c^4)*sqrt(a)+c^2*(3*a-b*c^4)*sqrt(b))/(a^(3/4)*(a+b*c^4)^2*sqrt(2))-1/2*b^(1/4)*d*arctan(1+b^(1/4)*(c+d*x)*sqrt(2)/a^(1/4))*((a-3*b*c^4)*sqrt(a)+c^2*(3*a-b*c^4)*sqrt(b))/(a^(3/4)*(a+b*c^4)^2*sqrt(2))],

# Integrands of the form x^m (a + b x + c x^2 + d x^3 + e x^4)^p when d^3 - 4 c d e + 8 b e^2=0
[(a+8*x-8*x^2+4*x^3-x^4)^4,x,3,-8/3*(3+a)^3*(-1+x)^3+4/5*(3-a)*(3+a)^2*(-1+x)^5+8/7*(3+a)*(5+3*a)*(-1+x)^7-2/9*(37+6*a-3*a^2)*(-1+x)^9-8/11*(5+3*a)*(-1+x)^11+4/13*(3-a)*(-1+x)^13+8/15*(-1+x)^15+1/17*(-1+x)^17+(3+a)^4*x],
[(a+8*x-8*x^2+4*x^3-x^4)^3,x,2,a^3*x+12*a^2*x^2+8*(8-a)*a*x^3+(128-96*a+3*a^2)*x^4-3/5*(512-128*a+a^2)*x^5+8*(48-5*a)*x^6-32/7*(70-3*a)*x^7+3*(64-a)*x^8-1/3*(256-a)*x^9+28*x^10-72/11*x^11+x^12-1/13*x^13],
[(a+8*x-8*x^2+4*x^3-x^4)^2,x,2,a^2*x+8*a*x^2+16/3*(4-a)*x^3-2*(16-a)*x^4+2/5*(64-a)*x^5-40/3*x^6+32/7*x^7-x^8+1/9*x^9],
[a+8*x-8*x^2+4*x^3-x^4,x,1,a*x+4*x^2-8/3*x^3+x^4-1/5*x^5],
[1/(a+8*x-8*x^2+4*x^3-x^4),x,4,-1/2*arctan((-1+x)/sqrt(1-sqrt(4+a)))/(sqrt(4+a)*sqrt(1-sqrt(4+a)))+1/2*arctan((-1+x)/sqrt(1+sqrt(4+a)))/(sqrt(4+a)*sqrt(1+sqrt(4+a)))],
[1/(a+8*x-8*x^2+4*x^3-x^4)^2,x,5,1/4*(5+a+(-1+x)^2)*(-1+x)/((12+7*a+a^2)*(3+a-2*(-1+x)^2-(-1+x)^4))-1/8*arctan((-1+x)/sqrt(1-sqrt(4+a)))*(10+3*a+sqrt(4+a))/((3+a)*(4+a)^(3/2)*sqrt(1-sqrt(4+a)))+1/8*arctan((-1+x)/sqrt(1+sqrt(4+a)))*(10+3*a-sqrt(4+a))/((3+a)*(4+a)^(3/2)*sqrt(1+sqrt(4+a)))],
[1/(a+8*x-8*x^2+4*x^3-x^4)^3,x,6,1/8*(5+a+(-1+x)^2)*(-1+x)/((12+7*a+a^2)*(3+a-2*(-1+x)^2-(-1+x)^4)^2)+1/32*((6+a)*(25+7*a)+6*(7+2*a)*(-1+x)^2)*(-1+x)/((12+7*a+a^2)^2*(3+a-2*(-1+x)^2-(-1+x)^4))-3/64*arctan((-1+x)/sqrt(1-sqrt(4+a)))*(80+7*a^2+14*sqrt(4+a)+a*(47+4*sqrt(4+a)))/((3+a)^2*(4+a)^(5/2)*sqrt(1-sqrt(4+a)))-3/64*arctan((-1+x)/sqrt(1+sqrt(4+a)))*(14+4*a+(-80-47*a-7*a^2)/sqrt(4+a))/((3+a)^2*(4+a)^2*sqrt(1+sqrt(4+a)))],
[x*(a+8*x-8*x^2+4*x^3-x^4)^4,x,2,1/2*a^4*x^2+32/3*a^3*x^3+8*(12-a)*a^2*x^4+16/5*a*(128-48*a+a^2)*x^5+2/3*(1024-1536*a+192*a^2-a^3)*x^6-32/7*(512-288*a+15*a^2)*x^7+8*(128-3*a)*(4-a)*x^8-16/3*(896-128*a+a^2)*x^9+1/5*(20480-1536*a+3*a^2)*x^10-32/11*(928-35*a)*x^11+8/3*(524-9*a)*x^12-16/13*(464-3*a)*x^13+2/7*(640-a)*x^14-224/5*x^15+8*x^16-16/17*x^17+1/18*x^18],
[x*(a+8*x-8*x^2+4*x^3-x^4)^3,x,2,1/2*a^3*x^2+8*a^2*x^3+6*(8-a)*a*x^4+4/5*(128-96*a+3*a^2)*x^5-1/2*(512-128*a+a^2)*x^6+48/7*(48-5*a)*x^7-4*(70-3*a)*x^8+8/3*(64-a)*x^9-3/10*(256-a)*x^10+280/11*x^11-6*x^12+12/13*x^13-1/14*x^14],
[x*(a+8*x-8*x^2+4*x^3-x^4)^2,x,2,1/2*a^2*x^2+16/3*a*x^3+4*(4-a)*x^4-8/5*(16-a)*x^5+1/3*(64-a)*x^6-80/7*x^7+4*x^8-8/9*x^9+1/10*x^10],
[x*(a+8*x-8*x^2+4*x^3-x^4),x,2,1/2*a*x^2+8/3*x^3-2*x^4+4/5*x^5-1/6*x^6],
[x/(a+8*x-8*x^2+4*x^3-x^4),x,8,1/2*arctanh((1+(-1+x)^2)/sqrt(4+a))/sqrt(4+a)-1/2*arctan((-1+x)/sqrt(1-sqrt(4+a)))/(sqrt(4+a)*sqrt(1-sqrt(4+a)))+1/2*arctan((-1+x)/sqrt(1+sqrt(4+a)))/(sqrt(4+a)*sqrt(1+sqrt(4+a)))],
[x/(a+8*x-8*x^2+4*x^3-x^4)^2,x,10,1/4*(1+(-1+x)^2)/((4+a)*(3+a-2*(-1+x)^2-(-1+x)^4))+1/4*(5+a+(-1+x)^2)*(-1+x)/((12+7*a+a^2)*(3+a-2*(-1+x)^2-(-1+x)^4))+1/4*arctanh((1+(-1+x)^2)/sqrt(4+a))/(4+a)^(3/2)-1/8*arctan((-1+x)/sqrt(1-sqrt(4+a)))*(10+3*a+sqrt(4+a))/((3+a)*(4+a)^(3/2)*sqrt(1-sqrt(4+a)))+1/8*arctan((-1+x)/sqrt(1+sqrt(4+a)))*(10+3*a-sqrt(4+a))/((3+a)*(4+a)^(3/2)*sqrt(1+sqrt(4+a)))],
[x/(a+8*x-8*x^2+4*x^3-x^4)^3,x,12,1/8*(1+(-1+x)^2)/((4+a)*(3+a-2*(-1+x)^2-(-1+x)^4)^2)+3/16*(1+(-1+x)^2)/((4+a)^2*(3+a-2*(-1+x)^2-(-1+x)^4))+1/8*(5+a+(-1+x)^2)*(-1+x)/((12+7*a+a^2)*(3+a-2*(-1+x)^2-(-1+x)^4)^2)+1/32*((6+a)*(25+7*a)+6*(7+2*a)*(-1+x)^2)*(-1+x)/((12+7*a+a^2)^2*(3+a-2*(-1+x)^2-(-1+x)^4))+3/16*arctanh((1+(-1+x)^2)/sqrt(4+a))/(4+a)^(5/2)-3/64*arctan((-1+x)/sqrt(1-sqrt(4+a)))*(80+7*a^2+14*sqrt(4+a)+a*(47+4*sqrt(4+a)))/((3+a)^2*(4+a)^(5/2)*sqrt(1-sqrt(4+a)))-3/64*arctan((-1+x)/sqrt(1+sqrt(4+a)))*(14+4*a+(-80-47*a-7*a^2)/sqrt(4+a))/((3+a)^2*(4+a)^2*sqrt(1+sqrt(4+a)))],
[x^2*(a+8*x-8*x^2+4*x^3-x^4)^4,x,2,1/3*a^4*x^3+8*a^3*x^4+32/5*(12-a)*a^2*x^5+8/3*a*(128-48*a+a^2)*x^6+4/7*(1024-1536*a+192*a^2-a^3)*x^7-4*(512-288*a+15*a^2)*x^8+64/9*(128-3*a)*(4-a)*x^9-24/5*(896-128*a+a^2)*x^10+2/11*(20480-1536*a+3*a^2)*x^11-8/3*(928-35*a)*x^12+32/13*(524-9*a)*x^13-8/7*(464-3*a)*x^14+4/15*(640-a)*x^15-42*x^16+128/17*x^17-8/9*x^18+1/19*x^19],
[x^2*(a+8*x-8*x^2+4*x^3-x^4)^3,x,2,1/3*a^3*x^3+6*a^2*x^4+24/5*(8-a)*a*x^5+2/3*(128-96*a+3*a^2)*x^6-3/7*(512-128*a+a^2)*x^7+6*(48-5*a)*x^8-32/9*(70-3*a)*x^9+12/5*(64-a)*x^10-3/11*(256-a)*x^11+70/3*x^12-72/13*x^13+6/7*x^14-1/15*x^15],
[x^2*(a+8*x-8*x^2+4*x^3-x^4)^2,x,2,1/3*a^2*x^3+4*a*x^4+16/5*(4-a)*x^5-4/3*(16-a)*x^6+2/7*(64-a)*x^7-10*x^8+32/9*x^9-4/5*x^10+1/11*x^11],
[x^2*(a+8*x-8*x^2+4*x^3-x^4),x,2,1/3*a*x^3+2*x^4-8/5*x^5+2/3*x^6-1/7*x^7],
[x^2/(a+8*x-8*x^2+4*x^3-x^4),x,9,arctanh((1+(-1+x)^2)/sqrt(4+a))/sqrt(4+a)-1/2*arctan((-1+x)/sqrt(1-sqrt(4+a)))/sqrt(1-sqrt(4+a))-1/2*arctan((-1+x)/sqrt(1+sqrt(4+a)))/sqrt(1+sqrt(4+a))],
[x^2/(a+8*x-8*x^2+4*x^3-x^4)^2,x,11,1/2*(1+(-1+x)^2)/((4+a)*(3+a-2*(-1+x)^2-(-1+x)^4))+1/4*(4+a)*(2+(-1+x)^2)*(-1+x)/((12+7*a+a^2)*(3+a-2*(-1+x)^2-(-1+x)^4))+1/2*arctanh((1+(-1+x)^2)/sqrt(4+a))/(4+a)^(3/2)-1/8*arctan((-1+x)/sqrt(1-sqrt(4+a)))*(4+a+sqrt(4+a))/((3+a)*(4+a)*sqrt(1-sqrt(4+a)))-1/8*arctan((-1+x)/sqrt(1+sqrt(4+a)))*(4+a-sqrt(4+a))/((3+a)*(4+a)*sqrt(1+sqrt(4+a)))],

