# Maple integration test file: "1 Algebraic functions\1.3 Miscellaneous\1.3.2 Algebraic functions.txt"

lst:=[

# Integrands of the form u (a+b x^3)^p

# Integrands of the form (c+d x)^m (a+b x^3)^p

# Integrands of the form 1 / ((c+d x) Sqrt[a+b x^3]) with b c^3 - 4 a d^3=0
[1/((2^(2/3)+x)*sqrt(1+x^3)),x,4,2/3*arctan((1+2^(1/3)*x)*sqrt(3)/sqrt(1+x^3))/sqrt(3)+2/3*2^(1/3)*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[1/((2^(2/3)-x)*sqrt(1-x^3)),x,4,-2/3*arctan((1-2^(1/3)*x)*sqrt(3)/sqrt(1-x^3))/sqrt(3)-2/3*2^(1/3)*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[1/((2^(2/3)-x)*sqrt(-1+x^3)),x,4,-2/3*arctanh((1-2^(1/3)*x)*sqrt(3)/sqrt(-1+x^3))/sqrt(3)-2/3*2^(1/3)*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[1/((2^(2/3)+x)*sqrt(-1-x^3)),x,4,2/3*arctanh((1+2^(1/3)*x)*sqrt(3)/sqrt(-1-x^3))/sqrt(3)+2/3*2^(1/3)*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[1/((2^(2/3)*a^(1/3)+b^(1/3)*x)*sqrt(a+b*x^3)),x,4,2/3*arctan(a^(1/6)*(a^(1/3)+2^(1/3)*b^(1/3)*x)*sqrt(3)/sqrt(a+b*x^3))/(b^(1/3)*sqrt(3)*sqrt(a))+2/3*2^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^(1/3)*b^(1/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[1/((2^(2/3)*a^(1/3)-b^(1/3)*x)*sqrt(a-b*x^3)),x,4,-2/3*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*b^(1/3)*x)*sqrt(3)/sqrt(a-b*x^3))/(b^(1/3)*sqrt(3)*sqrt(a))-2/3*2^(1/3)*(a^(1/3)-b^(1/3)*x)*EllipticF((-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^(1/3)*b^(1/3)*sqrt(a-b*x^3)*sqrt(a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[1/((2^(2/3)*a^(1/3)-b^(1/3)*x)*sqrt(-a+b*x^3)),x,4,-2/3*arctanh(a^(1/6)*(a^(1/3)-2^(1/3)*b^(1/3)*x)*sqrt(3)/sqrt(-a+b*x^3))/(b^(1/3)*sqrt(3)*sqrt(a))-2/3*2^(1/3)*(a^(1/3)-b^(1/3)*x)*EllipticF((-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2-sqrt(3))/(3^(1/4)*a^(1/3)*b^(1/3)*sqrt(-a+b*x^3)*sqrt(-a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))],
[1/((2^(2/3)*a^(1/3)+b^(1/3)*x)*sqrt(-a-b*x^3)),x,4,2/3*arctanh(a^(1/6)*(a^(1/3)+2^(1/3)*b^(1/3)*x)*sqrt(3)/sqrt(-a-b*x^3))/(b^(1/3)*sqrt(3)*sqrt(a))+2/3*2^(1/3)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2-sqrt(3))/(3^(1/4)*a^(1/3)*b^(1/3)*sqrt(-a-b*x^3)*sqrt(-a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))],
[1/((c+d*x)*sqrt(c^3+4*d^3*x^3)),x,4,2/3*arctan((c+2*d*x)*sqrt(3)*sqrt(c)/sqrt(c^3+4*d^3*x^3))/(c^(3/2)*d*sqrt(3))+2/3*2^(1/3)*(c+2^(2/3)*d*x)*EllipticF((2^(2/3)*d*x+c*(1-sqrt(3)))/(2^(2/3)*d*x+c*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((c^2-2^(2/3)*c*d*x+2*2^(1/3)*d^2*x^2)/(2^(2/3)*d*x+c*(1+sqrt(3)))^2)/(3^(1/4)*c*d*sqrt(c^3+4*d^3*x^3)*sqrt(c*(c+2^(2/3)*d*x)/(2^(2/3)*d*x+c*(1+sqrt(3)))^2))],

# Integrands of the form 1 / ((c+d x) Sqrt[a+b x^3]) with b^2 c^6-20 a b c^3 d^3-8 a^2 d^6=0
[1/((1+x+sqrt(3))*sqrt(1+x^3)),x,4,(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(3/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))+arctan((1+x)*sqrt(3+2*sqrt(3))/sqrt(1+x^3))/sqrt(3*(3+2*sqrt(3)))],
[1/((1-x+sqrt(3))*sqrt(1-x^3)),x,4,-(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(3/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))-arctan((1-x)*sqrt(3+2*sqrt(3))/sqrt(1-x^3))/sqrt(3*(3+2*sqrt(3)))],
[1/((1-x+sqrt(3))*sqrt(-1+x^3)),x,4,-(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(3/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))-arctanh((1-x)*sqrt(3+2*sqrt(3))/sqrt(-1+x^3))/sqrt(3*(3+2*sqrt(3)))],
[1/((1+x+sqrt(3))*sqrt(-1-x^3)),x,4,(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(3/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))+arctanh((1+x)*sqrt(3+2*sqrt(3))/sqrt(-1-x^3))/sqrt(3*(3+2*sqrt(3)))],

# Integrands of the form 1 / ((c+d x) Sqrt[a+b x^3])
[1/((3+x)*sqrt(1+x^3)),x,8,(1+x)*arctan(sqrt(13/2)*sqrt((1+x)/(1+x+sqrt(3))^2)/sqrt((1-x+x^2)/(1+x+sqrt(3))^2))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(26)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))+4*3^(1/4)*(1+x)*EllipticPi((-1-x+sqrt(3))/(1+x+sqrt(3)),97-56*sqrt(3),sqrt(-7-4*sqrt(3)))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(1+x^3)*sqrt(2-sqrt(3))*sqrt((1+x)/(1+x+sqrt(3))^2))+2*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)*sqrt(26+15*sqrt(3))/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[1/((3+x)*sqrt(1-x^3)),x,8,-1/2*(1-x)*arctanh(1/2*sqrt(7)*sqrt((1-x)/(1-x+sqrt(3))^2)/sqrt((1+x+x^2)/(1-x+sqrt(3))^2))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(7)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))+4/13*3^(1/4)*(1-x)*EllipticPi((-1+x+sqrt(3))/(1-x+sqrt(3)),1/169*(553+304*sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))-2*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*(4+sqrt(3))*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[1/((3+x)*sqrt(-1+x^3)),x,8,-2/13*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(62-35*sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))-1/2*(1-x)*arctanh(1/2*sqrt(7)*sqrt((1-x)/(1-x+sqrt(3))^2)/sqrt((1+x+x^2)/(1-x+sqrt(3))^2))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(7)*sqrt(-1+x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))+4/13*3^(1/4)*(1-x)*EllipticPi((-1+x+sqrt(3))/(1-x+sqrt(3)),1/169*(553+304*sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(-1+x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[1/((3+x)*sqrt(-1-x^3)),x,8,2*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(-1-x^3)*sqrt(2-sqrt(3))*sqrt((-1-x)/(1+x-sqrt(3))^2))+(1+x)*arctan(sqrt(13/2)*sqrt((1+x)/(1+x+sqrt(3))^2)/sqrt((1-x+x^2)/(1+x+sqrt(3))^2))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(26)*sqrt(-1-x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))+4*3^(1/4)*(1+x)*EllipticPi((-1-x+sqrt(3))/(1+x+sqrt(3)),97-56*sqrt(3),sqrt(-7-4*sqrt(3)))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(-1-x^3)*sqrt(2-sqrt(3))*sqrt((1+x)/(1+x+sqrt(3))^2))],

# Integrands of the form 1 / ((c+d x) (a+b x^3)^(1/3)) when b c^3+a d^3=0
[1/((c+d*x)*(-c^3+d^3*x^3)^(1/3)),x,1,1/4*log((c-d*x)*(c+d*x)^2)/(2^(1/3)*c*d)-3/4*log(d*(c-d*x)+2^(2/3)*d*(-c^3+d^3*x^3)^(1/3))/(2^(1/3)*c*d)+1/2*arctan((1-2^(1/3)*(c-d*x)/(-c^3+d^3*x^3)^(1/3))/sqrt(3))*sqrt(3)/(2^(1/3)*c*d)],

# Integrands of the form 1 / ((c+d x) (a+b x^3)^(1/3)) when 2 b c^3-a d^3=0
[1/((c+d*x)*(2*c^3+d^3*x^3)^(1/3)),x,3,-1/2*log(c+d*x)/(c*d)-1/4*log(-d*x+(2*c^3+d^3*x^3)^(1/3))/(c*d)+3/4*log(d*(2*c+d*x)-d*(2*c^3+d^3*x^3)^(1/3))/(c*d)+1/2*arctan((1+2*d*x/(2*c^3+d^3*x^3)^(1/3))/sqrt(3))/(c*d*sqrt(3))-1/2*arctan((1+2*(2*c+d*x)/(2*c^3+d^3*x^3)^(1/3))/sqrt(3))*sqrt(3)/(c*d)],