# Integrands of the form x^m P6[x]^p

# Integrands of the form x^m (a+b x^2+c x^3+d x^4+e x^6)^p when b^2-3 a d=0 and b^3-27 a^2 e=0

# p>0

# p<0
[x^4/(27*a^3+27*a^2*b*x^2+27*a^2*c*x^3+9*a*b^2*x^4+b^3*x^6),x,14,-1/18*log(3*a+3*a^(2/3)*c^(1/3)*x+b*x^2)/(a^(2/3)*b^2*c^(1/3))+1/6*log(3*a-3*(-1)^(1/3)*a^(2/3)*c^(1/3)*x+b*x^2)/((1+(-1)^(1/3))^2*a^(2/3)*b^2*c^(1/3))+1/18*(-1)^(1/3)*log(3*a+3*(-1)^(2/3)*a^(2/3)*c^(1/3)*x+b*x^2)/(a^(2/3)*b^2*c^(1/3))-1/9*(2*b-3*a^(1/3)*c^(2/3))*arctan((3*a^(2/3)*c^(1/3)+2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b-3*a^(1/3)*c^(2/3))))/(a^(5/6)*b^2*c^(2/3)*sqrt(3)*sqrt(4*b-3*a^(1/3)*c^(2/3)))-1/3*(-1)^(2/3)*(2*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3))*arctan((3*(-1)^(2/3)*a^(2/3)*c^(1/3)+2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3))))/((1-(-1)^(1/3))*(1+(-1)^(1/3))^2*a^(5/6)*b^2*c^(2/3)*sqrt(3)*sqrt(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3)))-1/3*(-1)^(1/3)*(2*(-1)^(1/3)*b+3*a^(1/3)*c^(2/3))*arctan((3*(-1)^(1/3)*a^(2/3)*c^(1/3)-2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3))))/((1+(-1)^(1/3))^2*a^(5/6)*b^2*c^(2/3)*sqrt(3)*sqrt(4*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3)))],
[x^3/(27*a^3+27*a^2*b*x^2+27*a^2*c*x^3+9*a*b^2*x^4+b^3*x^6),x,14,1/54*log(3*a+3*a^(2/3)*c^(1/3)*x+b*x^2)/(a^(4/3)*b*c^(2/3))-1/18*(-1)^(2/3)*log(3*a-3*(-1)^(1/3)*a^(2/3)*c^(1/3)*x+b*x^2)/((1+(-1)^(1/3))^2*a^(4/3)*b*c^(2/3))+1/54*(-1)^(2/3)*log(3*a+3*(-1)^(2/3)*a^(2/3)*c^(1/3)*x+b*x^2)/(a^(4/3)*b*c^(2/3))-1/9*arctan((3*a^(2/3)*c^(1/3)+2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b-3*a^(1/3)*c^(2/3))))/(a^(7/6)*b*c^(1/3)*sqrt(3)*sqrt(4*b-3*a^(1/3)*c^(2/3)))+1/3*(-1)^(1/3)*arctan((3*(-1)^(2/3)*a^(2/3)*c^(1/3)+2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3))))/((1-(-1)^(1/3))*(1+(-1)^(1/3))^2*a^(7/6)*b*c^(1/3)*sqrt(3)*sqrt(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3)))-1/3*arctan((3*(-1)^(1/3)*a^(2/3)*c^(1/3)-2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3))))/((1+(-1)^(1/3))^2*a^(7/6)*b*c^(1/3)*sqrt(3)*sqrt(4*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3)))],
[x^2/(27*a^3+27*a^2*b*x^2+27*a^2*c*x^3+9*a*b^2*x^4+b^3*x^6),x,8,2/27*arctan((3*a^(2/3)*c^(1/3)+2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b-3*a^(1/3)*c^(2/3))))/(a^(11/6)*c^(2/3)*sqrt(3)*sqrt(4*b-3*a^(1/3)*c^(2/3)))+2/9*(-1)^(2/3)*arctan((3*(-1)^(2/3)*a^(2/3)*c^(1/3)+2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3))))/((1-(-1)^(1/3))*(1+(-1)^(1/3))^2*a^(11/6)*c^(2/3)*sqrt(3)*sqrt(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3)))+2/9*(-1)^(2/3)*arctan((3*(-1)^(1/3)*a^(2/3)*c^(1/3)-2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3))))/((1+(-1)^(1/3))^2*a^(11/6)*c^(2/3)*sqrt(3)*sqrt(4*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3)))],
[x/(27*a^3+27*a^2*b*x^2+27*a^2*c*x^3+9*a*b^2*x^4+b^3*x^6),x,14,-1/162*log(3*a+3*a^(2/3)*c^(1/3)*x+b*x^2)/(a^(7/3)*c^(2/3))+1/54*(-1)^(2/3)*log(3*a-3*(-1)^(1/3)*a^(2/3)*c^(1/3)*x+b*x^2)/((1+(-1)^(1/3))^2*a^(7/3)*c^(2/3))-1/162*(-1)^(2/3)*log(3*a+3*(-1)^(2/3)*a^(2/3)*c^(1/3)*x+b*x^2)/(a^(7/3)*c^(2/3))-1/27*arctan((3*a^(2/3)*c^(1/3)+2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b-3*a^(1/3)*c^(2/3))))/(a^(13/6)*c^(1/3)*sqrt(3)*sqrt(4*b-3*a^(1/3)*c^(2/3)))+1/9*(-1)^(1/3)*arctan((3*(-1)^(2/3)*a^(2/3)*c^(1/3)+2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3))))/((1-(-1)^(1/3))*(1+(-1)^(1/3))^2*a^(13/6)*c^(1/3)*sqrt(3)*sqrt(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3)))-1/9*arctan((3*(-1)^(1/3)*a^(2/3)*c^(1/3)-2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3))))/((1+(-1)^(1/3))^2*a^(13/6)*c^(1/3)*sqrt(3)*sqrt(4*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3)))],
[1/(27*a^3+27*a^2*b*x^2+27*a^2*c*x^3+9*a*b^2*x^4+b^3*x^6),x,14,1/162*log(3*a+3*a^(2/3)*c^(1/3)*x+b*x^2)/(a^(8/3)*c^(1/3))-1/54*log(3*a-3*(-1)^(1/3)*a^(2/3)*c^(1/3)*x+b*x^2)/((1+(-1)^(1/3))^2*a^(8/3)*c^(1/3))-1/162*(-1)^(1/3)*log(3*a+3*(-1)^(2/3)*a^(2/3)*c^(1/3)*x+b*x^2)/(a^(8/3)*c^(1/3))-1/81*(2*b-3*a^(1/3)*c^(2/3))*arctan((3*a^(2/3)*c^(1/3)+2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b-3*a^(1/3)*c^(2/3))))/(a^(17/6)*c^(2/3)*sqrt(3)*sqrt(4*b-3*a^(1/3)*c^(2/3)))-1/27*(2*(-1)^(2/3)*b-3*a^(1/3)*c^(2/3))*arctan((3*(-1)^(2/3)*a^(2/3)*c^(1/3)+2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3))))/((1-(-1)^(1/3))*(1+(-1)^(1/3))^2*a^(17/6)*c^(2/3)*sqrt(3)*sqrt(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3)))-1/27*(-1)^(1/3)*(2*(-1)^(1/3)*b+3*a^(1/3)*c^(2/3))*arctan((3*(-1)^(1/3)*a^(2/3)*c^(1/3)-2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3))))/((1+(-1)^(1/3))^2*a^(17/6)*c^(2/3)*sqrt(3)*sqrt(4*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3)))],
[1/(x*(27*a^3+27*a^2*b*x^2+27*a^2*c*x^3+9*a*b^2*x^4+b^3*x^6)),x,14,1/27*log(x)/a^3-1/486*(3*a^(1/3)-b/c^(2/3))*log(3*a+3*a^(2/3)*c^(1/3)*x+b*x^2)/a^(10/3)-1/486*(3*a^(1/3)-(-1)^(2/3)*b/c^(2/3))*log(3*a+3*(-1)^(2/3)*a^(2/3)*c^(1/3)*x+b*x^2)/a^(10/3)-1/972*log(3*a-3*(-1)^(1/3)*a^(2/3)*c^(1/3)*x+b*x^2)*(b+6*a^(1/3)*c^(2/3)+I*b*sqrt(3))/(a^(10/3)*c^(2/3))+1/27*(b-a^(1/3)*c^(2/3))*arctan((3*a^(2/3)*c^(1/3)+2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b-3*a^(1/3)*c^(2/3))))/(a^(19/6)*c^(1/3)*sqrt(3)*sqrt(4*b-3*a^(1/3)*c^(2/3)))+1/9*(-1)^(2/3)*((-1)^(2/3)*b-a^(1/3)*c^(2/3))*arctan((3*(-1)^(2/3)*a^(2/3)*c^(1/3)+2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3))))/((1-(-1)^(1/3))*(1+(-1)^(1/3))^2*a^(19/6)*c^(1/3)*sqrt(3)*sqrt(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3)))+1/9*(b-(-1)^(2/3)*a^(1/3)*c^(2/3))*arctan((3*(-1)^(1/3)*a^(2/3)*c^(1/3)-2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3))))/((1+(-1)^(1/3))^2*a^(19/6)*c^(1/3)*sqrt(3)*sqrt(4*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3)))],
[1/(x^2*(27*a^3+27*a^2*b*x^2+27*a^2*c*x^3+9*a*b^2*x^4+b^3*x^6)),x,14,(-1/27)/(a^3*x)-1/486*(2*b-3*a^(1/3)*c^(2/3))*log(3*a+3*a^(2/3)*c^(1/3)*x+b*x^2)/(a^(11/3)*c^(1/3))+1/162*(2*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3))*log(3*a-3*(-1)^(1/3)*a^(2/3)*c^(1/3)*x+b*x^2)/((1+(-1)^(1/3))^2*a^(11/3)*c^(1/3))+1/486*(-1)^(1/3)*(2*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3))*log(3*a+3*(-1)^(2/3)*a^(2/3)*c^(1/3)*x+b*x^2)/(a^(11/3)*c^(1/3))+1/243*(2*b^2-12*a^(1/3)*b*c^(2/3)+9*a^(2/3)*c^(4/3))*arctan((3*a^(2/3)*c^(1/3)+2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b-3*a^(1/3)*c^(2/3))))/(a^(23/6)*c^(2/3)*sqrt(3)*sqrt(4*b-3*a^(1/3)*c^(2/3)))+1/81*(-1)^(2/3)*(2*b^2+12*(-1)^(1/3)*a^(1/3)*b*c^(2/3)+9*(-1)^(2/3)*a^(2/3)*c^(4/3))*arctan((3*(-1)^(2/3)*a^(2/3)*c^(1/3)+2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3))))/((1-(-1)^(1/3))*(1+(-1)^(1/3))^2*a^(23/6)*c^(2/3)*sqrt(3)*sqrt(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3)))+1/81*(2*(-1)^(2/3)*b^2+12*(-1)^(1/3)*a^(1/3)*b*c^(2/3)+9*a^(2/3)*c^(4/3))*arctan((3*(-1)^(1/3)*a^(2/3)*c^(1/3)-2*b*x)/(sqrt(3)*sqrt(a)*sqrt(4*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3))))/((1+(-1)^(1/3))^2*a^(23/6)*c^(2/3)*sqrt(3)*sqrt(4*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3)))],
[x^5/(216+108*x^2+324*x^3+18*x^4+x^6),x,14,1/108*(18-(-2)^(2/3)*3^(1/3))*log(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)+1/108*(18-2^(2/3)*3^(1/3))*log(6+3*2^(2/3)*3^(1/3)*x+x^2)+1/216*log(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)*(36+2^(2/3)*3^(1/3)*(1+I*sqrt(3)))+(3/2)^(1/6)*(1-(-3)^(2/3)*2^(1/3))*arctan(2^(1/6)*(3*(-3)^(1/3)-2^(1/3)*x)/sqrt(3*(4-3*(-3)^(2/3)*2^(1/3))))/((1+(-1)^(1/3))^2*sqrt(4-3*(-3)^(2/3)*2^(1/3)))-(1-2^(1/3)*3^(2/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/(2^(1/6)*3^(5/6)*sqrt(-4+3*2^(1/3)*3^(2/3)))-(-2)^(1/3)*(1+(-2)^(1/3)*3^(2/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))/(3^(5/6)*sqrt(8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3)))],
[x^4/(216+108*x^2+324*x^3+18*x^4+x^6),x,14,1/6*log(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)/(2^(2/3)*3^(1/3)*(1+(-1)^(1/3))^2)+1/18*(-1/3)^(1/3)*log(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)/2^(2/3)-1/18*log(6+3*2^(2/3)*3^(1/3)*x+x^2)/(2^(2/3)*3^(1/3))+1/9*(-1)^(2/3)*(3*(-3)^(2/3)-2^(2/3))*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/sqrt(6*(4-3*(-3)^(2/3)*2^(1/3))))/(3^(1/6)*(1+(-1)^(1/3))^2*sqrt(2*(4-3*(-3)^(2/3)*2^(1/3))))-1/27*(9-2^(2/3)*3^(1/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/sqrt(6*(-4+3*2^(1/3)*3^(2/3)))+1/27*(9-(-2)^(2/3)*3^(1/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))/sqrt(3*(8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3)))],
[x^3/(216+108*x^2+324*x^3+18*x^4+x^6),x,14,-1/36*(-1)^(2/3)*log(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)/(2^(1/3)*3^(2/3)*(1+(-1)^(1/3))^2)+1/108*(-1)^(2/3)*log(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)/(2^(1/3)*3^(2/3))+1/108*log(6+3*2^(2/3)*3^(1/3)*x+x^2)/(2^(1/3)*3^(2/3))-1/6*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/sqrt(6*(4-3*(-3)^(2/3)*2^(1/3))))/(2^(1/6)*3^(5/6)*(1+(-1)^(1/3))^2*sqrt(4-3*(-3)^(2/3)*2^(1/3)))+1/18*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/(2^(1/6)*3^(5/6)*sqrt(-4+3*2^(1/3)*3^(2/3)))+1/9*(-1)^(1/3)*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))/(2^(2/3)*3^(5/6)*sqrt(8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3)))],
[x^2/(216+108*x^2+324*x^3+18*x^4+x^6),x,8,1/27*(-1)^(2/3)*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/sqrt(6*(4-3*(-3)^(2/3)*2^(1/3))))/(2^(5/6)*3^(1/6)*(1+(-1)^(1/3))^2*sqrt(4-3*(-3)^(2/3)*2^(1/3)))-1/81*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/(2^(5/6)*3^(1/6)*sqrt(-4+3*2^(1/3)*3^(2/3)))+1/81*(-1)^(2/3)*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))/(2^(1/3)*3^(1/6)*sqrt(8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3)))],
[x/(216+108*x^2+324*x^3+18*x^4+x^6),x,14,1/216*(-1)^(2/3)*log(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)/(2^(1/3)*3^(2/3)*(1+(-1)^(1/3))^2)-1/648*(-1)^(2/3)*log(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)/(2^(1/3)*3^(2/3))-1/648*log(6+3*2^(2/3)*3^(1/3)*x+x^2)/(2^(1/3)*3^(2/3))-1/36*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/sqrt(6*(4-3*(-3)^(2/3)*2^(1/3))))/(2^(1/6)*3^(5/6)*(1+(-1)^(1/3))^2*sqrt(4-3*(-3)^(2/3)*2^(1/3)))+1/108*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/(2^(1/6)*3^(5/6)*sqrt(-4+3*2^(1/3)*3^(2/3)))+1/54*(-1)^(1/3)*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))/(2^(2/3)*3^(5/6)*sqrt(8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3)))],
[1/(216+108*x^2+324*x^3+18*x^4+x^6),x,14,-1/216*log(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)/(2^(2/3)*3^(1/3)*(1+(-1)^(1/3))^2)-1/648*(-1/3)^(1/3)*log(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)/2^(2/3)+1/648*log(6+3*2^(2/3)*3^(1/3)*x+x^2)/(2^(2/3)*3^(1/3))+1/324*(-1)^(2/3)*(3*(-3)^(2/3)-2^(2/3))*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/sqrt(6*(4-3*(-3)^(2/3)*2^(1/3))))/(3^(1/6)*(1+(-1)^(1/3))^2*sqrt(2*(4-3*(-3)^(2/3)*2^(1/3))))-1/972*(9-2^(2/3)*3^(1/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/sqrt(6*(-4+3*2^(1/3)*3^(2/3)))+1/972*(9-(-2)^(2/3)*3^(1/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))/sqrt(3*(8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3)))],
[1/(x*(216+108*x^2+324*x^3+18*x^4+x^6)),x,14,1/216*log(x)-1/23328*(18-(-2)^(2/3)*3^(1/3))*log(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)-1/23328*(18-2^(2/3)*3^(1/3))*log(6+3*2^(2/3)*3^(1/3)*x+x^2)-1/46656*log(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)*(36+2^(2/3)*3^(1/3)*(1+I*sqrt(3)))-1/216*(-1)^(2/3)*((-3)^(1/3)+3*2^(1/3))*arctan(2^(1/6)*(3*(-3)^(1/3)-2^(1/3)*x)/sqrt(3*(4-3*(-3)^(2/3)*2^(1/3))))/(6^(1/6)*(1+(-1)^(1/3))^2*sqrt(4-3*(-3)^(2/3)*2^(1/3)))-1/216*(1-2^(1/3)*3^(2/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/(2^(1/6)*3^(5/6)*sqrt(-4+3*2^(1/3)*3^(2/3)))+1/216*(-1)^(2/3)*((-2)^(2/3)-2*3^(2/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))/(2^(1/3)*3^(5/6)*sqrt(8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3)))],
[1/(x^2*(216+108*x^2+324*x^3+18*x^4+x^6)),x,14,(-1/216)/x-1/1296*(-1)^(2/3)*(9+(-3)^(1/3)*2^(2/3))*log(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)/(2^(1/3)*3^(2/3)*(1+(-1)^(1/3))^2)+1/7776*(3*(-6)^(2/3)+2*(-2)^(1/3))*log(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)/3^(1/3)-1/3888*(2^(2/3)-3*3^(2/3))*log(6+3*2^(2/3)*3^(1/3)*x+x^2)/6^(1/3)-1/1944*(-1)^(2/3)*(6*(-6)^(2/3)+27*(-3)^(1/3)-2^(1/3))*arctan(2^(1/6)*(3*(-3)^(1/3)-2^(1/3)*x)/sqrt(3*(4-3*(-3)^(2/3)*2^(1/3))))/(6^(1/6)*(1+(-1)^(1/3))^2*sqrt(4-3*(-3)^(2/3)*2^(1/3)))-1/5832*(2^(1/3)+27*3^(1/3)-6*6^(2/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/(6^(1/6)*sqrt(-4+3*2^(1/3)*3^(2/3)))-1/5832*(27*(-6)^(1/3)-(-2)^(2/3)+12*3^(2/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))/(3^(1/6)*sqrt(8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3)))],
[x^8/(216+108*x^2+324*x^3+18*x^4+x^6)^2,x,23,-1/162*(-1/3)^(1/3)*(9*(6+(-3)^(1/3)*2^(2/3))+(2-2^(2/3)*(6*(-6)^(2/3)+27*(-3)^(1/3)))*x)/(2^(2/3)*(1+(-1)^(1/3))^4*(4-3*(-3)^(2/3)*2^(1/3))*(6-3*(-3)^(1/3)*2^(2/3)*x+x^2))-1/729*(-1/3)^(1/3)*(9*(6-(-2)^(2/3)*3^(1/3))+(2+27*(-2)^(2/3)*3^(1/3)+12*(-2)^(1/3)*3^(2/3))*x)/(2^(2/3)*(8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))*(6+3*(-2)^(2/3)*3^(1/3)*x+x^2))+1/1458*(9*(6-2^(2/3)*3^(1/3))+(2+2^(2/3)*(27*3^(1/3)-6*6^(2/3)))*x)/(2^(2/3)*3^(1/3)*(4-3*2^(1/3)*3^(2/3))*(6+3*2^(2/3)*3^(1/3)*x+x^2))-1/162*(-1)^(1/3)*(2+27*(-2)^(2/3)*3^(1/3)+12*(-2)^(1/3)*3^(2/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))/(2^(1/6)*3^(5/6)*(1-(-1)^(1/3))^2*(1+(-1)^(1/3))^4*(4+3*(-2)^(1/3)*3^(2/3))^(3/2))-1/972*log(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)/(2^(1/3)*3^(2/3)*(1+(-1)^(1/3))^4)+1/972*I*log(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)/(2^(1/3)*3^(1/6)*(1+(-1)^(1/3))^5)-1/8748*log(6+3*2^(2/3)*3^(1/3)*x+x^2)/(2^(1/3)*3^(2/3))-1/81*(-1)^(1/3)*(6*(-6)^(2/3)+27*(-3)^(1/3)-2^(1/3))*arctan(2^(1/6)*(3*(-3)^(1/3)-2^(1/3)*x)/sqrt(3*(4-3*(-3)^(2/3)*2^(1/3))))/(3^(5/6)*(1+(-1)^(1/3))^4*(4-3*(-3)^(2/3)*2^(1/3))^(3/2)*sqrt(2))+1/81*(2^(1/3)+27*3^(1/3)-6*6^(2/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/(3^(5/6)*(1-(-1)^(1/3))^2*(1+(-1)^(1/3))^4*(-4+3*2^(1/3)*3^(2/3))^(3/2)*sqrt(2))+1/162*(I*2^(2/3)-9*3^(1/6)-3*I*3^(2/3))*arctan(2^(1/6)*(3*(-3)^(1/3)-2^(1/3)*x)/sqrt(3*(4-3*(-3)^(2/3)*2^(1/3))))/(2^(5/6)*3^(1/3)*(1+(-1)^(1/3))^5*sqrt(4-3*(-3)^(2/3)*2^(1/3)))-1/162*I*((-2)^(2/3)+6*3^(2/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))/(2^(5/6)*3^(1/3)*(1+(-1)^(1/3))^5*sqrt(4+3*(-2)^(1/3)*3^(2/3)))-1/1458*(1+3*2^(1/3)*3^(2/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/(2^(1/6)*3^(5/6)*sqrt(-4+3*2^(1/3)*3^(2/3)))],
[x^7/(216+108*x^2+324*x^3+18*x^4+x^6)^2,x,23,1/972*(-2*(2*(-1)^(1/3)*3^(2/3)+9*6^(1/3))+9*((-2)^(2/3)+2*(-1)^(1/3)*3^(2/3))*x)/(2^(2/3)*(1+(-1)^(1/3))^4*(4-3*(-3)^(2/3)*2^(1/3))*(6-3*(-3)^(1/3)*2^(2/3)*x+x^2))+1/4374*(-(-6)^(1/3)*(9*(-2)^(1/3)+2*3^(1/3))+9*(1+(-2)^(1/3)*3^(2/3))*x)/((8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))*(6+3*(-2)^(2/3)*3^(1/3)*x+x^2))+1/2916*(2*(2-3*2^(1/3)*3^(2/3))-3*(6-2^(2/3)*3^(1/3))*x)/(2^(2/3)*3^(1/3)*(4-3*2^(1/3)*3^(2/3))*(6+3*2^(2/3)*3^(1/3)*x+x^2))+1/648*I*log(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)/(2^(2/3)*3^(5/6)*(1+(-1)^(1/3))^5)-1/17496*log(6+3*2^(2/3)*3^(1/3)*x+x^2)/(2^(2/3)*3^(1/3))-1/54*(-1)^(1/3)*((-3)^(1/3)+3*2^(1/3))*arctan(2^(1/6)*(3*(-3)^(1/3)-2^(1/3)*x)/sqrt(3*(4-3*(-3)^(2/3)*2^(1/3))))/(3^(5/6)*(1+(-1)^(1/3))^4*(4-3*(-3)^(2/3)*2^(1/3))^(3/2)*sqrt(2))-1/1296*log(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)*(I+sqrt(3))/(2^(2/3)*3^(5/6)*(1+(-1)^(1/3))^5)+1/54*(1+(-2)^(1/3)*3^(2/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))/((1-(-1)^(1/3))^2*(1+(-1)^(1/3))^4*(4+3*(-2)^(1/3)*3^(2/3))^(3/2)*sqrt(6))+1/54*(1-2^(1/3)*3^(2/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/((1-(-1)^(1/3))^2*(1+(-1)^(1/3))^4*(-4+3*2^(1/3)*3^(2/3))^(3/2)*sqrt(6))+1/5832*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/sqrt(6*(4-3*(-3)^(2/3)*2^(1/3))))*(9*I+3^(1/3)*(2*I*2^(2/3)-9*3^(1/6)+2*2^(2/3)*sqrt(3)))/((1+(-1)^(1/3))^5*sqrt(2*(4-3*(-3)^(2/3)*2^(1/3))))+1/1944*(9*3^(1/6)+I*(4*2^(2/3)-3*3^(2/3)))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))/(3^(2/3)*(1+(-1)^(1/3))^5*sqrt(2*(4+3*(-2)^(1/3)*3^(2/3))))+1/26244*(2*2^(2/3)+3*3^(2/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/(3^(1/6)*sqrt(2*(-4+3*2^(1/3)*3^(2/3))))],
[x^6/(216+108*x^2+324*x^3+18*x^4+x^6)^2,x,14,1/2916*(9*(-2)^(2/3)+6^(1/3)*(9+(-3)^(1/3)*2^(2/3))*x)/(2^(2/3)*(1+(-1)^(1/3))^4*(4-3*(-3)^(2/3)*2^(1/3))*(6-3*(-3)^(1/3)*2^(2/3)*x+x^2))+1/13122*(9*2^(2/3)+(-1)^(1/3)*3^(2/3)*(2+3*(-2)^(1/3)*3^(2/3))*x)/(2^(2/3)*(8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))*(6+3*(-2)^(2/3)*3^(1/3)*x+x^2))+1/8748*(3*2^(2/3)*3^(1/3)-(2-3*2^(1/3)*3^(2/3))*x)/(2^(2/3)*3^(1/3)*(4-3*2^(1/3)*3^(2/3))*(6+3*2^(2/3)*3^(1/3)*x+x^2))+1/486*(-1)^(1/3)*(3*(-3)^(2/3)-2^(2/3))*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/sqrt(6*(4-3*(-3)^(2/3)*2^(1/3))))/(6^(5/6)*(1+(-1)^(1/3))^4*(4-3*(-3)^(2/3)*2^(1/3))^(3/2))+1/486*(3*(-3)^(2/3)+(-1)^(1/3)*2^(2/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))/(6^(5/6)*(1-(-1)^(1/3))^2*(1+(-1)^(1/3))^4*(4+3*(-2)^(1/3)*3^(2/3))^(3/2))-1/486*(2^(2/3)-3*3^(2/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/(6^(5/6)*(1-(-1)^(1/3))^2*(1+(-1)^(1/3))^4*(-4+3*2^(1/3)*3^(2/3))^(3/2))+1/5832*(-1/3)^(1/6)*log(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)/(2^(1/3)*(1+(-1)^(1/3))^5)-1/5832*I*log(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)/(2^(1/3)*3^(1/6)*(1+(-1)^(1/3))^5)+1/52488*log(6+3*2^(2/3)*3^(1/3)*x+x^2)/(2^(1/3)*3^(2/3))],
[x^5/(216+108*x^2+324*x^3+18*x^4+x^6)^2,x,17,1/1944*(-1/3)^(1/3)*(4-(-3)^(1/3)*2^(2/3)*x)/(2^(2/3)*(1+(-1)^(1/3))^4*(4-3*(-3)^(2/3)*2^(1/3))*(6-3*(-3)^(1/3)*2^(2/3)*x+x^2))+1/8748*(-1/3)^(1/3)*(4+(-2)^(2/3)*3^(1/3)*x)/(2^(2/3)*(8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))*(6+3*(-2)^(2/3)*3^(1/3)*x+x^2))+1/17496*(-4-2^(2/3)*3^(1/3)*x)/(2^(2/3)*3^(1/3)*(4-3*2^(1/3)*3^(2/3))*(6+3*2^(2/3)*3^(1/3)*x+x^2))+1/4374*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/sqrt(6*(4-3*(-3)^(2/3)*2^(1/3))))/((8-9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))^(3/2)*sqrt(3))-1/4374*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))/((8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))^(3/2)*sqrt(3))-1/8748*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/((-4+3*2^(1/3)*3^(2/3))^(3/2)*sqrt(6))-1/4374*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/sqrt(6*(4-3*(-3)^(2/3)*2^(1/3))))/(2^(5/6)*3^(1/6)*(1+(-1)^(1/3))^4*sqrt(4-3*(-3)^(2/3)*2^(1/3)))-1/1458*I*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))/(2^(5/6)*3^(2/3)*(1+(-1)^(1/3))^5*sqrt(4+3*(-2)^(1/3)*3^(2/3)))-1/39366*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/(2^(5/6)*3^(1/6)*sqrt(-4+3*2^(1/3)*3^(2/3)))],
[x^4/(216+108*x^2+324*x^3+18*x^4+x^6)^2,x,23,1/5832*(-1/3)^(1/3)*(3*(-3)^(1/3)*2^(2/3)-2*x)/(2^(2/3)*(1+(-1)^(1/3))^4*(4-3*(-3)^(2/3)*2^(1/3))*(6-3*(-3)^(1/3)*2^(2/3)*x+x^2))-1/26244*(-1/3)^(1/3)*(3*(-2)^(2/3)*3^(1/3)+2*x)/(2^(2/3)*(8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))*(6+3*(-2)^(2/3)*3^(1/3)*x+x^2))+1/52488*(-3*3^(1/3)-2^(1/3)*x)/((9*2^(1/3)-4*3^(1/3))*(6+3*2^(2/3)*3^(1/3)*x+x^2))+1/729*(-1)^(1/3)*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/sqrt(6*(4-3*(-3)^(2/3)*2^(1/3))))/(2^(2/3)*3^(5/6)*(1+(-1)^(1/3))^4*(8-9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))^(3/2))-1/2916*(-1)^(1/3)*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))/(2^(1/6)*3^(5/6)*(1-(-1)^(1/3))^2*(1+(-1)^(1/3))^4*(4+3*(-2)^(1/3)*3^(2/3))^(3/2))+1/26244*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/(2^(1/6)*3^(5/6)*(-4+3*2^(1/3)*3^(2/3))^(3/2))-1/34992*log(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)/(2^(1/3)*3^(2/3)*(1+(-1)^(1/3))^4)+1/34992*I*log(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)/(2^(1/3)*3^(1/6)*(1+(-1)^(1/3))^5)-1/314928*log(6+3*2^(2/3)*3^(1/3)*x+x^2)/(2^(1/3)*3^(2/3))-1/5832*I*arctan(2^(1/6)*(3*(-3)^(1/3)-2^(1/3)*x)/sqrt(3*(4-3*(-3)^(2/3)*2^(1/3))))/(2^(1/6)*3^(1/3)*(1+(-1)^(1/3))^5*sqrt(4-3*(-3)^(2/3)*2^(1/3)))-1/11664*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))*(I+sqrt(3))/(2^(1/6)*3^(1/3)*(1+(-1)^(1/3))^5*sqrt(4+3*(-2)^(1/3)*3^(2/3)))+1/52488*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/(2^(1/6)*3^(5/6)*sqrt(-4+3*2^(1/3)*3^(2/3)))],
[x^3/(216+108*x^2+324*x^3+18*x^4+x^6)^2,x,23,1/157464*((-6)^(1/3)*(2*(-3)^(1/3)+9*2^(1/3))-3*x)/((8-9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))*(6-3*(-3)^(1/3)*2^(2/3)*x+x^2))+1/157464*(-(-6)^(1/3)*(9*(-2)^(1/3)+2*3^(1/3))-3*x)/((8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))*(6+3*(-2)^(2/3)*3^(1/3)*x+x^2))+1/104976*(-2*2^(1/3)+3*6^(2/3)+3^(1/3)*x)/((9*2^(1/3)-4*3^(1/3))*(6+3*2^(2/3)*3^(1/3)*x+x^2))-1/23328*I*log(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)/(2^(2/3)*3^(5/6)*(1+(-1)^(1/3))^5)+1/629856*log(6+3*2^(2/3)*3^(1/3)*x+x^2)/(2^(2/3)*3^(1/3))+1/26244*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/sqrt(6*(4-3*(-3)^(2/3)*2^(1/3))))/((8-9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))^(3/2)*sqrt(3))-1/26244*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))/((8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))^(3/2)*sqrt(3))+1/46656*log(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)*(I+sqrt(3))/(2^(2/3)*3^(5/6)*(1+(-1)^(1/3))^5)-1/52488*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/((-4+3*2^(1/3)*3^(2/3))^(3/2)*sqrt(6))-1/209952*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/sqrt(6*(4-3*(-3)^(2/3)*2^(1/3))))*(9*I-3^(1/3)*(2*I*2^(2/3)+9*3^(1/6)+2*2^(2/3)*sqrt(3)))/((1+(-1)^(1/3))^5*sqrt(2*(4-3*(-3)^(2/3)*2^(1/3))))+1/209952*(9*I+3^(1/3)*(4*I*2^(2/3)-9*3^(1/6)))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))/((1+(-1)^(1/3))^5*sqrt(2*(4+3*(-2)^(1/3)*3^(2/3))))+1/944784*(2*2^(2/3)-3*3^(2/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/(3^(1/6)*sqrt(2*(-4+3*2^(1/3)*3^(2/3))))],
[x^2/(216+108*x^2+324*x^3+18*x^4+x^6)^2,x,23,1/104976*(-27*((-2)^(2/3)+2*(-1)^(1/3)*3^(2/3))+6^(1/3)*(9+(-3)^(1/3)*2^(2/3))*x)/(2^(2/3)*(1+(-1)^(1/3))^4*(4-3*(-3)^(2/3)*2^(1/3))*(6-3*(-3)^(1/3)*2^(2/3)*x+x^2))+1/472392*(-27*(2^(2/3)+2*(-1)^(1/3)*3^(2/3))+(-1)^(1/3)*3^(2/3)*(2+3*(-2)^(1/3)*3^(2/3))*x)/(2^(2/3)*(8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))*(6+3*(-2)^(2/3)*3^(1/3)*x+x^2))+1/314928*(9*(6-2^(2/3)*3^(1/3))-(2-3*2^(1/3)*3^(2/3))*x)/(2^(2/3)*3^(1/3)*(4-3*2^(1/3)*3^(2/3))*(6+3*2^(2/3)*3^(1/3)*x+x^2))+1/17496*(3*(-3)^(2/3)+(-1)^(1/3)*2^(2/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))/(6^(5/6)*(1-(-1)^(1/3))^2*(1+(-1)^(1/3))^4*(4+3*(-2)^(1/3)*3^(2/3))^(3/2))-1/17496*(2^(2/3)-3*3^(2/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/(6^(5/6)*(1-(-1)^(1/3))^2*(1+(-1)^(1/3))^4*(-4+3*2^(1/3)*3^(2/3))^(3/2))-1/209952*I*log(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)/(2^(1/3)*3^(1/6)*(1+(-1)^(1/3))^5)+1/1889568*log(6+3*2^(2/3)*3^(1/3)*x+x^2)/(2^(1/3)*3^(2/3))-1/8748*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/sqrt(6*(4-3*(-3)^(2/3)*2^(1/3))))*(1+3*2^(1/3)*3^(2/3)+I*sqrt(3))/(2^(2/3)*3^(5/6)*(1+(-1)^(1/3))^4*(8-9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))^(3/2))+1/419904*log(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)*(I+sqrt(3))/(2^(1/3)*3^(1/6)*(1+(-1)^(1/3))^5)+1/17496*I*arctan(2^(1/6)*(3*(-3)^(1/3)-2^(1/3)*x)/sqrt(3*(4-3*(-3)^(2/3)*2^(1/3))))/(2^(1/6)*3^(1/3)*(1+(-1)^(1/3))^5*sqrt(4-3*(-3)^(2/3)*2^(1/3)))+1/34992*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/sqrt(6*(4+3*(-2)^(1/3)*3^(2/3))))*(I+sqrt(3))/(2^(1/6)*3^(1/3)*(1+(-1)^(1/3))^5*sqrt(4+3*(-2)^(1/3)*3^(2/3)))-1/157464*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/sqrt(3*(-4+3*2^(1/3)*3^(2/3))))/(2^(1/6)*3^(5/6)*sqrt(-4+3*2^(1/3)*3^(2/3)))],