# Integrands of the form 1 / ((c+d x) (a+b x^3)^(2/3)) when 2 b c^3-a d^3=0
[1/((c+d*x)*(2*c^3+d^3*x^3)^(2/3)),x,1,-1/2*log(c+d*x)/(c^2*d)-1/4*log(d*x-(2*c^3+d^3*x^3)^(1/3))/(c^2*d)+3/4*log(d*(2*c+d*x)-d*(2*c^3+d^3*x^3)^(1/3))/(c^2*d)-1/2*arctan((1+2*d*x/(2*c^3+d^3*x^3)^(1/3))/sqrt(3))/(c^2*d*sqrt(3))+1/2*arctan((1+2*(2*c+d*x)/(2*c^3+d^3*x^3)^(1/3))/sqrt(3))*sqrt(3)/(c^2*d)],
[1/((1+2^(1/3)*x)*(1+x^3)^(2/3)),x,1,-log(1+2^(1/3)*x)/2^(2/3)-1/2*log(x-(1+x^3)^(1/3))/2^(2/3)+3/2*log(2+2^(1/3)*x-2^(1/3)*(1+x^3)^(1/3))/2^(2/3)-arctan((1+2*x/(1+x^3)^(1/3))/sqrt(3))/(2^(2/3)*sqrt(3))+arctan((1+2*(2^(2/3)+x)/(1+x^3)^(1/3))/sqrt(3))*sqrt(3)/2^(2/3)],
[1/((1-2^(1/3)*x)*(1-x^3)^(2/3)),x,1,log(1-2^(1/3)*x)/2^(2/3)+1/2*log(-x-(1-x^3)^(1/3))/2^(2/3)-3/2*log(-2+2^(1/3)*x+2^(1/3)*(1-x^3)^(1/3))/2^(2/3)+arctan((1-2*x/(1-x^3)^(1/3))/sqrt(3))/(2^(2/3)*sqrt(3))-arctan((1+(2*2^(2/3)-2*x)/(1-x^3)^(1/3))/sqrt(3))*sqrt(3)/2^(2/3)],

# Integrands of the form (c+d x)^m (a+b x^3)^(p/3)

# p>0
[(c+d*x)^4*(a+b*x^3)^(1/3),x,11,3/2*a*c^2*d^2*(a+b*x^3)^(1/3)/b+1/18*a*d^4*x^2*(a+b*x^3)^(1/3)/b+1/30*(a+b*x^3)^(1/3)*(15*c^4*x+40*c^3*d*x^2+45*c^2*d^2*x^3+24*c*d^3*x^4+5*d^4*x^5)+1/2*a*c^4*x*(1+b*x^3/a)^(2/3)*hypergeom([1/3,2/3],[4/3],-b*x^3/a)/(a+b*x^3)^(2/3)+1/5*a*c*d^3*x^4*(1+b*x^3/a)^(2/3)*hypergeom([2/3,4/3],[7/3],-b*x^3/a)/(a+b*x^3)^(2/3)-2/3*a*c^3*d*log(b^(1/3)*x-(a+b*x^3)^(1/3))/b^(2/3)+1/18*a^2*d^4*log(b^(1/3)*x-(a+b*x^3)^(1/3))/b^(5/3)-4/3*a*c^3*d*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(2/3)*sqrt(3))+1/9*a^2*d^4*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(5/3)*sqrt(3))],
[(c+d*x)^3*(a+b*x^3)^(1/3),x,9,3/4*a*c*d^2*(a+b*x^3)^(1/3)/b+1/10*a*d^3*x*(a+b*x^3)^(1/3)/b+1/20*(a+b*x^3)^(1/3)*(10*c^3*x+20*c^2*d*x^2+15*c*d^2*x^3+4*d^3*x^4)+1/10*a*(5*b*c^3-a*d^3)*x*(1+b*x^3/a)^(2/3)*hypergeom([1/3,2/3],[4/3],-b*x^3/a)/(b*(a+b*x^3)^(2/3))-1/2*a*c^2*d*log(b^(1/3)*x-(a+b*x^3)^(1/3))/b^(2/3)-a*c^2*d*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(2/3)*sqrt(3))],
[(c+d*x)^2*(a+b*x^3)^(1/3),x,8,1/4*a*d^2*(a+b*x^3)^(1/3)/b+1/12*(a+b*x^3)^(1/3)*(6*c^2*x+8*c*d*x^2+3*d^2*x^3)+1/2*a*c^2*x*(1+b*x^3/a)^(2/3)*hypergeom([1/3,2/3],[4/3],-b*x^3/a)/(a+b*x^3)^(2/3)-1/3*a*c*d*log(b^(1/3)*x-(a+b*x^3)^(1/3))/b^(2/3)-2/3*a*c*d*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(2/3)*sqrt(3))],
[(c+d*x)*(a+b*x^3)^(1/3),x,6,1/6*(3*c*x+2*d*x^2)*(a+b*x^3)^(1/3)+1/2*a*c*x*(1+b*x^3/a)^(2/3)*hypergeom([1/3,2/3],[4/3],-b*x^3/a)/(a+b*x^3)^(2/3)-1/6*a*d*log(b^(1/3)*x-(a+b*x^3)^(1/3))/b^(2/3)-1/3*a*d*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(2/3)*sqrt(3))],
[(a+b*x^3)^(1/3)/(c+d*x),x,13,(a+b*x^3)^(1/3)/d+x*(a+b*x^3)^(1/3)*AppellF1(1/3,-1/3,1,4/3,-b*x^3/a,-d^3*x^3/c^3)/(c*(1+b*x^3/a)^(1/3))+1/3*(b*c^3-a*d^3)^(1/3)*log(c^3+d^3*x^3)/d^2+1/2*b^(1/3)*c*log(b^(1/3)*x-(a+b*x^3)^(1/3))/d^2-1/2*(b*c^3-a*d^3)^(1/3)*log((b*c^3-a*d^3)^(1/3)*x/c-(a+b*x^3)^(1/3))/d^2-1/2*(b*c^3-a*d^3)^(1/3)*log((b*c^3-a*d^3)^(1/3)+d*(a+b*x^3)^(1/3))/d^2+b^(1/3)*c*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(d^2*sqrt(3))-(b*c^3-a*d^3)^(1/3)*arctan((1+2*(b*c^3-a*d^3)^(1/3)*x/(c*(a+b*x^3)^(1/3)))/sqrt(3))/(d^2*sqrt(3))+(b*c^3-a*d^3)^(1/3)*arctan((1-2*d*(a+b*x^3)^(1/3)/(b*c^3-a*d^3)^(1/3))/sqrt(3))/(d^2*sqrt(3))],
[(a+b*x^3)^(1/3)/(c+d*x)^2,x,20,-c^2*(a+b*x^3)^(1/3)/(d*(c^3+d^3*x^3))-d*x^2*(a+b*x^3)^(1/3)/(c^3+d^3*x^3)+x*(a+b*x^3)^(1/3)*AppellF1(1/3,-1/3,2,4/3,-b*x^3/a,-d^3*x^3/c^3)/(c^2*(1+b*x^3/a)^(1/3))-1/2*d^3*x^4*(a+b*x^3)^(1/3)*AppellF1(4/3,-1/3,2,7/3,-b*x^3/a,-d^3*x^3/c^3)/(c^5*(1+b*x^3/a)^(1/3))-1/6*b*c^2*log(c^3+d^3*x^3)/(d^2*(b*c^3-a*d^3)^(2/3))-1/9*a*d*log(c^3+d^3*x^3)/(c*(b*c^3-a*d^3)^(2/3))-1/18*(3*b*c^3-2*a*d^3)*log(c^3+d^3*x^3)/(c*d^2*(b*c^3-a*d^3)^(2/3))-1/2*b^(1/3)*log(b^(1/3)*x-(a+b*x^3)^(1/3))/d^2+1/3*a*d*log((b*c^3-a*d^3)^(1/3)*x/c-(a+b*x^3)^(1/3))/(c*(b*c^3-a*d^3)^(2/3))+1/6*(3*b*c^3-2*a*d^3)*log((b*c^3-a*d^3)^(1/3)*x/c-(a+b*x^3)^(1/3))/(c*d^2*(b*c^3-a*d^3)^(2/3))+1/2*b*c^2*log((b*c^3-a*d^3)^(1/3)+d*(a+b*x^3)^(1/3))/(d^2*(b*c^3-a*d^3)^(2/3))-b^(1/3)*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(d^2*sqrt(3))+2/3*a*d*arctan((1+2*(b*c^3-a*d^3)^(1/3)*x/(c*(a+b*x^3)^(1/3)))/sqrt(3))/(c*(b*c^3-a*d^3)^(2/3)*sqrt(3))+1/3*(3*b*c^3-2*a*d^3)*arctan((1+2*(b*c^3-a*d^3)^(1/3)*x/(c*(a+b*x^3)^(1/3)))/sqrt(3))/(c*d^2*(b*c^3-a*d^3)^(2/3)*sqrt(3))-b*c^2*arctan((1-2*d*(a+b*x^3)^(1/3)/(b*c^3-a*d^3)^(1/3))/sqrt(3))/(d^2*(b*c^3-a*d^3)^(2/3)*sqrt(3))],