# Integrands of the form P[x]^p Q[x]

# Integrands of the form P1[x]^p Q[x]

# Integrands of the form P1[x]^p Q5[x]

#  Can cancel GCD of numerator and denominator. 
[(a^2*c+a^2*d*x+2*a*b*c*x^2+2*a*b*d*x^3+b^2*c*x^4+b^2*d*x^5)/(c+d*x),x,2,a^2*x+2/3*a*b*x^3+1/5*b^2*x^5],
[(a^2*c+a^2*d*x+2*a*b*c*x^2+2*a*b*d*x^3+b^2*c*x^4+b^2*d*x^5)/(c+d*x)^2,x,4,-b*c*(b*c^2+2*a*d^2)*x/d^4+1/2*b*(b*c^2+2*a*d^2)*x^2/d^3-1/3*b^2*c*x^3/d^2+1/4*b^2*x^4/d+(b*c^2+a*d^2)^2*log(c+d*x)/d^5],

# Integrands of the form P2[x]^p Q[x]

# Integrands of the form P2[x]^p Q1[x]
[(b+2*c*x)*(b*x+c*x^2)^13,x,1,1/14*(b*x+c*x^2)^14],
[x^14*(b+2*c*x^2)*(b*x+c*x^3)^13,x,3,1/28*x^28*(b+c*x^2)^14],
[x^28*(b+2*c*x^3)*(b*x+c*x^4)^13,x,3,1/42*x^42*(b+c*x^3)^14],
[x^(14*(-1+n))*(b+2*c*x^n)*(b*x+c*x^(1+n))^13,x,3,1/14*x^(14*n)*(b+c*x^n)^14/n],
[(b+2*c*x)/(b*x+c*x^2),x,1,log(b*x+c*x^2)],
[(b+2*c*x^2)/(b*x+c*x^3),x,4,log(x)+1/2*log(b+c*x^2)],
[(b+2*c*x^3)/(b*x+c*x^4),x,4,log(x)+1/3*log(b+c*x^3)],
[(b+2*c*x^n)/(b*x+c*x^(1+n)),x,4,log(x)+log(b+c*x^n)/n],
[(b+2*c*x)/(b*x+c*x^2)^8,x,1,(-1/7)/(b*x+c*x^2)^7],
[(b+2*c*x^2)/(x^7*(b*x+c*x^3)^8),x,3,(-1/14)/(x^14*(b+c*x^2)^7)],
[(b+2*c*x^3)/(x^14*(b*x+c*x^4)^8),x,3,(-1/21)/(x^21*(b+c*x^3)^7)],
[(b+2*c*x^n)/(x^(7*(-1+n))*(b*x+c*x^(1+n))^8),x,3,(-1/7)/(n*x^(7*n)*(b+c*x^n)^7)],
[(b+2*c*x)*(b*x+c*x^2)^p,x,1,(b*x+c*x^2)^(1+p)/(1+p)],
[x^(1+p)*(b+2*c*x^2)*(b*x+c*x^3)^p,x,1,1/2*x^(1+p)*(b*x+c*x^3)^(1+p)/(1+p)],
[b*x^(1+p)*(b*x+c*x^3)^p+2*c*x^(3+p)*(b*x+c*x^3)^p,x,-7,1/2*x^(1+p)*(b*x+c*x^3)^(1+p)/(1+p)],
[x^(2*(1+p))*(b+2*c*x^3)*(b*x+c*x^4)^p,x,1,1/3*x^(2*(1+p))*(b*x+c*x^4)^(1+p)/(1+p)],
[x^((-1+n)*(1+p))*(b+2*c*x^n)*(b*x+c*x^(1+n))^p,x,1,(b*x+c*x^(1+n))^(1+p)/(n*(1+p)*x^((1-n)*(1+p)))],

# Integrands of the form P2[x]^p Q5[x]

#  Can cancel GCD of numerator and denominator. 
[(a^2*c+a^2*d*x+2*a*b*c*x^2+2*a*b*d*x^3+b^2*c*x^4+b^2*d*x^5)/(a+b*x^2),x,2,a*c*x+1/2*a*d*x^2+1/3*b*c*x^3+1/4*b*d*x^4],
[(a^2*c+a^2*d*x+2*a*b*c*x^2+2*a*b*d*x^3+b^2*c*x^4+b^2*d*x^5)/(a+b*x^2)^2,x,3,c*x+1/2*d*x^2],
[(a^2*c+a^2*d*x+2*a*b*c*x^2+2*a*b*d*x^3+b^2*c*x^4+b^2*d*x^5)/(a+b*x^2)^3,x,5,1/2*d*log(a+b*x^2)/b+c*arctan(x*sqrt(b)/sqrt(a))/(sqrt(a)*sqrt(b))],

# Integrands of the form P3[x]^p Q[x]

# Integrands of the form P3[x]^p Q2[x]
[(b+2*c*x+3*d*x^2)*(a+b*x+c*x^2+d*x^3)^n,x,1,(a+b*x+c*x^2+d*x^3)^(1+n)/(1+n)],
[(b+2*c*x+3*d*x^2)*(b*x+c*x^2+d*x^3)^n,x,1,(b*x+c*x^2+d*x^3)^(1+n)/(1+n)],
[x^n*(b+c*x+d*x^2)^n*(b+2*c*x+3*d*x^2),x,1,x^(1+n)*(b+c*x+d*x^2)^(1+n)/(1+n)],
[(b+3*d*x^2)*(a+b*x+d*x^3)^n,x,1,(a+b*x+d*x^3)^(1+n)/(1+n)],
[(b+3*d*x^2)*(b*x+d*x^3)^n,x,1,(b*x+d*x^3)^(1+n)/(1+n)],
[x^n*(b+d*x^2)^n*(b+3*d*x^2),x,1,x^(1+n)*(b+d*x^2)^(1+n)/(1+n)],
[(2*c*x+3*d*x^2)*(a+c*x^2+d*x^3)^n,x,1,(a+c*x^2+d*x^3)^(1+n)/(1+n)],
[(2*c*x+3*d*x^2)*(c*x^2+d*x^3)^n,x,1,(c*x^2+d*x^3)^(1+n)/(1+n)],
[x^n*(c*x+d*x^2)^n*(2*c*x+3*d*x^2),x,2,x^(1+n)*(c*x+d*x^2)^(1+n)/(1+n)],
[x^(2*n)*(c+d*x)^n*(2*c*x+3*d*x^2),x,1,x^(2*(1+n))*(c+d*x)^(1+n)/(1+n)],
[x*(2*c+3*d*x)*(a+c*x^2+d*x^3)^n,x,1,(a+c*x^2+d*x^3)^(1+n)/(1+n)],
[x*(2*c+3*d*x)*(c*x^2+d*x^3)^n,x,1,(c*x^2+d*x^3)^(1+n)/(1+n)],
[(b+2*c*x+3*d*x^2)*(a+b*x+c*x^2+d*x^3)^7,x,1,1/8*(a+b*x+c*x^2+d*x^3)^8],
[(b+2*c*x+3*d*x^2)*(b*x+c*x^2+d*x^3)^7,x,1,1/8*(b*x+c*x^2+d*x^3)^8],
[x^7*(b+c*x+d*x^2)^7*(b+2*c*x+3*d*x^2),x,1,1/8*x^8*(b+c*x+d*x^2)^8],
[(b+3*d*x^2)*(a+b*x+d*x^3)^7,x,1,1/8*(a+b*x+d*x^3)^8],
[(b+3*d*x^2)*(b*x+d*x^3)^7,x,1,1/8*(b*x+d*x^3)^8],
[x^7*(b+d*x^2)^7*(b+3*d*x^2),x,2,1/8*x^8*(b+d*x^2)^8],
[(2*c*x+3*d*x^2)*(a+c*x^2+d*x^3)^7,x,1,1/8*(a+c*x^2+d*x^3)^8],
[(2*c*x+3*d*x^2)*(c*x^2+d*x^3)^7,x,1,1/8*(c*x^2+d*x^3)^8],
[x^7*(c*x+d*x^2)^7*(2*c*x+3*d*x^2),x,2,1/8*x^16*(c+d*x)^8],
[x^14*(c+d*x)^7*(2*c*x+3*d*x^2),x,1,1/8*x^16*(c+d*x)^8],
[x*(2*c+3*d*x)*(a+c*x^2+d*x^3)^7,x,1,1/8*(a+c*x^2+d*x^3)^8],
[x*(2*c+3*d*x)*(c*x^2+d*x^3)^7,x,2,1/8*x^16*(c+d*x)^8],
[x^8*(2*c+3*d*x)*(c*x+d*x^2)^7,x,1,1/8*x^8*(c*x+d*x^2)^8],
[x^15*(c+d*x)^7*(2*c+3*d*x),x,1,1/8*x^16*(c+d*x)^8],
[(a+b*x)*(1+(a*x+1/2*b*x^2)^4),x,2,a*x+1/2*b*x^2+1/160*x^5*(2*a+b*x)^5],
[(a+b*x)*(1+(c+a*x+1/2*b*x^2)^4),x,2,a*x+1/2*b*x^2+1/5*(c+a*x+1/2*b*x^2)^5],
[(a+b*x)*(1+(a*x+1/2*b*x^2)^n),x,2,a*x+1/2*b*x^2+(a*x+1/2*b*x^2)^(1+n)/(1+n)],
[(a+b*x)*(1+(c+a*x+1/2*b*x^2)^n),x,2,a*x+1/2*b*x^2+(c+a*x+1/2*b*x^2)^(1+n)/(1+n)],
[(a+c*x^2)*(1+(a*x+1/3*c*x^3)^5),x,2,a*x+1/3*c*x^3+1/6*(a*x+1/3*c*x^3)^6],
[(a+c*x^2)*(1+(d+a*x+1/3*c*x^3)^5),x,2,a*x+1/3*c*x^3+1/6*(d+a*x+1/3*c*x^3)^6],
[(b*x+c*x^2)*(1+(1/2*b*x^2+1/3*c*x^3)^5),x,2,1/2*b*x^2+1/3*c*x^3+1/279936*x^12*(3*b+2*c*x)^6],
[(b*x+c*x^2)*(1+(d+1/2*b*x^2+1/3*c*x^3)^5),x,2,1/2*b*x^2+1/3*c*x^3+1/6*(d+1/2*b*x^2+1/3*c*x^3)^6],
[(a+b*x+c*x^2)*(1+(a*x+1/2*b*x^2+1/3*c*x^3)^5),x,2,a*x+1/2*b*x^2+1/3*c*x^3+1/6*(a*x+1/2*b*x^2+1/3*c*x^3)^6],
[(a+b*x+c*x^2)*(1+(d+a*x+1/2*b*x^2+1/3*c*x^3)^5),x,2,a*x+1/2*b*x^2+1/3*c*x^3+1/6*(d+a*x+1/2*b*x^2+1/3*c*x^3)^6],
[(a+c*x^2)*(1+(a*x+1/3*c*x^3)^n),x,2,a*x+1/3*c*x^3+(a*x+1/3*c*x^3)^(1+n)/(1+n)],
[(b*x+c*x^2)*(1+(1/2*b*x^2+1/3*c*x^3)^n),x,2,1/2*b*x^2+1/3*c*x^3+(1/2*b*x^2+1/3*c*x^3)^(1+n)/(1+n)],
[(a+b*x+c*x^2)*(1+(a*x+1/2*b*x^2+1/3*c*x^3)^n),x,2,a*x+1/2*b*x^2+1/3*c*x^3+(a*x+1/2*b*x^2+1/3*c*x^3)^(1+n)/(1+n)],
[(-4+4*x+x^2)*(5-12*x+6*x^2+x^3),x,1,1/6*(5-12*x+6*x^2+x^3)^2],
[(2*x+x^3)*(1+4*x^2+x^4),x,1,1/8*(1+4*x^2+x^4)^2],
[(1+2*x)*(x+x^2)^3*(-18+7*(x+x^2)^3)^2,x,-3,81*x^4*(1+x)^4-36*x^7*(1+x)^7+49/10*x^10*(1+x)^10],
[x^3*(1+x)^3*(1+2*x)*(-18+7*x^3*(1+x)^3)^2,x,-2,81*x^4*(1+x)^4-36*x^7*(1+x)^7+49/10*x^10*(1+x)^10],
[(2-x^2)/(1-6*x+x^3)^5,x,1,1/12/(1-6*x+x^3)^4],
[(2*x+x^2)/(4+3*x^2+x^3),x,1,1/3*log(4+3*x^2+x^3)],
[(1+x+x^3)/(4*x+2*x^2+x^4),x,1,1/4*log(4*x+2*x^2+x^4)],