# p<0
[(c+d*x)^4/(a+b*x^3)^(1/3),x,10,3*c^2*d^2*(a+b*x^3)^(2/3)/b+4/3*c*d^3*x*(a+b*x^3)^(2/3)/b+2*c^3*d*x^2*(1+b*x^3/a)^(1/3)*hypergeom([1/3,2/3],[5/3],-b*x^3/a)/(a+b*x^3)^(1/3)+1/5*d^4*x^5*(1+b*x^3/a)^(1/3)*hypergeom([1/3,5/3],[8/3],-b*x^3/a)/(a+b*x^3)^(1/3)-1/2*c^4*log(-b^(1/3)*x+(a+b*x^3)^(1/3))/b^(1/3)+2/3*a*c*d^3*log(-b^(1/3)*x+(a+b*x^3)^(1/3))/b^(4/3)+c^4*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(1/3)*sqrt(3))-4/3*a*c*d^3*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(4/3)*sqrt(3))],
[(c+d*x)^3/(a+b*x^3)^(1/3),x,8,3/2*c*d^2*(a+b*x^3)^(2/3)/b+1/3*d^3*x*(a+b*x^3)^(2/3)/b+3/2*c^2*d*x^2*(1+b*x^3/a)^(1/3)*hypergeom([1/3,2/3],[5/3],-b*x^3/a)/(a+b*x^3)^(1/3)-1/2*c^3*log(-b^(1/3)*x+(a+b*x^3)^(1/3))/b^(1/3)+1/6*a*d^3*log(-b^(1/3)*x+(a+b*x^3)^(1/3))/b^(4/3)+c^3*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(1/3)*sqrt(3))-1/3*a*d^3*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(4/3)*sqrt(3))],
[(c+d*x)^2/(a+b*x^3)^(1/3),x,7,1/2*d^2*(a+b*x^3)^(2/3)/b+c*d*x^2*(1+b*x^3/a)^(1/3)*hypergeom([1/3,2/3],[5/3],-b*x^3/a)/(a+b*x^3)^(1/3)-1/2*c^2*log(-b^(1/3)*x+(a+b*x^3)^(1/3))/b^(1/3)+c^2*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(1/3)*sqrt(3))],
[(c+d*x)/(a+b*x^3)^(1/3),x,5,1/2*d*x^2*(1+b*x^3/a)^(1/3)*hypergeom([1/3,2/3],[5/3],-b*x^3/a)/(a+b*x^3)^(1/3)-1/2*c*log(-b^(1/3)*x+(a+b*x^3)^(1/3))/b^(1/3)+c*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(1/3)*sqrt(3))],
[1/((c+d*x)*(a+b*x^3)^(1/3)),x,10,-1/2*d*x^2*(1+b*x^3/a)^(1/3)*AppellF1(2/3,1/3,1,5/3,-b*x^3/a,-d^3*x^3/c^3)/(c^2*(a+b*x^3)^(1/3))+1/3*log(c^3+d^3*x^3)/(b*c^3-a*d^3)^(1/3)-1/2*log((b*c^3-a*d^3)^(1/3)*x/c-(a+b*x^3)^(1/3))/(b*c^3-a*d^3)^(1/3)-1/2*log((b*c^3-a*d^3)^(1/3)+d*(a+b*x^3)^(1/3))/(b*c^3-a*d^3)^(1/3)+arctan((1+2*(b*c^3-a*d^3)^(1/3)*x/(c*(a+b*x^3)^(1/3)))/sqrt(3))/((b*c^3-a*d^3)^(1/3)*sqrt(3))-arctan((1-2*d*(a+b*x^3)^(1/3)/(b*c^3-a*d^3)^(1/3))/sqrt(3))/((b*c^3-a*d^3)^(1/3)*sqrt(3))],
[1/((c+d*x)^2*(a+b*x^3)^(1/3)),x,17,c^2*d^2*(a+b*x^3)^(2/3)/((b*c^3-a*d^3)*(c^3+d^3*x^3))-c*d^3*x*(a+b*x^3)^(2/3)/((b*c^3-a*d^3)*(c^3+d^3*x^3))-d*x^2*(1+b*x^3/a)^(1/3)*AppellF1(2/3,1/3,2,5/3,-b*x^3/a,-d^3*x^3/c^3)/(c^3*(a+b*x^3)^(1/3))+1/5*d^4*x^5*(1+b*x^3/a)^(1/3)*AppellF1(5/3,1/3,2,8/3,-b*x^3/a,-d^3*x^3/c^3)/(c^6*(a+b*x^3)^(1/3))+1/6*b*c^2*log(c^3+d^3*x^3)/(b*c^3-a*d^3)^(4/3)+1/9*a*d^3*log(c^3+d^3*x^3)/(c*(b*c^3-a*d^3)^(4/3))+1/18*(3*b*c^3-2*a*d^3)*log(c^3+d^3*x^3)/(c*(b*c^3-a*d^3)^(4/3))-1/3*a*d^3*log((b*c^3-a*d^3)^(1/3)*x/c-(a+b*x^3)^(1/3))/(c*(b*c^3-a*d^3)^(4/3))-1/6*(3*b*c^3-2*a*d^3)*log((b*c^3-a*d^3)^(1/3)*x/c-(a+b*x^3)^(1/3))/(c*(b*c^3-a*d^3)^(4/3))-1/2*b*c^2*log((b*c^3-a*d^3)^(1/3)+d*(a+b*x^3)^(1/3))/(b*c^3-a*d^3)^(4/3)+2/3*a*d^3*arctan((1+2*(b*c^3-a*d^3)^(1/3)*x/(c*(a+b*x^3)^(1/3)))/sqrt(3))/(c*(b*c^3-a*d^3)^(4/3)*sqrt(3))+1/3*(3*b*c^3-2*a*d^3)*arctan((1+2*(b*c^3-a*d^3)^(1/3)*x/(c*(a+b*x^3)^(1/3)))/sqrt(3))/(c*(b*c^3-a*d^3)^(4/3)*sqrt(3))-b*c^2*arctan((1-2*d*(a+b*x^3)^(1/3)/(b*c^3-a*d^3)^(1/3))/sqrt(3))/((b*c^3-a*d^3)^(4/3)*sqrt(3))],
[1/((c+d*x)^3*(a+b*x^3)^(1/3)),x,32,3/2*c^4*d^2*(a+b*x^3)^(2/3)/((b*c^3-a*d^3)*(c^3+d^3*x^3)^2)-3/2*c^3*d^3*x*(a+b*x^3)^(2/3)/((b*c^3-a*d^3)*(c^3+d^3*x^3)^2)+4/3*b*c^4*d^2*(a+b*x^3)^(2/3)/((b*c^3-a*d^3)^2*(c^3+d^3*x^3))-1/3*c*d^2*(b*c^3-3*a*d^3)*(a+b*x^3)^(2/3)/((b*c^3-a*d^3)^2*(c^3+d^3*x^3))+1/18*d^3*(3*b*c^3-7*a*d^3)*x*(a+b*x^3)^(2/3)/((b*c^3-a*d^3)^2*(c^3+d^3*x^3))-1/18*d^3*(9*b*c^3-5*a*d^3)*x*(a+b*x^3)^(2/3)/((b*c^3-a*d^3)^2*(c^3+d^3*x^3))-7/18*d^3*(3*b*c^3+a*d^3)*x*(a+b*x^3)^(2/3)/((b*c^3-a*d^3)^2*(c^3+d^3*x^3))-3/2*d*x^2*(1+b*x^3/a)^(1/3)*AppellF1(2/3,1/3,3,5/3,-b*x^3/a,-d^3*x^3/c^3)/(c^4*(a+b*x^3)^(1/3))+6/5*d^4*x^5*(1+b*x^3/a)^(1/3)*AppellF1(5/3,1/3,3,8/3,-b*x^3/a,-d^3*x^3/c^3)/(c^7*(a+b*x^3)^(1/3))+2/9*b^2*c^4*log(c^3+d^3*x^3)/(b*c^3-a*d^3)^(7/3)+1/27*a^2*d^6*log(c^3+d^3*x^3)/(c^2*(b*c^3-a*d^3)^(7/3))-1/18*b*c*(b*c^3-3*a*d^3)*log(c^3+d^3*x^3)/(b*c^3-a*d^3)^(7/3)+7/54*a*d^3*(3*b*c^3-a*d^3)*log(c^3+d^3*x^3)/(c^2*(b*c^3-a*d^3)^(7/3))+1/54*(9*b^2*c^6-12*a*b*c^3*d^3+5*a^2*d^6)*log(c^3+d^3*x^3)/(c^2*(b*c^3-a*d^3)^(7/3))-1/9*a^2*d^6*log((b*c^3-a*d^3)^(1/3)*x/c-(a+b*x^3)^(1/3))/(c^2*(b*c^3-a*d^3)^(7/3))-7/18*a*d^3*(3*b*c^3-a*d^3)*log((b*c^3-a*d^3)^(1/3)*x/c-(a+b*x^3)^(1/3))/(c^2*(b*c^3-a*d^3)^(7/3))-1/18*(9*b^2*c^6-12*a*b*c^3*d^3+5*a^2*d^6)*log((b*c^3-a*d^3)^(1/3)*x/c-(a+b*x^3)^(1/3))/(c^2*(b*c^3-a*d^3)^(7/3))-2/3*b^2*c^4*log((b*c^3-a*d^3)^(1/3)+d*(a+b*x^3)^(1/3))/(b*c^3-a*d^3)^(7/3)+1/6*b*c*(b*c^3-3*a*d^3)*log((b*c^3-a*d^3)^(1/3)+d*(a+b*x^3)^(1/3))/(b*c^3-a*d^3)^(7/3)+2/9*a^2*d^6*arctan((1+2*(b*c^3-a*d^3)^(1/3)*x/(c*(a+b*x^3)^(1/3)))/sqrt(3))/(c^2*(b*c^3-a*d^3)^(7/3)*sqrt(3))+7/9*a*d^3*(3*b*c^3-a*d^3)*arctan((1+2*(b*c^3-a*d^3)^(1/3)*x/(c*(a+b*x^3)^(1/3)))/sqrt(3))/(c^2*(b*c^3-a*d^3)^(7/3)*sqrt(3))+1/9*(9*b^2*c^6-12*a*b*c^3*d^3+5*a^2*d^6)*arctan((1+2*(b*c^3-a*d^3)^(1/3)*x/(c*(a+b*x^3)^(1/3)))/sqrt(3))/(c^2*(b*c^3-a*d^3)^(7/3)*sqrt(3))-4/3*b^2*c^4*arctan((1-2*d*(a+b*x^3)^(1/3)/(b*c^3-a*d^3)^(1/3))/sqrt(3))/((b*c^3-a*d^3)^(7/3)*sqrt(3))+1/3*b*c*(b*c^3-3*a*d^3)*arctan((1-2*d*(a+b*x^3)^(1/3)/(b*c^3-a*d^3)^(1/3))/sqrt(3))/((b*c^3-a*d^3)^(7/3)*sqrt(3))],
[(c+d*x)^4/(a+b*x^3)^(2/3),x,10,6*c^2*d^2*(a+b*x^3)^(1/3)/b+1/3*d^4*x^2*(a+b*x^3)^(1/3)/b+c^4*x*(1+b*x^3/a)^(2/3)*hypergeom([1/3,2/3],[4/3],-b*x^3/a)/(a+b*x^3)^(2/3)+c*d^3*x^4*(1+b*x^3/a)^(2/3)*hypergeom([2/3,4/3],[7/3],-b*x^3/a)/(a+b*x^3)^(2/3)-2*c^3*d*log(b^(1/3)*x-(a+b*x^3)^(1/3))/b^(2/3)+1/3*a*d^4*log(b^(1/3)*x-(a+b*x^3)^(1/3))/b^(5/3)-4*c^3*d*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(2/3)*sqrt(3))+2/3*a*d^4*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(5/3)*sqrt(3))],
[(c+d*x)^3/(a+b*x^3)^(2/3),x,8,3*c*d^2*(a+b*x^3)^(1/3)/b+1/2*d^3*x*(a+b*x^3)^(1/3)/b+1/2*(2*b*c^3-a*d^3)*x*(1+b*x^3/a)^(2/3)*hypergeom([1/3,2/3],[4/3],-b*x^3/a)/(b*(a+b*x^3)^(2/3))-3/2*c^2*d*log(b^(1/3)*x-(a+b*x^3)^(1/3))/b^(2/3)-c^2*d*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))*sqrt(3)/b^(2/3)],
[(c+d*x)^2/(a+b*x^3)^(2/3),x,7,d^2*(a+b*x^3)^(1/3)/b+c^2*x*(1+b*x^3/a)^(2/3)*hypergeom([1/3,2/3],[4/3],-b*x^3/a)/(a+b*x^3)^(2/3)-c*d*log(b^(1/3)*x-(a+b*x^3)^(1/3))/b^(2/3)-2*c*d*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(2/3)*sqrt(3))],
[(c+d*x)/(a+b*x^3)^(2/3),x,5,c*x*(1+b*x^3/a)^(2/3)*hypergeom([1/3,2/3],[4/3],-b*x^3/a)/(a+b*x^3)^(2/3)-1/2*d*log(b^(1/3)*x-(a+b*x^3)^(1/3))/b^(2/3)-d*arctan((1+2*b^(1/3)*x/(a+b*x^3)^(1/3))/sqrt(3))/(b^(2/3)*sqrt(3))],
[1/((c+d*x)*(a+b*x^3)^(2/3)),x,10,x*(1+b*x^3/a)^(2/3)*AppellF1(1/3,2/3,1,4/3,-b*x^3/a,-d^3*x^3/c^3)/(c*(a+b*x^3)^(2/3))-1/3*d*log(c^3+d^3*x^3)/(b*c^3-a*d^3)^(2/3)+1/2*d*log((b*c^3-a*d^3)^(1/3)*x/c-(a+b*x^3)^(1/3))/(b*c^3-a*d^3)^(2/3)+1/2*d*log((b*c^3-a*d^3)^(1/3)+d*(a+b*x^3)^(1/3))/(b*c^3-a*d^3)^(2/3)+d*arctan((1+2*(b*c^3-a*d^3)^(1/3)*x/(c*(a+b*x^3)^(1/3)))/sqrt(3))/((b*c^3-a*d^3)^(2/3)*sqrt(3))-d*arctan((1-2*d*(a+b*x^3)^(1/3)/(b*c^3-a*d^3)^(1/3))/sqrt(3))/((b*c^3-a*d^3)^(2/3)*sqrt(3))],
[1/((c+d*x)^2*(a+b*x^3)^(2/3)),x,18,c^2*d^2*(a+b*x^3)^(1/3)/((b*c^3-a*d^3)*(c^3+d^3*x^3))+d^4*x^2*(a+b*x^3)^(1/3)/((b*c^3-a*d^3)*(c^3+d^3*x^3))+x*(1+b*x^3/a)^(2/3)*AppellF1(1/3,2/3,2,4/3,-b*x^3/a,-d^3*x^3/c^3)/(c^2*(a+b*x^3)^(2/3))-1/2*d^3*x^4*(1+b*x^3/a)^(2/3)*AppellF1(4/3,2/3,2,7/3,-b*x^3/a,-d^3*x^3/c^3)/(c^5*(a+b*x^3)^(2/3))-1/3*b*c^2*d*log(c^3+d^3*x^3)/(b*c^3-a*d^3)^(5/3)-1/9*a*d^4*log(c^3+d^3*x^3)/(c*(b*c^3-a*d^3)^(5/3))-1/9*d*(3*b*c^3-a*d^3)*log(c^3+d^3*x^3)/(c*(b*c^3-a*d^3)^(5/3))+1/3*a*d^4*log((b*c^3-a*d^3)^(1/3)*x/c-(a+b*x^3)^(1/3))/(c*(b*c^3-a*d^3)^(5/3))+1/3*d*(3*b*c^3-a*d^3)*log((b*c^3-a*d^3)^(1/3)*x/c-(a+b*x^3)^(1/3))/(c*(b*c^3-a*d^3)^(5/3))+b*c^2*d*log((b*c^3-a*d^3)^(1/3)+d*(a+b*x^3)^(1/3))/(b*c^3-a*d^3)^(5/3)+2/3*a*d^4*arctan((1+2*(b*c^3-a*d^3)^(1/3)*x/(c*(a+b*x^3)^(1/3)))/sqrt(3))/(c*(b*c^3-a*d^3)^(5/3)*sqrt(3))+2/3*d*(3*b*c^3-a*d^3)*arctan((1+2*(b*c^3-a*d^3)^(1/3)*x/(c*(a+b*x^3)^(1/3)))/sqrt(3))/(c*(b*c^3-a*d^3)^(5/3)*sqrt(3))-2*b*c^2*d*arctan((1-2*d*(a+b*x^3)^(1/3)/(b*c^3-a*d^3)^(1/3))/sqrt(3))/((b*c^3-a*d^3)^(5/3)*sqrt(3))],
[1/((c+d*x)^3*(a+b*x^3)^(2/3)),x,30,3/2*c^4*d^2*(a+b*x^3)^(1/3)/((b*c^3-a*d^3)*(c^3+d^3*x^3)^2)+3/2*c^2*d^4*x^2*(a+b*x^3)^(1/3)/((b*c^3-a*d^3)*(c^3+d^3*x^3)^2)+5/3*b*c^4*d^2*(a+b*x^3)^(1/3)/((b*c^3-a*d^3)^2*(c^3+d^3*x^3))-1/6*c*d^2*(b*c^3-6*a*d^3)*(a+b*x^3)^(1/3)/((b*c^3-a*d^3)^2*(c^3+d^3*x^3))+1/6*d^4*(9*b*c^3-4*a*d^3)*x^2*(a+b*x^3)^(1/3)/(c*(b*c^3-a*d^3)^2*(c^3+d^3*x^3))+1/3*d^4*(3*b*c^3+2*a*d^3)*x^2*(a+b*x^3)^(1/3)/(c*(b*c^3-a*d^3)^2*(c^3+d^3*x^3))+x*(1+b*x^3/a)^(2/3)*AppellF1(1/3,2/3,3,4/3,-b*x^3/a,-d^3*x^3/c^3)/(c^3*(a+b*x^3)^(2/3))-7/4*d^3*x^4*(1+b*x^3/a)^(2/3)*AppellF1(4/3,2/3,3,7/3,-b*x^3/a,-d^3*x^3/c^3)/(c^6*(a+b*x^3)^(2/3))+1/7*d^6*x^7*(1+b*x^3/a)^(2/3)*AppellF1(7/3,2/3,3,10/3,-b*x^3/a,-d^3*x^3/c^3)/(c^9*(a+b*x^3)^(2/3))-5/9*b^2*c^4*d*log(c^3+d^3*x^3)/(b*c^3-a*d^3)^(8/3)+1/18*b*c*d*(b*c^3-6*a*d^3)*log(c^3+d^3*x^3)/(b*c^3-a*d^3)^(8/3)-1/9*a*d^4*(6*b*c^3-a*d^3)*log(c^3+d^3*x^3)/(c^2*(b*c^3-a*d^3)^(8/3))-1/18*d*(9*b^2*c^6-6*a*b*c^3*d^3+2*a^2*d^6)*log(c^3+d^3*x^3)/(c^2*(b*c^3-a*d^3)^(8/3))+1/3*a*d^4*(6*b*c^3-a*d^3)*log((b*c^3-a*d^3)^(1/3)*x/c-(a+b*x^3)^(1/3))/(c^2*(b*c^3-a*d^3)^(8/3))+1/6*d*(9*b^2*c^6-6*a*b*c^3*d^3+2*a^2*d^6)*log((b*c^3-a*d^3)^(1/3)*x/c-(a+b*x^3)^(1/3))/(c^2*(b*c^3-a*d^3)^(8/3))+5/3*b^2*c^4*d*log((b*c^3-a*d^3)^(1/3)+d*(a+b*x^3)^(1/3))/(b*c^3-a*d^3)^(8/3)-1/6*b*c*d*(b*c^3-6*a*d^3)*log((b*c^3-a*d^3)^(1/3)+d*(a+b*x^3)^(1/3))/(b*c^3-a*d^3)^(8/3)+2/3*a*d^4*(6*b*c^3-a*d^3)*arctan((1+2*(b*c^3-a*d^3)^(1/3)*x/(c*(a+b*x^3)^(1/3)))/sqrt(3))/(c^2*(b*c^3-a*d^3)^(8/3)*sqrt(3))+1/3*d*(9*b^2*c^6-6*a*b*c^3*d^3+2*a^2*d^6)*arctan((1+2*(b*c^3-a*d^3)^(1/3)*x/(c*(a+b*x^3)^(1/3)))/sqrt(3))/(c^2*(b*c^3-a*d^3)^(8/3)*sqrt(3))-10/3*b^2*c^4*d*arctan((1-2*d*(a+b*x^3)^(1/3)/(b*c^3-a*d^3)^(1/3))/sqrt(3))/((b*c^3-a*d^3)^(8/3)*sqrt(3))+1/3*b*c*d*(b*c^3-6*a*d^3)*arctan((1-2*d*(a+b*x^3)^(1/3)/(b*c^3-a*d^3)^(1/3))/sqrt(3))/((b*c^3-a*d^3)^(8/3)*sqrt(3))],