# Integrands of the form P3[x]^p Q3[x]
[(b*c-a*d-2*a*e*x-b*e*x^2-3*a*f*x^2-2*b*f*x^3)/(c+d*x+e*x^2+f*x^3)^2,x,3,a/(c+d*x+e*x^2+f*x^3)+b*x/(c+d*x+e*x^2+f*x^3)],

# Integrands of the form P4[x]^p Q[x]

# Integrands of the form P4[x]^p Q3[x]
[(A+B*x+C*x^2+D*x^3)/(a+b*x+c*x^2+b*x^3+a*x^4),x,9,-1/4*log(2*a+2*a*x^2+x*(b-sqrt(8*a^2+b^2-4*a*c)))*(2*a*(A-C)+D*(b-sqrt(8*a^2+b^2-4*a*c)))/(a*sqrt(8*a^2+b^2-4*a*c))+1/4*log(2*a+2*a*x^2+x*(b+sqrt(8*a^2+b^2-4*a*c)))*(2*a*(A-C)+D*(b+sqrt(8*a^2+b^2-4*a*c)))/(a*sqrt(8*a^2+b^2-4*a*c))+arctan((b+4*a*x-sqrt(8*a^2+b^2-4*a*c))/(sqrt(2)*sqrt(4*a^2+2*a*c-b*(b-sqrt(8*a^2+b^2-4*a*c)))))*(4*a^2*B+b*D*(b-sqrt(8*a^2+b^2-4*a*c))-a*(b*C+2*c*D+A*(b-sqrt(8*a^2+b^2-4*a*c))-C*sqrt(8*a^2+b^2-4*a*c)))/(a*sqrt(2)*sqrt(8*a^2+b^2-4*a*c)*sqrt(4*a^2+2*a*c-b*(b-sqrt(8*a^2+b^2-4*a*c))))-arctan((b+4*a*x+sqrt(8*a^2+b^2-4*a*c))/(sqrt(2)*sqrt(4*a^2+2*a*c-b*(b+sqrt(8*a^2+b^2-4*a*c)))))*(4*a^2*B+b*D*(b+sqrt(8*a^2+b^2-4*a*c))-a*(b*C+2*c*D+C*sqrt(8*a^2+b^2-4*a*c)+A*(b+sqrt(8*a^2+b^2-4*a*c))))/(a*sqrt(2)*sqrt(8*a^2+b^2-4*a*c)*sqrt(4*a^2+2*a*c-b*(b+sqrt(8*a^2+b^2-4*a*c))))],
[(2+x-4*x^2+2*x^3)/(1-x+x^2-x^3+x^4),x,3,-2*log(2+2*x^2-x*(1-sqrt(5)))/(1-sqrt(5))-2*log(2+2*x^2-x*(1+sqrt(5)))/(1+sqrt(5))],

#  Integrands of the form (a + b x + c x^2 + d x^3 + e x^4)^p (A+B x+C x^2+D x^3) when d^3 - 4 c d e + 8 b e^2=0 
[(3*x+3*x^2+x^3)/(1+4*x+6*x^2+4*x^3+x^4),x,4,1/3/(1+x)^3+log(1+x)],
[(-1+3*x-3*x^2+x^3)/(1+4*x+6*x^2+4*x^3+x^4),x,3,8/3/(1+x)^3+(-6)/(1+x)^2+6/(1+x)+log(1+x)],

# Integrands of the form P4[x]^p Q8[x]
[(9-40*x-18*x^2+174*x^4+24*x^5+26*x^6-39*x^8)/(3+2*x^2+x^4)^3,x,6,2*(1-2*x^2)/(3+2*x^2+x^4)^2-2*x*(18+13*x^2)/(3+2*x^2+x^4)^2+13*x/(3+2*x^2+x^4)],

# Integrands of the form P5[x]^p Q[x]

# Integrands of the form P5[x]^p Q5[x]
[(-1+4*x^5)/(1+x+x^5)^2,x,1,-x/(1+x+x^5)],

# Integrands of the form P6[x]^p Q[x]

# Integrands of the form P3[x^2]^p Q1[x^2]
[(1+x^2)/(1-7*x^2+7*x^4-x^6)^2,x,15,1/16*x/(1-x^2)+1/32*x*(29-5*x^2)/(1-6*x^2+x^4)+1/4*arctanh(x)+1/64*(arctanh(x*(-1+sqrt(2)))*(3-2*sqrt(2))-arctanh(x*(1+sqrt(2)))*(3+2*sqrt(2))),1/32/(1-x)+(-1/32)/(1+x)+1/64*(12+5*x)/(1-2*x-x^2)+1/64*(-12+5*x)/(1+2*x-x^2)+1/4*arctanh(x)-3/256*log(1-x+sqrt(2))*(2-3*sqrt(2))+3/256*log(1+x+sqrt(2))*(2-3*sqrt(2))-5/64*arctanh((1-x)/sqrt(2))/sqrt(2)+5/64*arctanh((1+x)/sqrt(2))/sqrt(2)-3/256*log(1-x-sqrt(2))*(2+3*sqrt(2))+3/256*log(1+x-sqrt(2))*(2+3*sqrt(2))],

# Integrands of the form x^m P[x]^p Q[x]

# Integrands of the form x^m P3[x]^p Q[x]

# Integrands of the form x^m P3[x]^p Q3[x]
[x^m*(a+b*x+c*x^2+d*x^3)^p*(a*(1+m)+x*(b*(2+m+p)+x*(c*(3+m+2*p)+d*(4+m+3*p)*x))),x,1,x^(1+m)*(a+b*x+c*x^2+d*x^3)^(1+p)],
[x^2*(a+b*x+c*x^2+d*x^3)^p*(3*a+b*(4+p)*x+c*(5+2*p)*x^2+d*(6+3*p)*x^3),x,1,x^3*(a+b*x+c*x^2+d*x^3)^(1+p)],
[x*(a+b*x+c*x^2+d*x^3)^p*(2*a+b*(3+p)*x+c*(4+2*p)*x^2+d*(5+3*p)*x^3),x,1,x^2*(a+b*x+c*x^2+d*x^3)^(1+p)],
[(a+b*x+c*x^2+d*x^3)^p*(a+b*(2+p)*x+c*(3+2*p)*x^2+d*(4+3*p)*x^3),x,1,x*(a+b*x+c*x^2+d*x^3)^(1+p)],
[(a+b*x+c*x^2+d*x^3)^p*(b*(1+p)*x+c*(2+2*p)*x^2+d*(3+3*p)*x^3)/x,x,2,(a+b*x+c*x^2+d*x^3)^(1+p)],
[(a+b*x+c*x^2+d*x^3)^p*(-a+b*p*x+c*(1+2*p)*x^2+d*(2+3*p)*x^3)/x^2,x,1,(a+b*x+c*x^2+d*x^3)^(1+p)/x],
[(a+b*x+c*x^2+d*x^3)^p*(-2*a+b*(-1+p)*x+2*c*p*x^2+d*(1+3*p)*x^3)/x^3,x,1,(a+b*x+c*x^2+d*x^3)^(1+p)/x^2],
[(a+b*x+c*x^2+d*x^3)^p*(-3*a+b*(-2+p)*x+c*(-1+2*p)*x^2+3*d*p*x^3)/x^4,x,1,(a+b*x+c*x^2+d*x^3)^(1+p)/x^3],

# Integrands of the form x^m P4[x]^p Q[x]

# Integrands of the form x^m P4[x]^p Q3[x]
[x^4*(5+x+3*x^2+2*x^3)/(2+x+3*x^2+x^3+2*x^4),x,10,5/4*x-3/4*x^2+1/3*x^3+1/4*x^4+1/3*log(1+x+x^2)-13/48*log(2-x+2*x^2)+1/24*arctan((1-4*x)/sqrt(15))*sqrt(5/3)-10/3*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[x^3*(5+x+3*x^2+2*x^3)/(2+x+3*x^2+x^3+2*x^4),x,10,-3/2*x+1/2*x^2+1/3*x^3+2/3*log(1+x+x^2)-1/24*log(2-x+2*x^2)+5/12*arctan((1-4*x)/sqrt(15))*sqrt(5/3)+8/3*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[x^2*(5+x+3*x^2+2*x^3)/(2+x+3*x^2+x^3+2*x^4),x,10,x+1/2*x^2-log(1+x+x^2)+1/4*log(2-x+2*x^2)+1/6*arctan((1-4*x)/sqrt(15))*sqrt(5/3)+2/3*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[x*(5+x+3*x^2+2*x^3)/(2+x+3*x^2+x^3+2*x^4),x,10,x+1/3*log(1+x+x^2)+1/6*log(2-x+2*x^2)-1/3*arctan((1-4*x)/sqrt(15))*sqrt(5/3)-10/3*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[(5+x+3*x^2+2*x^3)/(2+x+3*x^2+x^3+2*x^4),x,10,2/3*log(1+x+x^2)-1/6*log(2-x+2*x^2)-1/3*arctan((1-4*x)/sqrt(15))*sqrt(5/3)+8/3*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[(5+x+3*x^2+2*x^3)/(x*(2+x+3*x^2+x^3+2*x^4)),x,13,5/2*log(x)-log(1+x+x^2)-1/4*log(2-x+2*x^2)+1/6*arctan((1-4*x)/sqrt(15))*sqrt(5/3)+2/3*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[(5+x+3*x^2+2*x^3)/(x^2*(2+x+3*x^2+x^3+2*x^4)),x,13,(-5/2)/x-3/4*log(x)+1/3*log(1+x+x^2)+1/24*log(2-x+2*x^2)+5/12*arctan((1-4*x)/sqrt(15))*sqrt(5/3)-10/3*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[(5+x+3*x^2+2*x^3)/(x^3*(2+x+3*x^2+x^3+2*x^4)),x,13,(-5/4)/x^2+3/4/x-15/8*log(x)+2/3*log(1+x+x^2)+13/48*log(2-x+2*x^2)+1/24*arctan((1-4*x)/sqrt(15))*sqrt(5/3)+8/3*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[x^3*(5+x+3*x^2+2*x^3)/(2+x+5*x^2+x^3+2*x^4),x,13,1/28*x^2*(7-5*I*sqrt(7))+1/42*x^3*(7-5*I*sqrt(7))+1/28*x^2*(7+5*I*sqrt(7))+1/42*x^3*(7+5*I*sqrt(7))-1/28*x*(35-9*I*sqrt(7))-1/28*x*(35+9*I*sqrt(7))+3/112*log(4+4*x^2+x*(1-I*sqrt(7)))*(7-11*I*sqrt(7))+3/112*log(4+4*x^2+x*(1+I*sqrt(7)))*(7+11*I*sqrt(7))-11/4*arctan((1+8*x+I*sqrt(7))/sqrt(2*(35-I*sqrt(7))))*(9*I-5*sqrt(7))/sqrt(14*(35-I*sqrt(7)))+11/4*arctan((1+8*x-I*sqrt(7))/sqrt(2*(35+I*sqrt(7))))*(9*I+5*sqrt(7))/sqrt(14*(35+I*sqrt(7)))],
[x^2*(5+x+3*x^2+2*x^3)/(2+x+5*x^2+x^3+2*x^4),x,13,1/14*x*(7-5*I*sqrt(7))+1/28*x^2*(7-5*I*sqrt(7))+1/14*x*(7+5*I*sqrt(7))+1/28*x^2*(7+5*I*sqrt(7))-1/56*log(4+4*x^2+x*(1+I*sqrt(7)))*(35-9*I*sqrt(7))-1/56*log(4+4*x^2+x*(1-I*sqrt(7)))*(35+9*I*sqrt(7))+1/2*arctan((1+8*x+I*sqrt(7))/sqrt(2*(35-I*sqrt(7))))*(53*I-sqrt(7))/sqrt(14*(35-I*sqrt(7)))-1/2*arctan((1+8*x-I*sqrt(7))/sqrt(2*(35+I*sqrt(7))))*(53*I+sqrt(7))/sqrt(14*(35+I*sqrt(7)))],
[x*(5+x+3*x^2+2*x^3)/(2+x+5*x^2+x^3+2*x^4),x,11,1/14*x*(7-5*I*sqrt(7))+1/28*log(4+4*x^2+x*(1+I*sqrt(7)))*(7-5*I*sqrt(7))+1/14*x*(7+5*I*sqrt(7))+1/28*log(4+4*x^2+x*(1-I*sqrt(7)))*(7+5*I*sqrt(7))+arctan((1+8*x+I*sqrt(7))/sqrt(2*(35-I*sqrt(7))))*(19*I-7*sqrt(7))/sqrt(14*(35-I*sqrt(7)))-arctan((1+8*x-I*sqrt(7))/sqrt(2*(35+I*sqrt(7))))*(19*I+7*sqrt(7))/sqrt(14*(35+I*sqrt(7)))],
[(5+x+3*x^2+2*x^3)/(2+x+5*x^2+x^3+2*x^4),x,9,1/28*log(4+4*x^2+x*(1+I*sqrt(7)))*(7-5*I*sqrt(7))+1/28*log(4+4*x^2+x*(1-I*sqrt(7)))*(7+5*I*sqrt(7))-arctan((1+8*x+I*sqrt(7))/sqrt(2*(35-I*sqrt(7))))*(19*I-7*sqrt(7))/sqrt(14*(35-I*sqrt(7)))+arctan((1+8*x-I*sqrt(7))/sqrt(2*(35+I*sqrt(7))))*(19*I+7*sqrt(7))/sqrt(14*(35+I*sqrt(7)))],
[(5+x+3*x^2+2*x^3)/(x*(2+x+5*x^2+x^3+2*x^4)),x,13,1/28*log(x)*(35-9*I*sqrt(7))-1/56*log(4*I+4*I*x^2+x*(I-sqrt(7)))*(35-9*I*sqrt(7))+1/28*log(x)*(35+9*I*sqrt(7))-1/56*log(4*I+4*I*x^2+x*(I+sqrt(7)))*(35+9*I*sqrt(7))-1/2*arctanh((I+8*I*x-sqrt(7))/sqrt(2*(35-I*sqrt(7))))*(53+I*sqrt(7))/sqrt(14*(35-I*sqrt(7)))+1/2*arctanh((I+8*I*x+sqrt(7))/sqrt(2*(35+I*sqrt(7))))*(53-I*sqrt(7))/sqrt(14*(35+I*sqrt(7)))],
[(5+x+3*x^2+2*x^3)/(x^2*(2+x+5*x^2+x^3+2*x^4)),x,13,1/28*(-35+9*I*sqrt(7))/x+1/28*(-35-9*I*sqrt(7))/x-3/56*log(x)*(7-11*I*sqrt(7))+3/112*log(4*I+4*I*x^2+x*(I+sqrt(7)))*(7-11*I*sqrt(7))-3/56*log(x)*(7+11*I*sqrt(7))+3/112*log(4*I+4*I*x^2+x*(I-sqrt(7)))*(7+11*I*sqrt(7))+11/4*arctanh((I+8*I*x-sqrt(7))/sqrt(2*(35-I*sqrt(7))))*(9+5*I*sqrt(7))/sqrt(14*(35-I*sqrt(7)))-11/4*arctanh((I+8*I*x+sqrt(7))/sqrt(2*(35+I*sqrt(7))))*(9-5*I*sqrt(7))/sqrt(14*(35+I*sqrt(7)))],
[(5+x+3*x^2+2*x^3)/(x^3*(2+x+5*x^2+x^3+2*x^4)),x,13,1/56*(-35+9*I*sqrt(7))/x^2-1/16*log(x)*(35-9*I*sqrt(7))+1/32*log(4*I+4*I*x^2+x*(I-sqrt(7)))*(35-9*I*sqrt(7))+1/56*(-35-9*I*sqrt(7))/x^2-1/16*log(x)*(35+9*I*sqrt(7))+1/32*log(4*I+4*I*x^2+x*(I+sqrt(7)))*(35+9*I*sqrt(7))+3/56*(7-11*I*sqrt(7))/x+3/56*(7+11*I*sqrt(7))/x+1/8*arctanh((I+8*I*x-sqrt(7))/sqrt(2*(35-I*sqrt(7))))*(355-73*I*sqrt(7))/sqrt(14*(35-I*sqrt(7)))-1/8*arctanh((I+8*I*x+sqrt(7))/sqrt(2*(35+I*sqrt(7))))*(355+73*I*sqrt(7))/sqrt(14*(35+I*sqrt(7)))],

# Integrands of the form x^m P6[x]^p Q2[x]

# Integrands of the form x^m P3[x^2]^p Q1[x^2]
[x^2*(3*a+b*x^2)/(a^2+2*a*b*x^2+b^2*x^4+c^2*x^6),x,2,arctan(c*x^3/(a+b*x^2))/c],

# Integrands requiring algebraic expansion

# Problems from Calculus textbooks and competitions

# Anton Calculus, 4th Edition
[(1-3*x^4)/((-2+x)*(1+x^2)^2),x,6,1/5*(-1+2*x)/(1+x^2)-46/25*arctan(x)-47/25*log(2-x)-14/25*log(1+x^2)],
[(-9-9*x+2*x^2)/(-9*x+x^3),x,3,-log(3-x)+log(x)+2*log(3+x)],
[(1+2*x^2+x^5)/(-x+x^3),x,3,x+1/3*x^3+2*log(1-x)-log(x)+log(1+x)],
[(3+2*x^2)/((-1+x)^2*x),x,2,5/(1-x)-log(1-x)+3*log(x)],
[(-1+2*x^2)/((-1+4*x)*(1+x^2)),x,5,3/17*arctan(x)-7/34*log(1-4*x)+6/17*log(1+x^2)],
[(-3+2*x-3*x^2+x^3)/(1+x^2),x,3,-3*x+1/2*x^2+1/2*log(1+x^2)],
[(x+10*x^2+6*x^3+x^4)/(10+6*x+x^2),x,6,1/3*x^3-3*arctan(3+x)+1/2*log(10+6*x+x^2)],
[1/(-18+27*x-7*x^2-3*x^3+x^4),x,2,1/8*log(1-x)-1/5*log(2-x)+1/12*log(3-x)-1/120*log(3+x)],
[(1+x^3)/(-2+x),x,2,4*x+x^2+1/3*x^3+9*log(2-x)],