# Integrands of the form (c+d x)^m (e+f x)^n (a+b x^3)^p

# Integrands of the form (e+f x) / ((c+d x) Sqrt[a+b x^3]) with b c^3 - 4 a d^3=0

# 2 d e+c f = 0
[(2^(2/3)-2*x)/((2^(2/3)+x)*sqrt(1+x^3)),x,2,2*2^(2/3)*arctan((1+2^(1/3)*x)*sqrt(3)/sqrt(1+x^3))/sqrt(3)],
[(2^(2/3)+2*x)/((2^(2/3)-x)*sqrt(1-x^3)),x,2,-2*2^(2/3)*arctan((1-2^(1/3)*x)*sqrt(3)/sqrt(1-x^3))/sqrt(3)],
[(2^(2/3)+2*x)/((2^(2/3)-x)*sqrt(-1+x^3)),x,2,-2*2^(2/3)*arctanh((1-2^(1/3)*x)*sqrt(3)/sqrt(-1+x^3))/sqrt(3)],
[(2^(2/3)-2*x)/((2^(2/3)+x)*sqrt(-1-x^3)),x,2,2*2^(2/3)*arctanh((1+2^(1/3)*x)*sqrt(3)/sqrt(-1-x^3))/sqrt(3)],
[(2^(2/3)*a^(1/3)-2*b^(1/3)*x)/((2^(2/3)*a^(1/3)+b^(1/3)*x)*sqrt(a+b*x^3)),x,2,2*2^(2/3)*arctan(a^(1/6)*(a^(1/3)+2^(1/3)*b^(1/3)*x)*sqrt(3)/sqrt(a+b*x^3))/(a^(1/6)*b^(1/3)*sqrt(3))],
[(2^(2/3)*a^(1/3)+2*b^(1/3)*x)/((2^(2/3)*a^(1/3)-b^(1/3)*x)*sqrt(a-b*x^3)),x,2,-2*2^(2/3)*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*b^(1/3)*x)*sqrt(3)/sqrt(a-b*x^3))/(a^(1/6)*b^(1/3)*sqrt(3))],
[(2^(2/3)*a^(1/3)+2*b^(1/3)*x)/((2^(2/3)*a^(1/3)-b^(1/3)*x)*sqrt(-a+b*x^3)),x,2,-2*2^(2/3)*arctanh(a^(1/6)*(a^(1/3)-2^(1/3)*b^(1/3)*x)*sqrt(3)/sqrt(-a+b*x^3))/(a^(1/6)*b^(1/3)*sqrt(3))],
[(2^(2/3)*a^(1/3)-2*b^(1/3)*x)/((2^(2/3)*a^(1/3)+b^(1/3)*x)*sqrt(-a-b*x^3)),x,2,2*2^(2/3)*arctanh(a^(1/6)*(a^(1/3)+2^(1/3)*b^(1/3)*x)*sqrt(3)/sqrt(-a-b*x^3))/(a^(1/6)*b^(1/3)*sqrt(3))],
[(c-2*d*x)/((c+d*x)*sqrt(c^3+4*d^3*x^3)),x,2,2*arctan((c+2*d*x)*sqrt(3)*sqrt(c)/sqrt(c^3+4*d^3*x^3))/(d*sqrt(3)*sqrt(c))],

# 2 d e+c f /= 0
[(2+3*x)/((2^(2/3)+x)*sqrt(1+x^3)),x,4,2/3*(2-3*2^(2/3))*arctan((1+2^(1/3)*x)*sqrt(3)/sqrt(1+x^3))/sqrt(3)+2/3*(3+2*2^(1/3))*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[(2+3*x)/((2^(2/3)-x)*sqrt(1-x^3)),x,4,-2/3*(2+3*2^(2/3))*arctan((1-2^(1/3)*x)*sqrt(3)/sqrt(1-x^3))/sqrt(3)+2/3*(3-2*2^(1/3))*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[(2+3*x)/((2^(2/3)-x)*sqrt(-1+x^3)),x,4,-2/3*(2+3*2^(2/3))*arctanh((1-2^(1/3)*x)*sqrt(3)/sqrt(-1+x^3))/sqrt(3)+2/3*(3-2*2^(1/3))*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[(2+3*x)/((2^(2/3)+x)*sqrt(-1-x^3)),x,4,2/3*(2-3*2^(2/3))*arctanh((1+2^(1/3)*x)*sqrt(3)/sqrt(-1-x^3))/sqrt(3)+2/3*(3+2*2^(1/3))*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[(e+f*x)/((2^(2/3)+x)*sqrt(1+x^3)),x,4,2/3*(e-2^(2/3)*f)*arctan((1+2^(1/3)*x)*sqrt(3)/sqrt(1+x^3))/sqrt(3)+2/3*(2^(1/3)*e+f)*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[(e+f*x)/((2^(2/3)-x)*sqrt(1-x^3)),x,4,-2/3*(e+2^(2/3)*f)*arctan((1-2^(1/3)*x)*sqrt(3)/sqrt(1-x^3))/sqrt(3)-2/3*(2^(1/3)*e-f)*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[(e+f*x)/((2^(2/3)-x)*sqrt(-1+x^3)),x,4,-2/3*(e+2^(2/3)*f)*arctanh((1-2^(1/3)*x)*sqrt(3)/sqrt(-1+x^3))/sqrt(3)-2/3*(2^(1/3)*e-f)*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[(e+f*x)/((2^(2/3)+x)*sqrt(-1-x^3)),x,4,2/3*(e-2^(2/3)*f)*arctanh((1+2^(1/3)*x)*sqrt(3)/sqrt(-1-x^3))/sqrt(3)+2/3*(2^(1/3)*e+f)*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[(e+f*x)/((2^(2/3)*a^(1/3)+b^(1/3)*x)*sqrt(a+b*x^3)),x,4,2/3*(b^(1/3)*e-2^(2/3)*a^(1/3)*f)*arctan(a^(1/6)*(a^(1/3)+2^(1/3)*b^(1/3)*x)*sqrt(3)/sqrt(a+b*x^3))/(b^(2/3)*sqrt(3)*sqrt(a))+2/3*(2^(1/3)*b^(1/3)*e+a^(1/3)*f)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^(1/3)*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(e+f*x)/((2^(2/3)*a^(1/3)-b^(1/3)*x)*sqrt(a-b*x^3)),x,4,-2/3*(b^(1/3)*e+2^(2/3)*a^(1/3)*f)*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*b^(1/3)*x)*sqrt(3)/sqrt(a-b*x^3))/(b^(2/3)*sqrt(3)*sqrt(a))-2/3*(2^(1/3)*b^(1/3)*e-a^(1/3)*f)*(a^(1/3)-b^(1/3)*x)*EllipticF((-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^(1/3)*b^(2/3)*sqrt(a-b*x^3)*sqrt(a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(e+f*x)/((2^(2/3)*a^(1/3)-b^(1/3)*x)*sqrt(-a+b*x^3)),x,4,-2/3*(b^(1/3)*e+2^(2/3)*a^(1/3)*f)*arctanh(a^(1/6)*(a^(1/3)-2^(1/3)*b^(1/3)*x)*sqrt(3)/sqrt(-a+b*x^3))/(b^(2/3)*sqrt(3)*sqrt(a))-2/3*(2^(1/3)*b^(1/3)*e-a^(1/3)*f)*(a^(1/3)-b^(1/3)*x)*EllipticF((-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2-sqrt(3))/(3^(1/4)*a^(1/3)*b^(2/3)*sqrt(-a+b*x^3)*sqrt(-a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))],
[(e+f*x)/((2^(2/3)*a^(1/3)+b^(1/3)*x)*sqrt(-a-b*x^3)),x,4,2/3*(b^(1/3)*e-2^(2/3)*a^(1/3)*f)*arctanh(a^(1/6)*(a^(1/3)+2^(1/3)*b^(1/3)*x)*sqrt(3)/sqrt(-a-b*x^3))/(b^(2/3)*sqrt(3)*sqrt(a))+2/3*(2^(1/3)*b^(1/3)*e+a^(1/3)*f)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2-sqrt(3))/(3^(1/4)*a^(1/3)*b^(2/3)*sqrt(-a-b*x^3)*sqrt(-a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))],
[(e+f*x)/((c+d*x)*sqrt(c^3+4*d^3*x^3)),x,4,2/3*(d*e-c*f)*arctan((c+2*d*x)*sqrt(3)*sqrt(c)/sqrt(c^3+4*d^3*x^3))/(c^(3/2)*d^2*sqrt(3))+1/3*2^(1/3)*(2*d*e+c*f)*(c+2^(2/3)*d*x)*EllipticF((2^(2/3)*d*x+c*(1-sqrt(3)))/(2^(2/3)*d*x+c*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((c^2-2^(2/3)*c*d*x+2*2^(1/3)*d^2*x^2)/(2^(2/3)*d*x+c*(1+sqrt(3)))^2)/(3^(1/4)*c*d^2*sqrt(c^3+4*d^3*x^3)*sqrt(c*(c+2^(2/3)*d*x)/(2^(2/3)*d*x+c*(1+sqrt(3)))^2))],

# e = 0
[x/((2^(2/3)+x)*sqrt(1+x^3)),x,4,-2/3*2^(2/3)*arctan((1+2^(1/3)*x)*sqrt(3)/sqrt(1+x^3))/sqrt(3)+2/3*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[x/((2^(2/3)-x)*sqrt(1-x^3)),x,4,-2/3*2^(2/3)*arctan((1-2^(1/3)*x)*sqrt(3)/sqrt(1-x^3))/sqrt(3)+2/3*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[x/((2^(2/3)-x)*sqrt(-1+x^3)),x,4,-2/3*2^(2/3)*arctanh((1-2^(1/3)*x)*sqrt(3)/sqrt(-1+x^3))/sqrt(3)+2/3*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[x/((2^(2/3)+x)*sqrt(-1-x^3)),x,4,-2/3*2^(2/3)*arctanh((1+2^(1/3)*x)*sqrt(3)/sqrt(-1-x^3))/sqrt(3)+2/3*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[x/((2^(2/3)*a^(1/3)+b^(1/3)*x)*sqrt(a+b*x^3)),x,4,-2/3*2^(2/3)*arctan(a^(1/6)*(a^(1/3)+2^(1/3)*b^(1/3)*x)*sqrt(3)/sqrt(a+b*x^3))/(a^(1/6)*b^(2/3)*sqrt(3))+2/3*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x/((2^(2/3)*a^(1/3)-b^(1/3)*x)*sqrt(a-b*x^3)),x,4,-2/3*2^(2/3)*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*b^(1/3)*x)*sqrt(3)/sqrt(a-b*x^3))/(a^(1/6)*b^(2/3)*sqrt(3))+2/3*(a^(1/3)-b^(1/3)*x)*EllipticF((-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(2/3)*sqrt(a-b*x^3)*sqrt(a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x/((2^(2/3)*a^(1/3)-b^(1/3)*x)*sqrt(-a+b*x^3)),x,4,-2/3*2^(2/3)*arctanh(a^(1/6)*(a^(1/3)-2^(1/3)*b^(1/3)*x)*sqrt(3)/sqrt(-a+b*x^3))/(a^(1/6)*b^(2/3)*sqrt(3))+2/3*(a^(1/3)-b^(1/3)*x)*EllipticF((-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2-sqrt(3))/(3^(1/4)*b^(2/3)*sqrt(-a+b*x^3)*sqrt(-a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))],
[x/((2^(2/3)*a^(1/3)+b^(1/3)*x)*sqrt(-a-b*x^3)),x,4,-2/3*2^(2/3)*arctanh(a^(1/6)*(a^(1/3)+2^(1/3)*b^(1/3)*x)*sqrt(3)/sqrt(-a-b*x^3))/(a^(1/6)*b^(2/3)*sqrt(3))+2/3*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2-sqrt(3))/(3^(1/4)*b^(2/3)*sqrt(-a-b*x^3)*sqrt(-a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))],
[x/((c+d*x)*sqrt(c^3+4*d^3*x^3)),x,4,-2/3*arctan((c+2*d*x)*sqrt(3)*sqrt(c)/sqrt(c^3+4*d^3*x^3))/(d^2*sqrt(3)*sqrt(c))+1/3*2^(1/3)*(c+2^(2/3)*d*x)*EllipticF((2^(2/3)*d*x+c*(1-sqrt(3)))/(2^(2/3)*d*x+c*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((c^2-2^(2/3)*c*d*x+2*2^(1/3)*d^2*x^2)/(2^(2/3)*d*x+c*(1+sqrt(3)))^2)/(3^(1/4)*d^2*sqrt(c^3+4*d^3*x^3)*sqrt(c*(c+2^(2/3)*d*x)/(2^(2/3)*d*x+c*(1+sqrt(3)))^2))],

# Integrands of the form (e+f x) / ((c+d x) Sqrt[a+b x^3]) with b c^3 + 8 a d^3=0

# 2 d e+c f = 0
[(1+x)/((2-x)*sqrt(1+x^3)),x,2,2/3*arctanh(1/3*(1+x)^2/sqrt(1+x^3))],
[(1-x)/((2+x)*sqrt(1-x^3)),x,2,-2/3*arctanh(1/3*(1-x)^2/sqrt(1-x^3))],
[(1-x)/((2+x)*sqrt(-1+x^3)),x,2,-2/3*arctan(1/3*(1-x)^2/sqrt(-1+x^3))],
[(1+x)/((2-x)*sqrt(-1-x^3)),x,2,2/3*arctan(1/3*(1+x)^2/sqrt(-1-x^3))],
[(a^(1/3)+b^(1/3)*x)/((2*a^(1/3)-b^(1/3)*x)*sqrt(a+b*x^3)),x,2,2/3*arctanh(1/3*(a^(1/3)+b^(1/3)*x)^2/(a^(1/6)*sqrt(a+b*x^3)))/(a^(1/6)*b^(1/3))],
[(a^(1/3)-b^(1/3)*x)/((2*a^(1/3)+b^(1/3)*x)*sqrt(a-b*x^3)),x,2,-2/3*arctanh(1/3*(a^(1/3)-b^(1/3)*x)^2/(a^(1/6)*sqrt(a-b*x^3)))/(a^(1/6)*b^(1/3))],
[(a^(1/3)-b^(1/3)*x)/((2*a^(1/3)+b^(1/3)*x)*sqrt(-a+b*x^3)),x,2,-2/3*arctan(1/3*(a^(1/3)-b^(1/3)*x)^2/(a^(1/6)*sqrt(-a+b*x^3)))/(a^(1/6)*b^(1/3))],
[(a^(1/3)+b^(1/3)*x)/((2*a^(1/3)-b^(1/3)*x)*sqrt(-a-b*x^3)),x,2,2/3*arctan(1/3*(a^(1/3)+b^(1/3)*x)^2/(a^(1/6)*sqrt(-a-b*x^3)))/(a^(1/6)*b^(1/3))],
[(c-2*d*x)/((c+d*x)*sqrt(c^3-8*d^3*x^3)),x,2,-2/3*arctanh(1/3*(c-2*d*x)^2/(sqrt(c)*sqrt(c^3-8*d^3*x^3)))/(d*sqrt(c))],