# Ayres Calculus, 1964 edition
[(3*x-4*x^2+3*x^3)/(1+x^2),x,4,-4*x+3/2*x^2+4*arctan(x)],
[(5+3*x)/(1-x-x^2+x^3),x,3,4/(1-x)+arctanh(x)],
[(-1-x-x^3+x^4)/(-x^2+x^3),x,3,(-1)/x+1/2*x^2-2*log(1-x)+2*log(x)],
[(2+x+x^2+x^3)/(2+3*x^2+x^4),x,6,arctan(x)+1/2*log(2+x^2)],
[(-4+8*x-4*x^2+4*x^3-x^4+x^5)/(2+x^2)^3,x,5,(-1)/(2+x^2)^2+1/2*log(2+x^2)-arctan(x/sqrt(2))/sqrt(2)],
[(-1-3*x+x^2)/(-2*x+x^2+x^3),x,3,-log(1-x)+1/2*log(x)+3/2*log(2+x)],
[(3-x+3*x^2-2*x^3+x^4)/(3*x-2*x^2+x^3),x,4,1/2*x^2+log(x)-1/2*log(3-2*x+x^2)],
[(-1+x+x^3)/(1+x^2)^2,x,4,-1/2*x/(1+x^2)-1/2*arctan(x)+1/2*log(1+x^2)],
[(1+2*x-x^2+8*x^3+x^4)/((x+x^2)*(1+x^3)),x,7,(-3)/(1+x)+log(x)-2*log(1+x)+log(1-x+x^2)-2*arctan((1-2*x)/sqrt(3))/sqrt(3)],
[(15-5*x+x^2+x^3)/((5+x^2)*(3+2*x+x^2)),x,7,1/2*log(3+2*x+x^2)+5*arctan((1+x)/sqrt(2))/sqrt(2)-arctan(x/sqrt(5))*sqrt(5)],
[(-3+25*x+23*x^2+32*x^3+15*x^4+7*x^5+x^6)/((1+x^2)^2*(2+x+x^2)^2),x,6,(-3)/(1+x^2)+1/(2+x+x^2)+log(1+x^2)-log(2+x+x^2)],

# Edwards and Penney Calculus
[1/((1+x^2)*(4+x^2)),x,3,-1/6*arctan(1/2*x)+1/3*arctan(x)],
[(a+b*x^3)/(1+x^2),x,5,1/2*b*x^2+a*arctan(x)-1/2*b*log(1+x^2)],
[(x+x^2)/((4+x)*(-4+x^2)),x,4,-1/2*arctanh(1/2*x)+log(4+x)],
[(4+x^2)/((1+x^2)*(2+x^2)),x,3,3*arctan(x)-arctan(x/sqrt(2))*sqrt(2)],
[(5-4*x+3*x^2+x^4)/((-1+x)^2*(1+x^2)),x,5,5/2/(1-x)+x+2*arctan(x)+1/2*log(1-x)+3/4*log(1+x^2)],
[(1+x^4)/(2+x^2),x,3,-2*x+1/3*x^3+5*arctan(x/sqrt(2))/sqrt(2)],
[(2+2*x+x^4)/(x^4+x^5),x,3,(-2/3)/x^3+log(1+x)],
[(-1-5*x+2*x^2)/(2-x-2*x^2+x^3),x,2,2*log(1-x)-log(2-x)+log(1+x)],
[(2+x+x^3)/(1+2*x^2+x^4),x,5,x/(1+x^2)+arctan(x)+1/2*log(1+x^2)],
[(1+2*x+x^2+x^3)/(1+2*x^2+x^4),x,5,(-1/2)/(1+x^2)+arctan(x)+1/2*log(1+x^2)],
[(3+4*x)/((1+x^2)*(2+x^2)),x,8,3*arctan(x)+2*log(1+x^2)-2*log(2+x^2)-3*arctan(x/sqrt(2))/sqrt(2)],
[(2+x)/((1+x^2)*(4+x^2)),x,8,-1/3*arctan(1/2*x)+2/3*arctan(x)+1/6*log(1+x^2)-1/6*log(4+x^2)],

# Grossman Calculus
[(2-x+x^3)/(-7-6*x+x^2),x,5,6*x+1/2*x^2+169/4*log(7-x)-1/4*log(1+x)],
[(-1+x^5)/(-1+x^2),x,4,1/2*x^2+1/4*x^4+log(1+x)],
[(5+2*x-x^2+x^3)/(1+x+x^2),x,6,-2*x+1/2*x^2+3/2*log(1+x+x^2)+11*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[(-3+x-2*x^3+x^4)/(10-8*x+2*x^2),x,6,3/2*x+1/2*x^2+1/6*x^3+6*arctan(2-x)+3/4*log(5-4*x+x^2)],
[(1+2*x+3*x^2+x^3)/((-3+x)*(-2+x)*(-1+x)),x,2,x+7/2*log(1-x)-25*log(2-x)+61/2*log(3-x)],
[(2-7*x+x^2-x^3+x^4)/(-24-14*x+x^2+x^3),x,2,-2*x+1/2*x^2+13/3*log(4-x)-22/3*log(2+x)+20*log(3+x)],
[(2+x^2)/((-1+x)^2*x*(1+x)),x,2,3/2/(1-x)-5/4*log(1-x)+2*log(x)-3/4*log(1+x)],
[(3+x^2+x^3)/(2+x^2)^2,x,4,1/4*(4+x)/(2+x^2)+1/2*log(2+x^2)+5/4*arctan(x/sqrt(2))/sqrt(2)],
[(-35+70*x-4*x^2+2*x^3)/((26-10*x+x^2)*(17-2*x+x^2)),x,10,-15033/1025*arctan(5-x)-4607/4100*arctan(1/4*(-1+x))+1003/1025*log(26-10*x+x^2)+22/1025*log(17-2*x+x^2)],
[(2+x^2)/((-5+x)*(-3+x)*(4+x)),x,2,-11/14*log(3-x)+3/2*log(5-x)+2/7*log(4+x)],
[x^4/((-1+x)*(2+x^2)),x,5,x+1/2*x^2+1/3*log(1-x)-2/3*log(2+x^2)-2/3*arctan(x/sqrt(2))*sqrt(2)],

# Spivak Calculus
[(-1+7*x+2*x^2)/(-1-x+x^2+x^3),x,2,(-3)/(1+x)+2*log(1-x)],
[(1+2*x)/(-1+3*x-3*x^2+x^3),x,2,(-3/2)/(1-x)^2+2/(1-x)],
[(5-5*x+7*x^2+x^3)/((-1+x)^2*(1+x)^3),x,2,1/(1-x)+(-2)/(1+x)^2],
[(1+3*x+3*x^2)/(1+2*x+2*x^2+x^3),x,6,log(1+x)+log(1+x+x^2)-2*arctan((1+2*x)/sqrt(3))/sqrt(3)],

# Stewart Calculus
[(-1+2*x+x^2)/(-2*x+3*x^2+2*x^3),x,3,1/10*log(1-2*x)+1/2*log(x)-1/10*log(2+x)],
[(1+4*x-2*x^2+x^4)/(1-x-x^2+x^3),x,2,2/(1-x)+x+1/2*x^2+log(1-x)-log(1+x)],
[(4-x+2*x^2)/(4*x+x^3),x,6,-1/2*arctan(1/2*x)+log(x)+1/2*log(4+x^2)],
[(1+x^2+x^3)/((-1+x)*x*(1+x^2)^3*(1+x+x^2)),x,14,1/8*(1+x)/(1+x^2)^2-3/8*(1-x)/(1+x^2)+3/16*x/(1+x^2)+7/16*arctan(x)+1/8*log(1-x)-log(x)+15/16*log(1+x^2)-1/2*log(1+x+x^2)-arctan((1+2*x)/sqrt(3))/sqrt(3)],
[(1-3*x+2*x^2-x^3)/(1+x^2)^2,x,4,1/2*(2-x)/(1+x^2)+3/2*arctan(x)-1/2*log(1+x^2)],
[(1-3*x+2*x^2-x^3)/(x*(1+x^2)^2),x,6,1/2*(-1-2*x)/(1+x^2)-2*arctan(x)+log(x)-1/2*log(1+x^2)],
[(1-x-x^2+x^3+x^4)/(-x+x^3),x,4,x+1/2*x^2-log(x)+1/2*log(1-x^2)],
[(2-4*x^2+x^3)/((1+x^2)*(2+x^2)),x,8,6*arctan(x)-1/2*log(1+x^2)+log(2+x^2)-5*arctan(x/sqrt(2))*sqrt(2)],
[(1+x^2+x^4)/((1+x^2)*(4+x^2)^2),x,6,-13/24*x/(4+x^2)+25/144*arctan(1/2*x)+1/9*arctan(x)],
[(1+x^2+x^3)/(2*x^2+x^3+x^4),x,7,(-1/2)/x-1/4*log(x)+5/8*log(2+x+x^2)+1/4*arctan((1+2*x)/sqrt(7))/sqrt(7)],
[(1-12*x+x^2+x^3)/(-12+x+x^2),x,5,1/2*x^2-2/7*arctanh(1/7*(1+2*x)),1/2*x^2+1/7*log(3-x)-1/7*log(4+x)],
[(-3+5*x+6*x^2)/(-3*x+2*x^2+x^3),x,3,2*log(1-x)+log(x)+3*log(3+x)],
[(-2+3*x+5*x^2)/(2*x^2+x^3),x,3,1/x+2*log(x)+3*log(2+x)],
[(18-2*x-4*x^2)/(-6+x+4*x^2+x^3),x,2,log(1-x)-2*log(2+x)-3*log(3+x)],
[(1+x-2*x^2+x^3)/(4+5*x^2+x^4),x,7,-3/2*arctan(1/2*x)+arctan(x)+1/2*log(4+x^2)],
[(-32+5*x-27*x^2+4*x^3)/(-70-299*x-286*x^2+50*x^3-13*x^4+30*x^5),x,6,-3146/80155*log(7-3*x)-334/323*log(1+2*x)+4822/4879*log(2+5*x)+11049/260015*log(5+x+x^2)+3988/13685*arctan((1+2*x)/sqrt(19))/sqrt(19)],
[(8-13*x^2-7*x^3+12*x^5)/(4-20*x+41*x^2-80*x^3+116*x^4-80*x^5+100*x^6),x,7,5828/9075/(2-5*x)+1/1452*(-313-502*x)/(1+2*x^2)-59096/99825*log(2-5*x)+2843/7986*log(1+2*x^2)+503/7986*arctan(x*sqrt(2))/sqrt(2),5828/9075/(2-5*x)+1/1452*(-313-502*x)/(1+2*x^2)-59096/99825*log(2-5*x)+2843/7986*log(1+2*x^2)-251/726*arctan(x*sqrt(2))/sqrt(2)+272/1331*arctan(x*sqrt(2))*sqrt(2)],

# Thomas Calculus, 8th Edition
[(9+x^4)/(x^2*(9+x^2)),x,3,(-1)/x+x-10/3*arctan(1/3*x)],
[(2*x+x^4)/(1+x^2),x,6,-x+1/3*x^3+arctan(x)+log(1+x^2)],
[(-x+x^3)/((-1+x)^2*(1+x^2)),x,5,arctan(x)+log(1-x)],
[(2+5*x+3*x^2+2*x^3)/(1+x+x^2),x,3,x+x^2+log(1+x+x^2)],
[(3-4*x-5*x^2+3*x^3)/(x^3*(-1+x+x^2)),x,5,3/2/x^2+(-1)/x+3*log(x)-1/10*log(1+2*x-sqrt(5))*(15-sqrt(5))-1/10*log(1+2*x+sqrt(5))*(15+sqrt(5))],
[(4+8*x+5*x^2+2*x^3)/(2+2*x+x^2)^2,x,5,(-1)/(2+2*x+x^2)-arctan(1+x)+log(2+2*x+x^2)],
[(-1+x)^4*x^4/(1+x^2),x,3,4*x-4/3*x^3+x^5-2/3*x^6+1/7*x^7-4*arctan(x)],
[(-20*x+4*x^2)/(9-10*x^2+x^4),x,-11,log(1-x)-1/2*log(3-x)+3/2*log(1+x)-2*log(3+x)],
[(-1+x+4*x^3)/((-1+x)*x^2*(1+x^2)),x,5,(-1)/x+arctan(x)+2*log(1-x)-log(1+x^2)],
[(1-3*x+2*x^2-4*x^3+x^4)/(1+x^2)^3,x,4,(-1/4)/(1+x^2)^2+2/(1+x^2)+arctan(x)],
[(1-3*x+2*x^2-4*x^3+x^4)/(1+3*x^2+3*x^4+x^6),x,5,(-1/4)/(1+x^2)^2+2/(1+x^2)+arctan(x)],

# North Texas University Integration Competition
[(1+x+2*x^2+2*x^3)/(x^2+x^3+x^4),x,4,(-1)/x+log(1+x+x^2)],

# Miscellaneous problems requiring algebraic expansion
[x^2*(c+d*x)^2/(a+b*x^3),x,10,2*c*d*x/b+1/2*d^2*x^2/b-1/3*a^(1/3)*d*(2*b^(1/3)*c-a^(1/3)*d)*log(a^(1/3)+b^(1/3)*x)/b^(5/3)+1/6*a^(1/3)*d*(2*b^(1/3)*c-a^(1/3)*d)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(5/3)+1/3*c^2*log(a+b*x^3)/b+a^(1/3)*d*(2*b^(1/3)*c+a^(1/3)*d)*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(5/3)*sqrt(3))],
[(-x+2*x^3+4*x^5)/(3+2*x^2+x^4)^2,x,6,1/8*(5-7*x^2)/(3+2*x^2+x^4)+9/8*arctan((1+x^2)/sqrt(2))/sqrt(2)],
[(x+x^5)/(1+2*x^2+2*x^4)^3,x,7,1/16*(3+4*x^2)/(1+2*x^2+2*x^4)^2+1/2*(1+2*x^2)/(1+2*x^2+2*x^4)+arctan(1+2*x^2)],
[(a+b*x+c*x^2)/(d+e*x^2+f*x^4),x,8,-b*arctanh((e+2*f*x^2)/sqrt(e^2-4*d*f))/sqrt(e^2-4*d*f)+arctan(x*sqrt(2)*sqrt(f)/sqrt(e-sqrt(e^2-4*d*f)))*(c+(-c*e+2*a*f)/sqrt(e^2-4*d*f))/(sqrt(2)*sqrt(f)*sqrt(e-sqrt(e^2-4*d*f)))+arctan(x*sqrt(2)*sqrt(f)/sqrt(e+sqrt(e^2-4*d*f)))*(c+(c*e-2*a*f)/sqrt(e^2-4*d*f))/(sqrt(2)*sqrt(f)*sqrt(e+sqrt(e^2-4*d*f)))],
[(d+e*x)^2/(a+b*x^2+c*x^4),x,8,-2*d*e*arctanh((b+2*c*x^2)/sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c)+arctan(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)))*(e^2+(2*c*d^2-b*e^2)/sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c)*sqrt(b-sqrt(b^2-4*a*c)))+arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))*(e^2+(-2*c*d^2+b*e^2)/sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c)*sqrt(b+sqrt(b^2-4*a*c)))],
[x^2/((a+b*x)*(c+d*x)),x,2,x/(b*d)+a^2*log(a+b*x)/(b^2*(b*c-a*d))-c^2*log(c+d*x)/(d^2*(b*c-a*d))],
[x^2/((c+d*x)*(a+b*x^2)),x,5,c^2*log(c+d*x)/(d*(b*c^2+a*d^2))+1/2*a*d*log(a+b*x^2)/(b*(b*c^2+a*d^2))-c*arctan(x*sqrt(b)/sqrt(a))*sqrt(a)/((b*c^2+a*d^2)*sqrt(b))],
[x^2/((c+d*x)*(a+b*x^3)),x,10,1/3*a^(1/3)*d*(b^(1/3)*c+a^(1/3)*d)*log(a^(1/3)+b^(1/3)*x)/(b^(2/3)*(b*c^3-a*d^3))-c^2*log(c+d*x)/(b*c^3-a*d^3)-1/6*a^(1/3)*d*(b^(1/3)*c+a^(1/3)*d)*log(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(2/3)*(b*c^3-a*d^3))+1/3*c^2*log(a+b*x^3)/(b*c^3-a*d^3)-a^(1/3)*d*arctan((a^(1/3)-2*b^(1/3)*x)/(a^(1/3)*sqrt(3)))/(b^(2/3)*(b^(2/3)*c^2+a^(1/3)*b^(1/3)*c*d+a^(2/3)*d^2)*sqrt(3))],
[x^2/((c+d*x)*(a+b*x^4)),x,16,c^2*d*log(c+d*x)/(b*c^4+a*d^4)-1/4*c^2*d*log(a+b*x^4)/(b*c^4+a*d^4)+1/2*d^3*arctan(x^2*sqrt(b)/sqrt(a))*sqrt(a)/((b*c^4+a*d^4)*sqrt(b))-1/2*c*arctan(1-b^(1/4)*x*sqrt(2)/a^(1/4))*(-d^2*sqrt(a)+c^2*sqrt(b))/(a^(1/4)*b^(1/4)*(b*c^4+a*d^4)*sqrt(2))+1/2*c*arctan(1+b^(1/4)*x*sqrt(2)/a^(1/4))*(-d^2*sqrt(a)+c^2*sqrt(b))/(a^(1/4)*b^(1/4)*(b*c^4+a*d^4)*sqrt(2))+1/4*c*log(-a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(d^2*sqrt(a)+c^2*sqrt(b))/(a^(1/4)*b^(1/4)*(b*c^4+a*d^4)*sqrt(2))-1/4*c*log(a^(1/4)*b^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(b))*(d^2*sqrt(a)+c^2*sqrt(b))/(a^(1/4)*b^(1/4)*(b*c^4+a*d^4)*sqrt(2))],
[x/((1-x)*(1+x)^2),x,3,1/2/(1+x)+1/2*arctanh(x)],
[x^2/((1-x^2)*(1+x^2)^2),x,2,-1/4*x/(1+x^2)+1/4*arctanh(x)],
[x^3/((1-x^3)*(1+x^3)^2),x,14,-1/6*x/(1+x^3)-1/12*log(1-x)-1/36*log(1+x)+1/72*log(1-x+x^2)+1/24*log(1+x+x^2)+1/12*arctan((1-2*x)/sqrt(3))/sqrt(3)+1/4*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[(9+x+3*x^2+x^3)/((1+x^2)*(3+x^2)),x,4,3*arctan(x)+1/2*log(3+x^2)],
[(3+x+x^2+x^3)/((1+x^2)*(3+x^2)),x,4,arctan(x)+1/2*log(3+x^2)],
[(-4+6*x-x^2+3*x^3)/((1+x^2)*(2+x^2)),x,6,-3*arctan(x)+3/2*log(1+x^2)+arctan(x/sqrt(2))*sqrt(2)],
[1/((4-4*x+x^2)*(5-4*x+x^2)),x,4,1/(2-x)+arctan(2-x)],
[(-3+x+x^2)/((-3+x)*x^2),x,2,(-1)/x+log(3-x)],
[(1+x+4*x^2)/(x+4*x^3),x,4,1/2*arctan(2*x)+log(x)],
[(1-x+3*x^2)/(-x^2+x^3),x,3,1/x+3*log(1-x)],
[(4+3*x+x^2)/(x+x^2),x,3,x+4*log(x)-2*log(1+x)],
[(4+x+3*x^2)/(x+x^3),x,6,arctan(x)+4*log(x)-1/2*log(1+x^2)],
[(7-4*x+8*x^2)/((1+4*x)*(1+x^2)),x,3,-arctan(x)+2*log(1+4*x)],
[x^2/((-1+x)*(1+2*x+x^2)),x,3,1/2/(1+x)+1/4*log(1-x)+3/4*log(1+x)],
[(-4+3*x+x^2)/((-1+2*x)^2*(3+2*x)),x,2,(-9/32)/(1-2*x)+41/128*log(1-2*x)-25/128*log(3+2*x)],
[(5-4*x+3*x^2)/((-1+x)*(1+x^2)),x,5,-3*arctan(x)+2*log(1-x)+1/2*log(1+x^2)],
[(-1-2*x+x^2)/((-1+x)^2*(1+x^2)),x,5,1/(-1+x)+arctan(x)+log(1-x)-1/2*log(1+x^2)],
[(5+x^3)/((10-6*x+x^2)*(1/2-x+x^2)),x,10,-261/221*arctan(1-2*x)-1026/221*arctan(3-x)+56/221*log(10-6*x+x^2)+109/442*log(1-2*x+2*x^2)],
[(4+3*x+x^2)/((-3+x)*(-2+x)*(-1+x)),x,2,4*log(1-x)-14*log(2-x)+11*log(3-x)],
[(1+16*x)/((5+x)^2*(-3+2*x)*(1+x+x^2)),x,6,(-79/273)/(5+x)+200/3211*log(3-2*x)+2731/24843*log(5+x)-481/5586*log(1+x+x^2)+451/2793*arctan((1+2*x)/sqrt(3))/sqrt(3)],
[(-1+x^3)/(1+x+x^2),x,2,-x+1/2*x^2],
[(-3+x^3)/(-7-6*x+x^2),x,5,6*x+1/2*x^2+85/2*log(7-x)+1/2*log(1+x)],
[(1+x^3)/(13+4*x+x^2)^2,x,5,1/18*(67+47*x)/(13+4*x+x^2)-61/54*arctan(1/3*(2+x))+1/2*log(13+4*x+x^2)],
[(-32+36*x-42*x^2+21*x^3-10*x^4+3*x^5)/(x*(1+x^2)*(4+x^2)^2),x,7,1/(4+x^2)+1/2*arctan(1/2*x)+2*arctan(x)-2*log(x)+log(4+x^2)],
[(-1+x^4+7*x^5+x^9)/(-7+6*x^4+x^8),x,17,1/2*x^2-1/2*arctanh(x^2)-1/2*arctan(1-x*sqrt(2)/7^(1/4))/(7^(3/4)*sqrt(2))+1/2*arctan(1+x*sqrt(2)/7^(1/4))/(7^(3/4)*sqrt(2))-1/4*log(x^2-7^(1/4)*x*sqrt(2)+sqrt(7))/(7^(3/4)*sqrt(2))+1/4*log(x^2+7^(1/4)*x*sqrt(2)+sqrt(7))/(7^(3/4)*sqrt(2))],
[(1+x^3+x^6)/(x+x^5),x,18,1/2*x^2-1/2*arctan(x^2)+log(x)-1/4*log(1+x^4)-1/2*arctan(1-x*sqrt(2))/sqrt(2)+1/2*arctan(1+x*sqrt(2))/sqrt(2)+1/4*log(1+x^2-x*sqrt(2))/sqrt(2)-1/4*log(1+x^2+x*sqrt(2))/sqrt(2)],
#  Note: This test problem formerly caused stack overflow because the degree of the polynomial