# 2 d e+c f /= 0
[(e+f*x)/((2-x)*sqrt(1+x^3)),x,4,2/9*(e+2*f)*arctanh(1/3*(1+x)^2/sqrt(1+x^3))+2/3*(e-f)*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[(e+f*x)/((2+x)*sqrt(1-x^3)),x,4,-2/9*(e-2*f)*arctanh(1/3*(1-x)^2/sqrt(1-x^3))-2/3*(e+f)*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[(e+f*x)/((2+x)*sqrt(-1+x^3)),x,4,-2/9*(e-2*f)*arctan(1/3*(1-x)^2/sqrt(-1+x^3))-2/3*(e+f)*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[(e+f*x)/((2-x)*sqrt(-1-x^3)),x,4,2/9*(e+2*f)*arctan(1/3*(1+x)^2/sqrt(-1-x^3))+2/3*(e-f)*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[(e+f*x)/((2*a^(1/3)-b^(1/3)*x)*sqrt(a+b*x^3)),x,4,2/9*(b^(1/3)*e+2*a^(1/3)*f)*arctanh(1/3*(a^(1/3)+b^(1/3)*x)^2/(a^(1/6)*sqrt(a+b*x^3)))/(b^(2/3)*sqrt(a))+2/3*(b^(1/3)*e-a^(1/3)*f)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^(1/3)*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(e+f*x)/((2*a^(1/3)+b^(1/3)*x)*sqrt(a-b*x^3)),x,4,-2/9*(b^(1/3)*e-2*a^(1/3)*f)*arctanh(1/3*(a^(1/3)-b^(1/3)*x)^2/(a^(1/6)*sqrt(a-b*x^3)))/(b^(2/3)*sqrt(a))-2/3*(b^(1/3)*e+a^(1/3)*f)*(a^(1/3)-b^(1/3)*x)*EllipticF((-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*a^(1/3)*b^(2/3)*sqrt(a-b*x^3)*sqrt(a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(e+f*x)/((2*a^(1/3)+b^(1/3)*x)*sqrt(-a+b*x^3)),x,4,-2/9*(b^(1/3)*e-2*a^(1/3)*f)*arctan(1/3*(a^(1/3)-b^(1/3)*x)^2/(a^(1/6)*sqrt(-a+b*x^3)))/(b^(2/3)*sqrt(a))-2/3*(b^(1/3)*e+a^(1/3)*f)*(a^(1/3)-b^(1/3)*x)*EllipticF((-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2-sqrt(3))/(3^(1/4)*a^(1/3)*b^(2/3)*sqrt(-a+b*x^3)*sqrt(-a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))],
[(e+f*x)/((2*a^(1/3)-b^(1/3)*x)*sqrt(-a-b*x^3)),x,4,2/9*(b^(1/3)*e+2*a^(1/3)*f)*arctan(1/3*(a^(1/3)+b^(1/3)*x)^2/(a^(1/6)*sqrt(-a-b*x^3)))/(b^(2/3)*sqrt(a))+2/3*(b^(1/3)*e-a^(1/3)*f)*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2-sqrt(3))/(3^(1/4)*a^(1/3)*b^(2/3)*sqrt(-a-b*x^3)*sqrt(-a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))],
[(e+f*x)/((c+d*x)*sqrt(c^3-8*d^3*x^3)),x,4,-2/9*(d*e-c*f)*arctanh(1/3*(c-2*d*x)^2/(sqrt(c)*sqrt(c^3-8*d^3*x^3)))/(c^(3/2)*d^2)-1/3*(2*d*e+c*f)*(c-2*d*x)*EllipticF((-2*d*x+c*(1-sqrt(3)))/(-2*d*x+c*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((c^2+2*c*d*x+4*d^2*x^2)/(-2*d*x+c*(1+sqrt(3)))^2)/(3^(1/4)*c*d^2*sqrt(c^3-8*d^3*x^3)*sqrt(c*(c-2*d*x)/(-2*d*x+c*(1+sqrt(3)))^2))],

# e = 0
[x/((2-x)*sqrt(1+x^3)),x,4,4/9*arctanh(1/3*(1+x)^2/sqrt(1+x^3))-2/3*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[x/((2+x)*sqrt(1-x^3)),x,4,4/9*arctanh(1/3*(1-x)^2/sqrt(1-x^3))-2/3*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[x/((2+x)*sqrt(-1+x^3)),x,4,4/9*arctan(1/3*(1-x)^2/sqrt(-1+x^3))-2/3*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[x/((2-x)*sqrt(-1-x^3)),x,4,4/9*arctan(1/3*(1+x)^2/sqrt(-1-x^3))-2/3*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[x/((2*a^(1/3)-b^(1/3)*x)*sqrt(a+b*x^3)),x,4,4/9*arctanh(1/3*(a^(1/3)+b^(1/3)*x)^2/(a^(1/6)*sqrt(a+b*x^3)))/(a^(1/6)*b^(2/3))-2/3*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x/((2*a^(1/3)+b^(1/3)*x)*sqrt(a-b*x^3)),x,4,4/9*arctanh(1/3*(a^(1/3)-b^(1/3)*x)^2/(a^(1/6)*sqrt(a-b*x^3)))/(a^(1/6)*b^(2/3))-2/3*(a^(1/3)-b^(1/3)*x)*EllipticF((-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(2/3)*sqrt(a-b*x^3)*sqrt(a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x/((2*a^(1/3)+b^(1/3)*x)*sqrt(-a+b*x^3)),x,4,4/9*arctan(1/3*(a^(1/3)-b^(1/3)*x)^2/(a^(1/6)*sqrt(-a+b*x^3)))/(a^(1/6)*b^(2/3))-2/3*(a^(1/3)-b^(1/3)*x)*EllipticF((-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2-sqrt(3))/(3^(1/4)*b^(2/3)*sqrt(-a+b*x^3)*sqrt(-a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))],
[x/((2*a^(1/3)-b^(1/3)*x)*sqrt(-a-b*x^3)),x,4,4/9*arctan(1/3*(a^(1/3)+b^(1/3)*x)^2/(a^(1/6)*sqrt(-a-b*x^3)))/(a^(1/6)*b^(2/3))-2/3*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2-sqrt(3))/(3^(1/4)*b^(2/3)*sqrt(-a-b*x^3)*sqrt(-a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))],
[x/((c+d*x)*sqrt(c^3-8*d^3*x^3)),x,4,2/9*arctanh(1/3*(c-2*d*x)^2/(sqrt(c)*sqrt(c^3-8*d^3*x^3)))/(d^2*sqrt(c))-1/3*(c-2*d*x)*EllipticF((-2*d*x+c*(1-sqrt(3)))/(-2*d*x+c*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((c^2+2*c*d*x+4*d^2*x^2)/(-2*d*x+c*(1+sqrt(3)))^2)/(3^(1/4)*d^2*sqrt(c^3-8*d^3*x^3)*sqrt(c*(c-2*d*x)/(-2*d*x+c*(1+sqrt(3)))^2))],