# 	was not properly reduced by the Ostrogradskiy-Hermite method code. 

#  {(a + 2*b*x - a*x^2)^4/(-1 + x^2)^5, x, 14, -((4*a*b*(3*a^2 - 2*b^2))/(3*(1 - x^2)^4)) + (11*a^4*x)/(5*(1 - x^2)^4) - (48*a^2*b^2*x)/(5*(1 - x^2)^4) + (6*b^4*x)/(5*(1 - x^2)^4) + (4*a*b*(9*a^2 - 8*b^2)*x^2)/(3*(1 - x^2)^4) - ((73*a^4 - 264*a^2*b^2 + 48*b^4)*x^3)/(15*(1 - x^2)^4) - (4*a*b*(3*a^2 - 2*b^2)*x^4)/(1 - x^2)^4 + (a^2*(11*a^2 - 24*b^2)*x^5)/(3*(1 - x^2)^4) + (4*a^3*b*x^6)/(1 - x^2)^4 - (a^4*x^7)/(1 - x^2)^4 - ((8*a^4 - 24*a^2*b^2 + 3*b^4)*x)/(15*(1 - x^2)^3) - ((8*a^4 - 24*a^2*b^2 + 3*b^4)*x)/(12*(1 - x^2)^2) - ((8*a^4 - 24*a^2*b^2 + 3*b^4)*x)/(8*(1 - x^2)) - (1/8)*(8*a^4 - 24*a^2*b^2 + 3*b^4)*ArcTanh[x]} 
[(1+x^2)/(-x+x^2),x,3,x+2*log(1-x)-log(x)],
[(1+x^3)/(-x+x^3),x,3,x+log(1-x)-log(x)],
[(1+x^3)/(-x^2+x^3),x,3,1/x+x+2*log(1-x)-log(x)],
[(-1+x^5)/(-x+x^3),x,3,x+1/3*x^3+log(x)-log(1+x)],
[(1+x^4)/(x^3+x^5),x,4,(-1/2)/x^2-log(x)+log(1+x^2)],

#  Integrands of the form (a+b*x^m)/(c*x^n+d*x^p+e*x^q) where m, n, p and q are integers 
[(1+x^2)/(x+2*x^2+x^3),x,4,2/(1+x)+log(x)],
[(1+x^5)/(-10*x-3*x^2+x^3),x,3,19*x+3/2*x^2+1/3*x^3+3126/35*log(5-x)-1/10*log(x)-31/14*log(2+x)],
[(15-5*x+x^2+x^3)/((5+x^2)*(3+2*x+x^2)),x,7,1/2*log(3+2*x+x^2)+5*arctan((1+x)/sqrt(2))/sqrt(2)-arctan(x/sqrt(5))*sqrt(5)],
[1/((1+x^2)*(3+10*x/(1+x^2))),x,4,-1/8*log(3+x)+1/8*log(1+3*x)],

#  Integrands of the form x^m/(a*x^n+b*x^p+c*x^q) where m, n, p and q are integers 
#  In some of the following examples gcd cancellation should occur without also partial fraction

# 	expansion, since that will result in unnecessary expansion. 
[x^3/(13+2/x+15*x),x,6,139/3375*x-13/450*x^2+1/45*x^3-16/567*log(2+3*x)+1/4375*log(1+5*x)],
[x^2/(13+2/x+15*x),x,6,-13/225*x+1/30*x^2+8/189*log(2+3*x)-1/875*log(1+5*x)],
[x/(13+2/x+15*x),x,5,1/15*x-4/63*log(2+3*x)+1/175*log(1+5*x)],
[1/(13+2/x+15*x),x,4,2/21*log(2+3*x)-1/35*log(1+5*x)],
[1/(x*(13+2/x+15*x)),x,4,-1/7*log(2+3*x)+1/7*log(1+5*x)],
[1/(x^2*(13+2/x+15*x)),x,6,1/2*log(x)+3/14*log(2+3*x)-5/7*log(1+5*x)],
[1/(x^3*(13+2/x+15*x)),x,4,(-1/2)/x-13/4*log(x)-9/28*log(2+3*x)+25/7*log(1+5*x)],
[1/(x^4*(13+2/x+15*x)),x,4,(-1/4)/x^2+13/4/x+139/8*log(x)+27/56*log(2+3*x)-125/7*log(1+5*x)],
[1/(x^5*(13+2/x+15*x)),x,4,(-1/6)/x^3+13/8/x^2+(-139/8)/x-1417/16*log(x)-81/112*log(2+3*x)+625/7*log(1+5*x)],
[x^2/(2-(1+x^2)^4),x,8,1/4*I*arctan(x/sqrt(1-I*2^(1/4)))*sqrt(1-I*2^(1/4))/2^(3/4)-1/4*I*arctan(x/sqrt(1+I*2^(1/4)))*sqrt(1+I*2^(1/4))/2^(3/4)+1/4*arctanh(x/sqrt(-1+2^(1/4)))*sqrt(-1+2^(1/4))/2^(3/4)-1/4*arctan(x/sqrt(1+2^(1/4)))*sqrt(1+2^(1/4))/2^(3/4)],
[x^2/(2-(1-x^2)^4),x,8,-1/4*I*arctanh(x/sqrt(1-I*2^(1/4)))*sqrt(1-I*2^(1/4))/2^(3/4)+1/4*I*arctanh(x/sqrt(1+I*2^(1/4)))*sqrt(1+I*2^(1/4))/2^(3/4)-1/4*arctan(x/sqrt(-1+2^(1/4)))*sqrt(-1+2^(1/4))/2^(3/4)+1/4*arctanh(x/sqrt(1+2^(1/4)))*sqrt(1+2^(1/4))/2^(3/4)],
[x^2/(2+(1+x^2)^4),x,8,1/4*(-1)^(1/4)*arctan(x/sqrt(1-(-2)^(1/4)))*sqrt(1-(-2)^(1/4))/2^(3/4)-1/4*(-1/2)^(3/4)*arctan(x/sqrt(1+I*(-2)^(1/4)))*sqrt(1+I*(-2)^(1/4))-1/4*(-1)^(1/4)*arctan(x/sqrt(1+(-2)^(1/4)))*sqrt(1+(-2)^(1/4))/2^(3/4)+1/8*I*arctan(x*sqrt((1+I)/(1+I+2^(3/4))))*((-2)^(1/4)+sqrt(2))*sqrt((1+I)/(1+I+2^(3/4)))],
[x^2/(2+(1-x^2)^4),x,8,-1/4*(-1)^(1/4)*arctanh(x/sqrt(1-(-2)^(1/4)))*sqrt(1-(-2)^(1/4))/2^(3/4)+1/4*(-1/2)^(3/4)*arctanh(x/sqrt(1+I*(-2)^(1/4)))*sqrt(1+I*(-2)^(1/4))+1/4*(-1)^(1/4)*arctanh(x/sqrt(1+(-2)^(1/4)))*sqrt(1+(-2)^(1/4))/2^(3/4)-1/8*I*arctanh(x*sqrt((1+I)/(1+I+2^(3/4))))*((-2)^(1/4)+sqrt(2))*sqrt((1+I)/(1+I+2^(3/4)))],
[(1-x^2)/(a+b*(1-x^2)^4),x,16,-1/4*arctan(b^(1/8)*x/sqrt((-a)^(1/4)-b^(1/4)))/(b^(3/8)*sqrt(-a)*sqrt((-a)^(1/4)-b^(1/4)))+1/4*arctanh(b^(1/8)*x/sqrt((-a)^(1/4)+b^(1/4)))/(b^(3/8)*sqrt(-a)*sqrt((-a)^(1/4)+b^(1/4)))-1/4*arctan((-b^(1/8)*x*sqrt(2)+sqrt(b^(1/4)+sqrt(sqrt(-a)+sqrt(b))))/sqrt(-b^(1/4)+sqrt(sqrt(-a)+sqrt(b))))*sqrt(-b^(1/4)+sqrt(sqrt(-a)+sqrt(b)))/(b^(3/8)*sqrt(2)*sqrt(-a)*sqrt(sqrt(-a)+sqrt(b)))+1/4*arctan((b^(1/8)*x*sqrt(2)+sqrt(b^(1/4)+sqrt(sqrt(-a)+sqrt(b))))/sqrt(-b^(1/4)+sqrt(sqrt(-a)+sqrt(b))))*sqrt(-b^(1/4)+sqrt(sqrt(-a)+sqrt(b)))/(b^(3/8)*sqrt(2)*sqrt(-a)*sqrt(sqrt(-a)+sqrt(b)))+1/8*log(b^(1/4)*x^2+sqrt(sqrt(-a)+sqrt(b))-b^(1/8)*x*sqrt(2)*sqrt(b^(1/4)+sqrt(sqrt(-a)+sqrt(b))))*sqrt(b^(1/4)+sqrt(sqrt(-a)+sqrt(b)))/(b^(3/8)*sqrt(2)*sqrt(-a)*sqrt(sqrt(-a)+sqrt(b)))-1/8*log(b^(1/4)*x^2+sqrt(sqrt(-a)+sqrt(b))+b^(1/8)*x*sqrt(2)*sqrt(b^(1/4)+sqrt(sqrt(-a)+sqrt(b))))*sqrt(b^(1/4)+sqrt(sqrt(-a)+sqrt(b)))/(b^(3/8)*sqrt(2)*sqrt(-a)*sqrt(sqrt(-a)+sqrt(b)))],
[(1-x^2)/(a+b*(-1+x^2)^4),x,17,-1/4*arctan(b^(1/8)*x/sqrt((-a)^(1/4)-b^(1/4)))/(b^(3/8)*sqrt(-a)*sqrt((-a)^(1/4)-b^(1/4)))+1/4*arctanh(b^(1/8)*x/sqrt((-a)^(1/4)+b^(1/4)))/(b^(3/8)*sqrt(-a)*sqrt((-a)^(1/4)+b^(1/4)))-1/4*arctan((-b^(1/8)*x*sqrt(2)+sqrt(b^(1/4)+sqrt(sqrt(-a)+sqrt(b))))/sqrt(-b^(1/4)+sqrt(sqrt(-a)+sqrt(b))))*sqrt(-b^(1/4)+sqrt(sqrt(-a)+sqrt(b)))/(b^(3/8)*sqrt(2)*sqrt(-a)*sqrt(sqrt(-a)+sqrt(b)))+1/4*arctan((b^(1/8)*x*sqrt(2)+sqrt(b^(1/4)+sqrt(sqrt(-a)+sqrt(b))))/sqrt(-b^(1/4)+sqrt(sqrt(-a)+sqrt(b))))*sqrt(-b^(1/4)+sqrt(sqrt(-a)+sqrt(b)))/(b^(3/8)*sqrt(2)*sqrt(-a)*sqrt(sqrt(-a)+sqrt(b)))+1/8*log(b^(1/4)*x^2+sqrt(sqrt(-a)+sqrt(b))-b^(1/8)*x*sqrt(2)*sqrt(b^(1/4)+sqrt(sqrt(-a)+sqrt(b))))*sqrt(b^(1/4)+sqrt(sqrt(-a)+sqrt(b)))/(b^(3/8)*sqrt(2)*sqrt(-a)*sqrt(sqrt(-a)+sqrt(b)))-1/8*log(b^(1/4)*x^2+sqrt(sqrt(-a)+sqrt(b))+b^(1/8)*x*sqrt(2)*sqrt(b^(1/4)+sqrt(sqrt(-a)+sqrt(b))))*sqrt(b^(1/4)+sqrt(sqrt(-a)+sqrt(b)))/(b^(3/8)*sqrt(2)*sqrt(-a)*sqrt(sqrt(-a)+sqrt(b)))],
[(1+x^2)^2/(a*x^6+b*(1+x^2)^3),x,-5,1/3*arctan(x*sqrt(a^(1/3)+b^(1/3))/b^(1/6))/(b^(5/6)*sqrt(a^(1/3)+b^(1/3)))+1/3*arctan(x*sqrt(-(-1)^(1/3)*a^(1/3)+b^(1/3))/b^(1/6))/(b^(5/6)*sqrt(-(-1)^(1/3)*a^(1/3)+b^(1/3)))+1/3*arctan(x*sqrt((-1)^(2/3)*a^(1/3)+b^(1/3))/b^(1/6))/(b^(5/6)*sqrt((-1)^(2/3)*a^(1/3)+b^(1/3)))],

# Integrands of the form (d+e x)^p (a+c x^4)^q

# q>0

# q<0
[(d+e*x)^3/(a+c*x^4),x,15,1/4*e^3*log(a+c*x^4)/c+3/2*d^2*e*arctan(x^2*sqrt(c)/sqrt(a))/(sqrt(a)*sqrt(c))-1/4*d*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-3*e^2*sqrt(a)+d^2*sqrt(c))/(a^(3/4)*c^(3/4)*sqrt(2))+1/4*d*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-3*e^2*sqrt(a)+d^2*sqrt(c))/(a^(3/4)*c^(3/4)*sqrt(2))-1/2*d*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(3*e^2*sqrt(a)+d^2*sqrt(c))/(a^(3/4)*c^(3/4)*sqrt(2))+1/2*d*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(3*e^2*sqrt(a)+d^2*sqrt(c))/(a^(3/4)*c^(3/4)*sqrt(2))],
[(d+e*x)^2/(a+c*x^4),x,13,d*e*arctan(x^2*sqrt(c)/sqrt(a))/(sqrt(a)*sqrt(c))-1/4*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*sqrt(a)+d^2*sqrt(c))/(a^(3/4)*c^(3/4)*sqrt(2))+1/4*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*sqrt(a)+d^2*sqrt(c))/(a^(3/4)*c^(3/4)*sqrt(2))-1/2*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*sqrt(a)+d^2*sqrt(c))/(a^(3/4)*c^(3/4)*sqrt(2))+1/2*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*sqrt(a)+d^2*sqrt(c))/(a^(3/4)*c^(3/4)*sqrt(2))],
[(d+e*x)/(a+c*x^4),x,13,-1/2*d*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(3/4)*c^(1/4)*sqrt(2))+1/2*d*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(3/4)*c^(1/4)*sqrt(2))-1/4*d*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(3/4)*c^(1/4)*sqrt(2))+1/4*d*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(3/4)*c^(1/4)*sqrt(2))+1/2*e*arctan(x^2*sqrt(c)/sqrt(a))/(sqrt(a)*sqrt(c))],
[1/(a+c*x^4),x,9,-1/2*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(3/4)*c^(1/4)*sqrt(2))+1/2*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(3/4)*c^(1/4)*sqrt(2))-1/4*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(3/4)*c^(1/4)*sqrt(2))+1/4*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(3/4)*c^(1/4)*sqrt(2))],
[1/((d+e*x)*(a+c*x^4)),x,17,e^3*log(d+e*x)/(c*d^4+a*e^4)-1/4*e^3*log(a+c*x^4)/(c*d^4+a*e^4)-1/2*d^2*e*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/((c*d^4+a*e^4)*sqrt(a))-1/4*c^(1/4)*d*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*sqrt(a)+d^2*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)*sqrt(2))+1/4*c^(1/4)*d*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*sqrt(a)+d^2*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)*sqrt(2))-1/2*c^(1/4)*d*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*sqrt(a)+d^2*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)*sqrt(2))+1/2*c^(1/4)*d*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*sqrt(a)+d^2*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)*sqrt(2))],
[1/((d+e*x)^2*(a+c*x^4)),x,17,-e^3/((c*d^4+a*e^4)*(d+e*x))+4*c*d^3*e^3*log(d+e*x)/(c*d^4+a*e^4)^2-c*d^3*e^3*log(a+c*x^4)/(c*d^4+a*e^4)^2-d*e*(c*d^4-a*e^4)*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/((c*d^4+a*e^4)^2*sqrt(a))-1/4*c^(1/4)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*(3*c*d^4-a*e^4)*sqrt(a)+d^2*(c*d^4-3*a*e^4)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^2*sqrt(2))+1/4*c^(1/4)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*(3*c*d^4-a*e^4)*sqrt(a)+d^2*(c*d^4-3*a*e^4)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^2*sqrt(2))-1/2*c^(1/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*(3*c*d^4-a*e^4)*sqrt(a)+d^2*(c*d^4-3*a*e^4)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^2*sqrt(2))+1/2*c^(1/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*(3*c*d^4-a*e^4)*sqrt(a)+d^2*(c*d^4-3*a*e^4)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^2*sqrt(2))],
[1/((d+e*x)^3*(a+c*x^4)),x,17,-1/2*e^3/((c*d^4+a*e^4)*(d+e*x)^2)-4*c*d^3*e^3/((c*d^4+a*e^4)^2*(d+e*x))+2*c*d^2*e^3*(5*c*d^4-3*a*e^4)*log(d+e*x)/(c*d^4+a*e^4)^3-1/2*c*d^2*e^3*(5*c*d^4-3*a*e^4)*log(a+c*x^4)/(c*d^4+a*e^4)^3-1/2*e*(3*c^2*d^8-12*a*c*d^4*e^4+a^2*e^8)*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/((c*d^4+a*e^4)^3*sqrt(a))-1/4*c^(3/4)*d*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(c^2*d^8-12*a*c*d^4*e^4+3*a^2*e^8-2*d^2*e^2*(3*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^3*sqrt(2))+1/4*c^(3/4)*d*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(c^2*d^8-12*a*c*d^4*e^4+3*a^2*e^8-2*d^2*e^2*(3*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^3*sqrt(2))-1/2*c^(3/4)*d*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(c^2*d^8-12*a*c*d^4*e^4+3*a^2*e^8+2*d^2*e^2*(3*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^3*sqrt(2))+1/2*c^(3/4)*d*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(c^2*d^8-12*a*c*d^4*e^4+3*a^2*e^8+2*d^2*e^2*(3*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^3*sqrt(2))],
[(d+e*x)^3/(a+c*x^4)^2,x,16,1/4*(-a*e^3+c*x*(d^3+3*d^2*e*x+3*d*e^2*x^2))/(a*c*(a+c*x^4))+3/4*d^2*e*arctan(x^2*sqrt(c)/sqrt(a))/(a^(3/2)*sqrt(c))-3/16*d*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*sqrt(a)+d^2*sqrt(c))/(a^(7/4)*c^(3/4)*sqrt(2))+3/16*d*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*sqrt(a)+d^2*sqrt(c))/(a^(7/4)*c^(3/4)*sqrt(2))-3/8*d*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*sqrt(a)+d^2*sqrt(c))/(a^(7/4)*c^(3/4)*sqrt(2))+3/8*d*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*sqrt(a)+d^2*sqrt(c))/(a^(7/4)*c^(3/4)*sqrt(2))],
[(d+e*x)^2/(a+c*x^4)^2,x,14,1/4*x*(d+e*x)^2/(a*(a+c*x^4))+1/2*d*e*arctan(x^2*sqrt(c)/sqrt(a))/(a^(3/2)*sqrt(c))-1/16*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*sqrt(a)+3*d^2*sqrt(c))/(a^(7/4)*c^(3/4)*sqrt(2))+1/16*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*sqrt(a)+3*d^2*sqrt(c))/(a^(7/4)*c^(3/4)*sqrt(2))-1/8*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*sqrt(a)+3*d^2*sqrt(c))/(a^(7/4)*c^(3/4)*sqrt(2))+1/8*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*sqrt(a)+3*d^2*sqrt(c))/(a^(7/4)*c^(3/4)*sqrt(2))],
[(d+e*x)/(a+c*x^4)^2,x,14,1/4*x*(d+e*x)/(a*(a+c*x^4))-3/8*d*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(7/4)*c^(1/4)*sqrt(2))+3/8*d*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(7/4)*c^(1/4)*sqrt(2))-3/16*d*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(7/4)*c^(1/4)*sqrt(2))+3/16*d*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(7/4)*c^(1/4)*sqrt(2))+1/4*e*arctan(x^2*sqrt(c)/sqrt(a))/(a^(3/2)*sqrt(c))],
[1/(a+c*x^4)^2,x,10,1/4*x/(a*(a+c*x^4))-3/8*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(7/4)*c^(1/4)*sqrt(2))+3/8*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(7/4)*c^(1/4)*sqrt(2))-3/16*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(7/4)*c^(1/4)*sqrt(2))+3/16*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(7/4)*c^(1/4)*sqrt(2))],
[1/((d+e*x)*(a+c*x^4)^2),x,31,1/4*(a*e^3+c*x*(d^3-d^2*e*x+d*e^2*x^2))/(a*(c*d^4+a*e^4)*(a+c*x^4))+e^7*log(d+e*x)/(c*d^4+a*e^4)^2-1/4*e^7*log(a+c*x^4)/(c*d^4+a*e^4)^2-1/4*d^2*e*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/(a^(3/2)*(c*d^4+a*e^4))-1/2*d^2*e^5*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/((c*d^4+a*e^4)^2*sqrt(a))-1/4*c^(1/4)*d*e^4*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*sqrt(a)+d^2*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^2*sqrt(2))+1/4*c^(1/4)*d*e^4*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*sqrt(a)+d^2*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^2*sqrt(2))-1/2*c^(1/4)*d*e^4*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*sqrt(a)+d^2*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^2*sqrt(2))+1/2*c^(1/4)*d*e^4*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*sqrt(a)+d^2*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^2*sqrt(2))-1/16*c^(1/4)*d*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*sqrt(a)+3*d^2*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)*sqrt(2))+1/16*c^(1/4)*d*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*sqrt(a)+3*d^2*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)*sqrt(2))-1/8*c^(1/4)*d*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*sqrt(a)+3*d^2*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)*sqrt(2))+1/8*c^(1/4)*d*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*sqrt(a)+3*d^2*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)*sqrt(2))],
[1/((d+e*x)^2*(a+c*x^4)^2),x,31,-e^7/((c*d^4+a*e^4)^2*(d+e*x))+1/4*c*(4*a*d^3*e^3+x*(d^2*(c*d^4-3*a*e^4)-2*d*e*(c*d^4-a*e^4)*x+e^2*(3*c*d^4-a*e^4)*x^2))/(a*(c*d^4+a*e^4)^2*(a+c*x^4))+8*c*d^3*e^7*log(d+e*x)/(c*d^4+a*e^4)^3-2*c*d^3*e^7*log(a+c*x^4)/(c*d^4+a*e^4)^3-1/2*d*e*(c*d^4-a*e^4)*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/(a^(3/2)*(c*d^4+a*e^4)^2)-d*e^5*(3*c*d^4-a*e^4)*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/((c*d^4+a*e^4)^3*sqrt(a))-1/16*c^(1/4)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*(3*c*d^4-a*e^4)*sqrt(a)+3*d^2*(c*d^4-3*a*e^4)*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)^2*sqrt(2))+1/16*c^(1/4)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*(3*c*d^4-a*e^4)*sqrt(a)+3*d^2*(c*d^4-3*a*e^4)*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)^2*sqrt(2))-1/8*c^(1/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*(3*c*d^4-a*e^4)*sqrt(a)+3*d^2*(c*d^4-3*a*e^4)*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)^2*sqrt(2))+1/8*c^(1/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*(3*c*d^4-a*e^4)*sqrt(a)+3*d^2*(c*d^4-3*a*e^4)*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)^2*sqrt(2))-1/4*c^(1/4)*e^4*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*(7*c*d^4-a*e^4)*sqrt(a)+d^2*(5*c*d^4-3*a*e^4)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^3*sqrt(2))+1/4*c^(1/4)*e^4*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*(7*c*d^4-a*e^4)*sqrt(a)+d^2*(5*c*d^4-3*a*e^4)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^3*sqrt(2))-1/2*c^(1/4)*e^4*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*(7*c*d^4-a*e^4)*sqrt(a)+d^2*(5*c*d^4-3*a*e^4)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^3*sqrt(2))+1/2*c^(1/4)*e^4*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*(7*c*d^4-a*e^4)*sqrt(a)+d^2*(5*c*d^4-3*a*e^4)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^3*sqrt(2))],
[1/((d+e*x)^3*(a+c*x^4)^2),x,31,-1/2*e^7/((c*d^4+a*e^4)^2*(d+e*x)^2)-8*c*d^3*e^7/((c*d^4+a*e^4)^3*(d+e*x))+1/4*c*(2*a*d^2*e^3*(5*c*d^4-3*a*e^4)+x*(d*(c^2*d^8-12*a*c*d^4*e^4+3*a^2*e^8)-e*(3*c^2*d^8-12*a*c*d^4*e^4+a^2*e^8)*x+2*c*d^3*e^2*(3*c*d^4-5*a*e^4)*x^2))/(a*(c*d^4+a*e^4)^3*(a+c*x^4))+12*c*d^2*e^7*(3*c*d^4-a*e^4)*log(d+e*x)/(c*d^4+a*e^4)^4-3*c*d^2*e^7*(3*c*d^4-a*e^4)*log(a+c*x^4)/(c*d^4+a*e^4)^4-1/4*e*(3*c^2*d^8-12*a*c*d^4*e^4+a^2*e^8)*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/(a^(3/2)*(c*d^4+a*e^4)^3)-1/2*e^5*(21*c^2*d^8-26*a*c*d^4*e^4+a^2*e^8)*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/((c*d^4+a*e^4)^4*sqrt(a))-1/16*c^(3/4)*d*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(3*c^2*d^8-36*a*c*d^4*e^4+9*a^2*e^8-2*d^2*e^2*(3*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)^3*sqrt(2))+1/16*c^(3/4)*d*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(3*c^2*d^8-36*a*c*d^4*e^4+9*a^2*e^8-2*d^2*e^2*(3*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)^3*sqrt(2))-1/8*c^(3/4)*d*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(3*c^2*d^8-36*a*c*d^4*e^4+9*a^2*e^8+2*d^2*e^2*(3*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)^3*sqrt(2))+1/8*c^(3/4)*d*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(3*c^2*d^8-36*a*c*d^4*e^4+9*a^2*e^8+2*d^2*e^2*(3*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)^3*sqrt(2))+1/4*c^(3/4)*d*e^4*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-3*(5*c^2*d^8-10*a*c*d^4*e^4+a^2*e^8)+4*d^2*e^2*(7*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^4*sqrt(2))-1/4*c^(3/4)*d*e^4*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-3*(5*c^2*d^8-10*a*c*d^4*e^4+a^2*e^8)+4*d^2*e^2*(7*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^4*sqrt(2))-1/2*c^(3/4)*d*e^4*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(3*(5*c^2*d^8-10*a*c*d^4*e^4+a^2*e^8)+4*d^2*e^2*(7*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^4*sqrt(2))+1/2*c^(3/4)*d*e^4*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(3*(5*c^2*d^8-10*a*c*d^4*e^4+a^2*e^8)+4*d^2*e^2*(7*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^4*sqrt(2))],
[(d+e*x)^3/(a+c*x^4)^3,x,15,1/32*x*(7*d^3+18*d^2*e*x+15*d*e^2*x^2)/(a^2*(a+c*x^4))+1/8*(-a*e^3+c*x*(d^3+3*d^2*e*x+3*d*e^2*x^2))/(a*c*(a+c*x^4)^2)+9/16*d^2*e*arctan(x^2*sqrt(c)/sqrt(a))/(a^(5/2)*sqrt(c))-3/128*d*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-5*e^2*sqrt(a)+7*d^2*sqrt(c))/(a^(11/4)*c^(3/4)*sqrt(2))+3/128*d*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-5*e^2*sqrt(a)+7*d^2*sqrt(c))/(a^(11/4)*c^(3/4)*sqrt(2))-3/64*d*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(5*e^2*sqrt(a)+7*d^2*sqrt(c))/(a^(11/4)*c^(3/4)*sqrt(2))+3/64*d*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(5*e^2*sqrt(a)+7*d^2*sqrt(c))/(a^(11/4)*c^(3/4)*sqrt(2))],
[(d+e*x)^2/(a+c*x^4)^3,x,15,1/8*x*(d+e*x)^2/(a*(a+c*x^4)^2)+1/32*x*(7*d^2+12*d*e*x+5*e^2*x^2)/(a^2*(a+c*x^4))+3/8*d*e*arctan(x^2*sqrt(c)/sqrt(a))/(a^(5/2)*sqrt(c))-1/128*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-5*e^2*sqrt(a)+21*d^2*sqrt(c))/(a^(11/4)*c^(3/4)*sqrt(2))+1/128*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-5*e^2*sqrt(a)+21*d^2*sqrt(c))/(a^(11/4)*c^(3/4)*sqrt(2))-1/64*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(5*e^2*sqrt(a)+21*d^2*sqrt(c))/(a^(11/4)*c^(3/4)*sqrt(2))+1/64*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(5*e^2*sqrt(a)+21*d^2*sqrt(c))/(a^(11/4)*c^(3/4)*sqrt(2))],
[(d+e*x)/(a+c*x^4)^3,x,15,1/8*x*(d+e*x)/(a*(a+c*x^4)^2)+1/32*x*(7*d+6*e*x)/(a^2*(a+c*x^4))-21/64*d*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(11/4)*c^(1/4)*sqrt(2))+21/64*d*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(11/4)*c^(1/4)*sqrt(2))-21/128*d*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(11/4)*c^(1/4)*sqrt(2))+21/128*d*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(11/4)*c^(1/4)*sqrt(2))+3/16*e*arctan(x^2*sqrt(c)/sqrt(a))/(a^(5/2)*sqrt(c))],
[1/(a+c*x^4)^3,x,11,1/8*x/(a*(a+c*x^4)^2)+7/32*x/(a^2*(a+c*x^4))-21/64*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(11/4)*c^(1/4)*sqrt(2))+21/64*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))/(a^(11/4)*c^(1/4)*sqrt(2))-21/128*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(11/4)*c^(1/4)*sqrt(2))+21/128*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))/(a^(11/4)*c^(1/4)*sqrt(2))],
[1/((d+e*x)*(a+c*x^4)^3),x,46,1/32*c*x*(7*d^3-6*d^2*e*x+5*d*e^2*x^2)/(a^2*(c*d^4+a*e^4)*(a+c*x^4))+1/8*(a*e^3+c*x*(d^3-d^2*e*x+d*e^2*x^2))/(a*(c*d^4+a*e^4)*(a+c*x^4)^2)+1/4*e^4*(a*e^3+c*x*(d^3-d^2*e*x+d*e^2*x^2))/(a*(c*d^4+a*e^4)^2*(a+c*x^4))+e^11*log(d+e*x)/(c*d^4+a*e^4)^3-1/4*e^11*log(a+c*x^4)/(c*d^4+a*e^4)^3-1/4*d^2*e^5*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/(a^(3/2)*(c*d^4+a*e^4)^2)-3/16*d^2*e*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/(a^(5/2)*(c*d^4+a*e^4))-1/2*d^2*e^9*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/((c*d^4+a*e^4)^3*sqrt(a))-1/4*c^(1/4)*d*e^8*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*sqrt(a)+d^2*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^3*sqrt(2))+1/4*c^(1/4)*d*e^8*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*sqrt(a)+d^2*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^3*sqrt(2))-1/2*c^(1/4)*d*e^8*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*sqrt(a)+d^2*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^3*sqrt(2))+1/2*c^(1/4)*d*e^8*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*sqrt(a)+d^2*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^3*sqrt(2))-1/16*c^(1/4)*d*e^4*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*sqrt(a)+3*d^2*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)^2*sqrt(2))+1/16*c^(1/4)*d*e^4*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*sqrt(a)+3*d^2*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)^2*sqrt(2))-1/8*c^(1/4)*d*e^4*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*sqrt(a)+3*d^2*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)^2*sqrt(2))+1/8*c^(1/4)*d*e^4*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*sqrt(a)+3*d^2*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)^2*sqrt(2))-1/128*c^(1/4)*d*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-5*e^2*sqrt(a)+21*d^2*sqrt(c))/(a^(11/4)*(c*d^4+a*e^4)*sqrt(2))+1/128*c^(1/4)*d*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-5*e^2*sqrt(a)+21*d^2*sqrt(c))/(a^(11/4)*(c*d^4+a*e^4)*sqrt(2))-1/64*c^(1/4)*d*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(5*e^2*sqrt(a)+21*d^2*sqrt(c))/(a^(11/4)*(c*d^4+a*e^4)*sqrt(2))+1/64*c^(1/4)*d*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(5*e^2*sqrt(a)+21*d^2*sqrt(c))/(a^(11/4)*(c*d^4+a*e^4)*sqrt(2))],
[1/((d+e*x)^2*(a+c*x^4)^3),x,46,-e^11/((c*d^4+a*e^4)^3*(d+e*x))+1/32*c*x*(7*d^2*(c*d^4-3*a*e^4)-12*d*e*(c*d^4-a*e^4)*x+5*e^2*(3*c*d^4-a*e^4)*x^2)/(a^2*(c*d^4+a*e^4)^2*(a+c*x^4))+1/8*c*(4*a*d^3*e^3+x*(d^2*(c*d^4-3*a*e^4)-2*d*e*(c*d^4-a*e^4)*x+e^2*(3*c*d^4-a*e^4)*x^2))/(a*(c*d^4+a*e^4)^2*(a+c*x^4)^2)+1/4*c*e^4*(8*a*d^3*e^3+x*(d^2*(5*c*d^4-3*a*e^4)-2*d*e*(3*c*d^4-a*e^4)*x+e^2*(7*c*d^4-a*e^4)*x^2))/(a*(c*d^4+a*e^4)^3*(a+c*x^4))+12*c*d^3*e^11*log(d+e*x)/(c*d^4+a*e^4)^4-3*c*d^3*e^11*log(a+c*x^4)/(c*d^4+a*e^4)^4-1/2*d*e^5*(3*c*d^4-a*e^4)*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/(a^(3/2)*(c*d^4+a*e^4)^3)-3/8*d*e*(c*d^4-a*e^4)*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/(a^(5/2)*(c*d^4+a*e^4)^2)-d*e^9*(5*c*d^4-a*e^4)*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/((c*d^4+a*e^4)^4*sqrt(a))-1/4*c^(1/4)*e^8*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(9*c^(3/2)*d^6+a^(3/2)*e^6-11*c*d^4*e^2*sqrt(a)-3*a*d^2*e^4*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^4*sqrt(2))+1/4*c^(1/4)*e^8*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(9*c^(3/2)*d^6+a^(3/2)*e^6-11*c*d^4*e^2*sqrt(a)-3*a*d^2*e^4*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^4*sqrt(2))-1/128*c^(1/4)*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-5*e^2*(3*c*d^4-a*e^4)*sqrt(a)+21*d^2*(c*d^4-3*a*e^4)*sqrt(c))/(a^(11/4)*(c*d^4+a*e^4)^2*sqrt(2))+1/128*c^(1/4)*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-5*e^2*(3*c*d^4-a*e^4)*sqrt(a)+21*d^2*(c*d^4-3*a*e^4)*sqrt(c))/(a^(11/4)*(c*d^4+a*e^4)^2*sqrt(2))-1/64*c^(1/4)*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(5*e^2*(3*c*d^4-a*e^4)*sqrt(a)+21*d^2*(c*d^4-3*a*e^4)*sqrt(c))/(a^(11/4)*(c*d^4+a*e^4)^2*sqrt(2))+1/64*c^(1/4)*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(5*e^2*(3*c*d^4-a*e^4)*sqrt(a)+21*d^2*(c*d^4-3*a*e^4)*sqrt(c))/(a^(11/4)*(c*d^4+a*e^4)^2*sqrt(2))-1/16*c^(1/4)*e^4*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*(7*c*d^4-a*e^4)*sqrt(a)+3*d^2*(5*c*d^4-3*a*e^4)*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)^3*sqrt(2))+1/16*c^(1/4)*e^4*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-e^2*(7*c*d^4-a*e^4)*sqrt(a)+3*d^2*(5*c*d^4-3*a*e^4)*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)^3*sqrt(2))-1/8*c^(1/4)*e^4*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*(7*c*d^4-a*e^4)*sqrt(a)+3*d^2*(5*c*d^4-3*a*e^4)*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)^3*sqrt(2))+1/8*c^(1/4)*e^4*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*(7*c*d^4-a*e^4)*sqrt(a)+3*d^2*(5*c*d^4-3*a*e^4)*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)^3*sqrt(2))-1/2*c^(1/4)*e^8*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*(11*c*d^4-a*e^4)*sqrt(a)+3*d^2*(3*c*d^4-a*e^4)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^4*sqrt(2))+1/2*c^(1/4)*e^8*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(e^2*(11*c*d^4-a*e^4)*sqrt(a)+3*d^2*(3*c*d^4-a*e^4)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^4*sqrt(2))],
[1/((d+e*x)^3*(a+c*x^4)^3),x,46,-1/2*e^11/((c*d^4+a*e^4)^3*(d+e*x)^2)-12*c*d^3*e^11/((c*d^4+a*e^4)^4*(d+e*x))+1/32*c*x*(7*d*(c^2*d^8-12*a*c*d^4*e^4+3*a^2*e^8)-6*e*(3*c^2*d^8-12*a*c*d^4*e^4+a^2*e^8)*x+10*c*d^3*e^2*(3*c*d^4-5*a*e^4)*x^2)/(a^2*(c*d^4+a*e^4)^3*(a+c*x^4))+1/8*c*(2*a*d^2*e^3*(5*c*d^4-3*a*e^4)+x*(d*(c^2*d^8-12*a*c*d^4*e^4+3*a^2*e^8)-e*(3*c^2*d^8-12*a*c*d^4*e^4+a^2*e^8)*x+2*c*d^3*e^2*(3*c*d^4-5*a*e^4)*x^2))/(a*(c*d^4+a*e^4)^3*(a+c*x^4)^2)+1/4*c*e^4*(12*a*d^2*e^3*(3*c*d^4-a*e^4)+x*(3*d*(5*c^2*d^8-10*a*c*d^4*e^4+a^2*e^8)-e*(21*c^2*d^8-26*a*c*d^4*e^4+a^2*e^8)*x+4*c*d^3*e^2*(7*c*d^4-5*a*e^4)*x^2))/(a*(c*d^4+a*e^4)^4*(a+c*x^4))+6*c*d^2*e^11*(13*c*d^4-3*a*e^4)*log(d+e*x)/(c*d^4+a*e^4)^5-3/2*c*d^2*e^11*(13*c*d^4-3*a*e^4)*log(a+c*x^4)/(c*d^4+a*e^4)^5-1/4*e^5*(21*c^2*d^8-26*a*c*d^4*e^4+a^2*e^8)*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/(a^(3/2)*(c*d^4+a*e^4)^4)-3/16*e*(3*c^2*d^8-12*a*c*d^4*e^4+a^2*e^8)*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/(a^(5/2)*(c*d^4+a*e^4)^3)-1/2*e^9*(55*c^2*d^8-40*a*c*d^4*e^4+a^2*e^8)*arctan(x^2*sqrt(c)/sqrt(a))*sqrt(c)/((c*d^4+a*e^4)^5*sqrt(a))+1/128*c^(3/4)*d*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-21*(c^2*d^8-12*a*c*d^4*e^4+3*a^2*e^8)+10*d^2*e^2*(3*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(11/4)*(c*d^4+a*e^4)^3*sqrt(2))-1/128*c^(3/4)*d*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-21*(c^2*d^8-12*a*c*d^4*e^4+3*a^2*e^8)+10*d^2*e^2*(3*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(11/4)*(c*d^4+a*e^4)^3*sqrt(2))-1/64*c^(3/4)*d*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(21*(c^2*d^8-12*a*c*d^4*e^4+3*a^2*e^8)+10*d^2*e^2*(3*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(11/4)*(c*d^4+a*e^4)^3*sqrt(2))+1/64*c^(3/4)*d*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(21*(c^2*d^8-12*a*c*d^4*e^4+3*a^2*e^8)+10*d^2*e^2*(3*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(11/4)*(c*d^4+a*e^4)^3*sqrt(2))+1/16*c^(3/4)*d*e^4*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-9*(5*c^2*d^8-10*a*c*d^4*e^4+a^2*e^8)+4*d^2*e^2*(7*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)^4*sqrt(2))-1/16*c^(3/4)*d*e^4*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(-9*(5*c^2*d^8-10*a*c*d^4*e^4+a^2*e^8)+4*d^2*e^2*(7*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)^4*sqrt(2))-1/8*c^(3/4)*d*e^4*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(9*(5*c^2*d^8-10*a*c*d^4*e^4+a^2*e^8)+4*d^2*e^2*(7*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)^4*sqrt(2))+1/8*c^(3/4)*d*e^4*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(9*(5*c^2*d^8-10*a*c*d^4*e^4+a^2*e^8)+4*d^2*e^2*(7*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(7/4)*(c*d^4+a*e^4)^4*sqrt(2))-3/4*c^(3/4)*d*e^8*log(-a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(15*c^2*d^8-16*a*c*d^4*e^4+a^2*e^8-2*d^2*e^2*(11*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^5*sqrt(2))+3/4*c^(3/4)*d*e^8*log(a^(1/4)*c^(1/4)*x*sqrt(2)+sqrt(a)+x^2*sqrt(c))*(15*c^2*d^8-16*a*c*d^4*e^4+a^2*e^8-2*d^2*e^2*(11*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^5*sqrt(2))-3/2*c^(3/4)*d*e^8*arctan(1-c^(1/4)*x*sqrt(2)/a^(1/4))*(15*c^2*d^8-16*a*c*d^4*e^4+a^2*e^8+2*d^2*e^2*(11*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^5*sqrt(2))+3/2*c^(3/4)*d*e^8*arctan(1+c^(1/4)*x*sqrt(2)/a^(1/4))*(15*c^2*d^8-16*a*c*d^4*e^4+a^2*e^8+2*d^2*e^2*(11*c*d^4-5*a*e^4)*sqrt(a)*sqrt(c))/(a^(3/4)*(c*d^4+a*e^4)^5*sqrt(2))],