# Integrands of the form (e+f x) / ((c+d x) Sqrt[a+b x^3]) with b^2 c^6-20 a b c^3 d^3-8 a^2 d^6=0

# 6 a d^4 e-c f (b c^3 - 22 a d^3) = 0
[(1+x+sqrt(3))/((1+x-sqrt(3))*sqrt(1+x^3)),x,2,-2*arctanh((1+x)*sqrt(-3+2*sqrt(3))/sqrt(1+x^3))/sqrt(-3+2*sqrt(3))],
[(1-x+sqrt(3))/((1-x-sqrt(3))*sqrt(1-x^3)),x,2,2*arctanh((1-x)*sqrt(-3+2*sqrt(3))/sqrt(1-x^3))/sqrt(-3+2*sqrt(3))],
[(1-x+sqrt(3))/((1-x-sqrt(3))*sqrt(-1+x^3)),x,2,2*arctan((1-x)*sqrt(-3+2*sqrt(3))/sqrt(-1+x^3))/sqrt(-3+2*sqrt(3))],
[(1+x+sqrt(3))/((1+x-sqrt(3))*sqrt(-1-x^3)),x,2,-2*arctan((1+x)*sqrt(-3+2*sqrt(3))/sqrt(-1-x^3))/sqrt(-3+2*sqrt(3))],
[(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))*sqrt(a+b*x^3)),x,2,-2*arctanh(a^(1/6)*(a^(1/3)+b^(1/3)*x)*sqrt(-3+2*sqrt(3))/sqrt(a+b*x^3))/(a^(1/6)*b^(1/3)*sqrt(-3+2*sqrt(3)))],
[(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/((-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))*sqrt(a-b*x^3)),x,2,2*arctanh(a^(1/6)*(a^(1/3)-b^(1/3)*x)*sqrt(-3+2*sqrt(3))/sqrt(a-b*x^3))/(a^(1/6)*b^(1/3)*sqrt(-3+2*sqrt(3)))],
[(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/((-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))*sqrt(-a+b*x^3)),x,2,2*arctan(a^(1/6)*(a^(1/3)-b^(1/3)*x)*sqrt(-3+2*sqrt(3))/sqrt(-a+b*x^3))/(a^(1/6)*b^(1/3)*sqrt(-3+2*sqrt(3)))],
[(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))*sqrt(-a-b*x^3)),x,2,-2*arctan(a^(1/6)*(a^(1/3)+b^(1/3)*x)*sqrt(-3+2*sqrt(3))/sqrt(-a-b*x^3))/(a^(1/6)*b^(1/3)*sqrt(-3+2*sqrt(3)))],
[(1+(b/a)^(1/3)*x+sqrt(3))/((1+(b/a)^(1/3)*x-sqrt(3))*sqrt(a+b*x^3)),x,2,-2*arctanh((1+(b/a)^(1/3)*x)*sqrt(a)*sqrt(-3+2*sqrt(3))/sqrt(a+b*x^3))/((b/a)^(1/3)*sqrt(a)*sqrt(-3+2*sqrt(3)))],
[(1-(b/a)^(1/3)*x+sqrt(3))/((1-(b/a)^(1/3)*x-sqrt(3))*sqrt(a-b*x^3)),x,2,2*arctanh((1-(b/a)^(1/3)*x)*sqrt(a)*sqrt(-3+2*sqrt(3))/sqrt(a-b*x^3))/((b/a)^(1/3)*sqrt(a)*sqrt(-3+2*sqrt(3)))],
[(1-(b/a)^(1/3)*x+sqrt(3))/((1-(b/a)^(1/3)*x-sqrt(3))*sqrt(-a+b*x^3)),x,2,2*arctan((1-(b/a)^(1/3)*x)*sqrt(a)*sqrt(-3+2*sqrt(3))/sqrt(-a+b*x^3))/((b/a)^(1/3)*sqrt(a)*sqrt(-3+2*sqrt(3)))],
[(1+(b/a)^(1/3)*x+sqrt(3))/((1+(b/a)^(1/3)*x-sqrt(3))*sqrt(-a-b*x^3)),x,2,-2*arctan((1+(b/a)^(1/3)*x)*sqrt(a)*sqrt(-3+2*sqrt(3))/sqrt(-a-b*x^3))/((b/a)^(1/3)*sqrt(a)*sqrt(-3+2*sqrt(3)))],
[(1+x-sqrt(3))/((1+x+sqrt(3))*sqrt(1+x^3)),x,2,-2*arctan((1+x)*sqrt(3+2*sqrt(3))/sqrt(1+x^3))/sqrt(3+2*sqrt(3))],
[(1-x-sqrt(3))/((1-x+sqrt(3))*sqrt(1-x^3)),x,2,2*arctan((1-x)*sqrt(3+2*sqrt(3))/sqrt(1-x^3))/sqrt(3+2*sqrt(3))],
[(1-x-sqrt(3))/((1-x+sqrt(3))*sqrt(-1+x^3)),x,2,2*arctanh((1-x)*sqrt(3+2*sqrt(3))/sqrt(-1+x^3))/sqrt(3+2*sqrt(3))],
[(1+x-sqrt(3))/((1+x+sqrt(3))*sqrt(-1-x^3)),x,2,-2*arctanh((1+x)*sqrt(3+2*sqrt(3))/sqrt(-1-x^3))/sqrt(3+2*sqrt(3))],
[(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/((b^(1/3)*x+a^(1/3)*(1+sqrt(3)))*sqrt(a+b*x^3)),x,2,-2*arctan(a^(1/6)*(a^(1/3)+b^(1/3)*x)*sqrt(3+2*sqrt(3))/sqrt(a+b*x^3))/(a^(1/6)*b^(1/3)*sqrt(3+2*sqrt(3)))],
[(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/((-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))*sqrt(a-b*x^3)),x,2,2*arctan(a^(1/6)*(a^(1/3)-b^(1/3)*x)*sqrt(3+2*sqrt(3))/sqrt(a-b*x^3))/(a^(1/6)*b^(1/3)*sqrt(3+2*sqrt(3)))],
[(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/((-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))*sqrt(-a+b*x^3)),x,2,2*arctanh(a^(1/6)*(a^(1/3)-b^(1/3)*x)*sqrt(3+2*sqrt(3))/sqrt(-a+b*x^3))/(a^(1/6)*b^(1/3)*sqrt(3+2*sqrt(3)))],
[(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/((b^(1/3)*x+a^(1/3)*(1+sqrt(3)))*sqrt(-a-b*x^3)),x,2,-2*arctanh(a^(1/6)*(a^(1/3)+b^(1/3)*x)*sqrt(3+2*sqrt(3))/sqrt(-a-b*x^3))/(a^(1/6)*b^(1/3)*sqrt(3+2*sqrt(3)))],
[(1+(b/a)^(1/3)*x-sqrt(3))/((1+(b/a)^(1/3)*x+sqrt(3))*sqrt(a+b*x^3)),x,2,-2*arctan((1+(b/a)^(1/3)*x)*sqrt(a)*sqrt(3+2*sqrt(3))/sqrt(a+b*x^3))/((b/a)^(1/3)*sqrt(a)*sqrt(3+2*sqrt(3)))],
[(1-(b/a)^(1/3)*x-sqrt(3))/((1-(b/a)^(1/3)*x+sqrt(3))*sqrt(a-b*x^3)),x,2,2*arctan((1-(b/a)^(1/3)*x)*sqrt(a)*sqrt(3+2*sqrt(3))/sqrt(a-b*x^3))/((b/a)^(1/3)*sqrt(a)*sqrt(3+2*sqrt(3)))],
[(1-(b/a)^(1/3)*x-sqrt(3))/((1-(b/a)^(1/3)*x+sqrt(3))*sqrt(-a+b*x^3)),x,2,2*arctanh((1-(b/a)^(1/3)*x)*sqrt(a)*sqrt(3+2*sqrt(3))/sqrt(-a+b*x^3))/((b/a)^(1/3)*sqrt(a)*sqrt(3+2*sqrt(3)))],
[(1+(b/a)^(1/3)*x-sqrt(3))/((1+(b/a)^(1/3)*x+sqrt(3))*sqrt(-a-b*x^3)),x,2,-2*arctanh((1+(b/a)^(1/3)*x)*sqrt(a)*sqrt(3+2*sqrt(3))/sqrt(-a-b*x^3))/((b/a)^(1/3)*sqrt(a)*sqrt(3+2*sqrt(3)))],

# 6 a d^4 e-c f (b c^3 - 22 a d^3) /= 0
[(1+x)/((1+x+sqrt(3))*sqrt(1+x^3)),x,4,(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))-arctan((1+x)*sqrt(3+2*sqrt(3))/sqrt(1+x^3))/sqrt(3+2*sqrt(3))],
[(1+x)/((1+x-sqrt(3))*sqrt(1+x^3)),x,4,(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))-arctanh((1+x)*sqrt(-3+2*sqrt(3))/sqrt(1+x^3))/sqrt(-3+2*sqrt(3))],
[(e+f*x)/((1+x+sqrt(3))*sqrt(1+x^3)),x,4,(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*(e-f*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(3/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))+arctan((1+x)*sqrt(3+2*sqrt(3))/sqrt(1+x^3))*(e-f-f*sqrt(3))/sqrt(3*(3+2*sqrt(3)))],
[(e+f*x)/((1-x+sqrt(3))*sqrt(1-x^3)),x,4,-(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*(e+f*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(3/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))-arctan((1-x)*sqrt(3+2*sqrt(3))/sqrt(1-x^3))*(e+f+f*sqrt(3))/sqrt(3*(3+2*sqrt(3)))],
[(e+f*x)/((1-x+sqrt(3))*sqrt(-1+x^3)),x,4,-(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*(e+f*(1-sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(3/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))-arctanh((1-x)*sqrt(3+2*sqrt(3))/sqrt(-1+x^3))*(e+f+f*sqrt(3))/sqrt(3*(3+2*sqrt(3)))],
[(e+f*x)/((1+x+sqrt(3))*sqrt(-1-x^3)),x,4,(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*(e-f*(1-sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(3/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))+arctanh((1+x)*sqrt(3+2*sqrt(3))/sqrt(-1-x^3))*(e-f*(1+sqrt(3)))/sqrt(3*(3+2*sqrt(3)))],
[(e+f*x)/((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))*sqrt(a+b*x^3)),x,4,-arctanh(a^(1/6)*(a^(1/3)+b^(1/3)*x)*sqrt(-3+2*sqrt(3))/sqrt(a+b*x^3))*(b^(1/3)*e-a^(1/3)*f*(1-sqrt(3)))/(b^(2/3)*sqrt(a)*sqrt(3*(-3+2*sqrt(3))))-(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(b^(1/3)*e-a^(1/3)*f*(1+sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*a^(1/3)*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(e+f*x)/((-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))*sqrt(a-b*x^3)),x,4,arctanh(a^(1/6)*(a^(1/3)-b^(1/3)*x)*sqrt(-3+2*sqrt(3))/sqrt(a-b*x^3))*(b^(1/3)*e+a^(1/3)*f*(1-sqrt(3)))/(b^(2/3)*sqrt(a)*sqrt(3*(-3+2*sqrt(3))))+(a^(1/3)-b^(1/3)*x)*EllipticF((-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(b^(1/3)*e+a^(1/3)*f*(1+sqrt(3)))*sqrt(2+sqrt(3))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(