# Integrands requiring algebraic simplification

#  Following pairs of integrands are equal. 
[(-1+x)/(1-x+x^2),x,4,1/2*log(1-x+x^2)+arctan((1-2*x)/sqrt(3))/sqrt(3)],
[(-1+x^2)/(1+x^3),x,5,1/2*log(1-x+x^2)+arctan((1-2*x)/sqrt(3))/sqrt(3)],
[(-4+3*x)/(4-2*x+x^2),x,4,3/2*log(4-2*x+x^2)+arctan((1-x)/sqrt(3))/sqrt(3)],
[(-8+2*x+3*x^2)/(8+x^3),x,5,3/2*log(4-2*x+x^2)+arctan((1-x)/sqrt(3))/sqrt(3)],
[(2+x)/(-1+2*x+x^2),x,3,1/4*log(1+x+sqrt(2))*(2-sqrt(2))+1/4*log(1+x-sqrt(2))*(2+sqrt(2))],
[(-4+x^2)/(2-5*x+x^3),x,4,1/4*log(1+x+sqrt(2))*(2-sqrt(2))+1/4*log(1+x-sqrt(2))*(2+sqrt(2))],
[2/(-1+4*x^2),x,2,-arctanh(2*x)],
[1/(-1+2*x)+(-1)/(1+2*x),x,1,1/2*log(1-2*x)-1/2*log(1+2*x)],
[x/(1-x^2)^5,x,1,1/8/(1-x^2)^4],
[(-1/32)/(-1+x)^5+3/64/(-1+x)^4+(-5/128)/(-1+x)^3+5/256/(-1+x)^2+(-1/32)/(1+x)^5+(-3/64)/(1+x)^4+(-5/128)/(1+x)^3+(-5/256)/(1+x)^2,x,1,1/8/(1-x^2)^4,1/128/(1-x)^4+1/64/(1-x)^3+5/256/(1-x)^2+5/256/(1-x)+1/128/(1+x)^4+1/64/(1+x)^3+5/256/(1+x)^2+5/256/(1+x)],
[(1+x^6)/(-1+x^6),x,11,x-2/3*arctanh(x)+1/6*log(1-x+x^2)-1/6*log(1+x+x^2)+arctan((1-2*x)/sqrt(3))/sqrt(3)-arctan((1+2*x)/sqrt(3))/sqrt(3)],
[(1/x^3+x^3)/((-1)/x^3+x^3),x,13,x-2/3*arctanh(x)+1/6*log(1-x+x^2)-1/6*log(1+x+x^2)+arctan((1-2*x)/sqrt(3))/sqrt(3)-arctan((1+2*x)/sqrt(3))/sqrt(3)],

# Miscellaneous rational function integration problems
[(-x+x^3)/(6+2*x),x,3,4*x-3/4*x^2+1/6*x^3-12*log(3+x)],
[(x+x^3)/(-1+x),x,3,2*x+1/2*x^2+1/3*x^3+2*log(1-x)],
[a*c+(b*c+d)*x,x,1,a*c*x+1/2*(b*c+d)*x^2],
[d*x+c*(a+b*x),x,1,1/2*d*x^2+1/2*c*(a+b*x)^2/b],
[(4+4*x)/(x^2*(1+x^2)),x,5,(-4)/x-4*arctan(x)+4*log(x)-2*log(1+x^2)],
[(24+8*x)/(x*(-4+x^2)),x,2,5*log(2-x)-6*log(x)+log(2+x)],
[(-1+x^2)/(-2*x+x^3),x,4,1/2*log(x)+1/4*log(2-x^2)],
[(1+x^2)/(3*x+x^3),x,1,1/3*log(3*x+x^3)],
[(a+3*b*x^2)/(a*x+b*x^3),x,1,log(a*x+b*x^3)],
[(-2+4*x)/(-x+x^3),x,3,log(1-x)+2*log(x)-3*log(1+x)],
[(4+x)/(4*x+x^3),x,6,1/2*arctan(1/2*x)+log(x)-1/2*log(4+x^2)],
[(-x+2*x^3)/(1-x^2+x^4),x,1,1/2*log(1-x^2+x^4)],
[(-3+x)/(2*x+3*x^2+x^3),x,3,-3/2*log(x)+4*log(1+x)-5/2*log(2+x)],
[(2+4*x)/(x^2+2*x^3+x^4),x,3,(-2)/(x*(1+x))],
[(1+x)/(-6*x+x^2+x^3),x,3,3/10*log(2-x)-1/6*log(x)-2/15*log(3+x)],
[(4*x^2+x^3)/(x+x^3),x,6,x-arctan(x)+2*log(1+x^2)],
[(x+2*x^3)/(x^2+x^4)^3,x,1,(-1/4)/(x^2+x^4)^2],
[(a*x^2+b*x^3)/(c*x^2+d*x^3),x,4,b*x/d-(b*c-a*d)*log(c+d*x)/d^2],
[(x+x^2)/(-2*x-x^2+x^3),x,2,log(2-x)],
[(1-5*x^2)/(x^3*(1+x^2)),x,3,(-1/2)/x^2-6*log(x)+3*log(1+x^2)],
[2*x/((-1+x)*(5+x^2)),x,6,1/3*log(1-x)-1/6*log(5+x^2)+1/3*arctan(x/sqrt(5))*sqrt(5)],
[(2+x^2)/(2+x),x,2,-2*x+1/2*x^2+6*log(2+x)],
[1/((-3+x)*(4+x^2)),x,5,-3/26*arctan(1/2*x)+1/13*log(3-x)-1/26*log(4+x^2)],
[(-2+3*x^6)/(x*(5+2*x^6)),x,3,-2/5*log(x)+19/60*log(5+2*x^6)],
[(3+2*x)/((-2+x)*(5+x)),x,2,log(2-x)+log(5+x)],
[x^4/(4+5*x^2+x^4),x,4,x-8/3*arctan(1/2*x)+1/3*arctan(x)],
[1/((1+x)*(2+x)^2*(3+x)^3),x,2,1/(2+x)+1/4/(3+x)^2+5/4/(3+x)+1/8*log(1+x)+2*log(2+x)-17/8*log(3+x)],
[x/(-1+x^2),x,1,1/2*log(1-x^2)],
[1/(-1+x^2)^2,x,2,1/2*x/(1-x^2)+1/2*arctanh(x)],
[x^2/(1+x^2)^2,x,2,-1/2*x/(1+x^2)+1/2*arctan(x)],
[1/(2+3*x),x,1,1/3*log(2+3*x)],
[1/(a^2+x^2),x,1,arctan(x/a)/a],
[1/(a+b*x^2),x,1,arctan(x*sqrt(b)/sqrt(a))/(sqrt(a)*sqrt(b))],
[1/(2-x+x^2),x,2,-2*arctan((1-2*x)/sqrt(7))/sqrt(7)],
[x^2*(4-x^2)^2,x,2,16/3*x^3-8/5*x^5+1/7*x^7],
[x*(1-x^3)^2,x,2,1/2*x^2-2/5*x^5+1/8*x^8],
[(-4+5*x^2+x^3)/x^2,x,2,4/x+5*x+1/2*x^2],
[(-1+x)/(3-4*x+3*x^2),x,4,1/6*log(3-4*x+3*x^2)+1/3*arctan((2-3*x)/sqrt(5))/sqrt(5)],
[(2+x^3)^2,x,2,4*x+x^4+1/7*x^7],
[(-4+x^2)/(2+x),x,2,-2*x+1/2*x^2],
[1/((2+x)*(1+x^2)),x,5,2/5*arctan(x)+1/5*log(2+x)-1/10*log(1+x^2)],
[1/((1+x)*(1+x^2)),x,5,1/2*arctan(x)+1/2*log(1+x)-1/4*log(1+x^2)],
[x/((1+x)*(1+x^2)),x,5,1/2*arctan(x)-1/2*log(1+x)+1/4*log(1+x^2)],
[(2*x+x^2)/(1+x)^2,x,2,x^2/(1+x),x+1/(1+x)],
[(-10+x^2)/(4+9*x^2+2*x^4),x,3,arctan(1/2*x)-3*arctan(x*sqrt(2))/sqrt(2)],
[(31+5*x)/(11-4*x+3*x^2),x,4,5/6*log(11-4*x+3*x^2)-103/3*arctan((2-3*x)/sqrt(29))/sqrt(29)],
[(-2+x^2+x^3)/x^4,x,2,2/3/x^3+(-1)/x+log(x)],
[(1+x+x^3)/x^2,x,2,(-1)/x+1/2*x^2+log(x)],
[(-2+x^2)/(x*(2+x^2)),x,3,-log(x)+log(2+x^2)],
[(-3+x)*(-7+4*x^2),x,2,21*x-4*x^3+1/16*(7-4*x^2)^2],
[(-2+7*x)^3,x,1,1/28*(2-7*x)^4],
[(-7+4*x^2)/(3+2*x),x,2,-3*x+x^2+log(3+2*x)],
[(1+x)/((-1+x)*x^2),x,2,1/x+2*log(1-x)-2*log(x)],
[1/(4*x^2+4*x^3+x^4),x,3,1/2*(1+x)/(1-(1+x)^2)+1/2*arctanh(1+x)],
[(1+x^2)/(1+x),x,2,-x+1/2*x^2+2*log(1+x)],
[(-1+3*x-3*x^2+x^3)/x^2,x,2,1/x-3*x+1/2*x^2+3*log(x)],
[(x+1/2*(3-sqrt(37)))*(x+1/2*(3+sqrt(37))),x,2,-7*x+3/2*x^2+1/3*x^3],
[(4+3*x^2+2*x^3)/(1+x)^4,x,2,(-5/3)/(1+x)^3+3/(1+x)+2*log(1+x)],
[x/((1+x)^2*(1+x^2)),x,3,1/2/(1+x)+1/2*arctan(x)],
[(7-2*x+3*x^2-x^3+x^4)/(2+x),x,2,-20*x+9/2*x^2-x^3+1/4*x^4+47*log(2+x)],
[(-1+x^3)/(-1+x),x,2,x+1/2*x^2+1/3*x^3],
[(2+2*x)/((-1+x)^3*(1+x^2)),x,3,(-1)/(1-x)^2+1/(-1+x)+arctan(x)],
[1/(b*x+c*(d+e*x)^2),x,3,-2*arctanh((b+2*c*e*(d+e*x))/(sqrt(b)*sqrt(b+4*c*d*e)))/(sqrt(b)*sqrt(b+4*c*d*e))],
[1/(a+b*x+c*(d+e*x)^2),x,3,-2*arctanh((b+2*c*e*(d+e*x))/sqrt(b^2+4*b*c*d*e-4*a*c*e^2))/sqrt(b^2+4*b*c*d*e-4*a*c*e^2)],
[x^2/(1+(-1+x^2)^2),x,10,-1/2*arctan((-2*x+sqrt(2*(1+sqrt(2))))/sqrt(2*(-1+sqrt(2))))*sqrt(1/2*(1+sqrt(2)))+1/2*arctan((2*x+sqrt(2*(1+sqrt(2))))/sqrt(2*(-1+sqrt(2))))*sqrt(1/2*(1+sqrt(2)))+1/4*log(x^2+sqrt(2)-x*sqrt(2*(1+sqrt(2))))/sqrt(2*(1+sqrt(2)))-1/4*log(x^2+sqrt(2)+x*sqrt(2*(1+sqrt(2))))/sqrt(2*(1+sqrt(2)))],

#  Following integrands are equal. 

#  Requires the Ostrogradskiy-Hermite method 
[(-15+36*x-5*x^2-12*x^3+34*x^4-140*x^5-15*x^6-8*x^7+30*x^9)/(3+x+x^4)^4,x,5,2/(3+x+x^4)^3-3*x/(3+x+x^4)^3+5*x^2/(3+x+x^4)^3+x^4/(3+x+x^4)^3-5*x^6/(3+x+x^4)^3],
[3*(-47+228*x+120*x^2+19*x^3)/(3+x+x^4)^4+(42-320*x-75*x^2-8*x^3)/(3+x+x^4)^3+30*x/(3+x+x^4)^2,x,-7,(2-3*x+5*x^2+x^4-5*x^6)/(3+x+x^4)^3],
[(-3+10*x+4*x^3-30*x^5)/(3+x+x^4)^3-3*(1+4*x^3)*(2-3*x+5*x^2+x^4-5*x^6)/(3+x+x^4)^4,x,-13,(2-3*x+5*x^2+x^4-5*x^6)/(3+x+x^4)^3]]:
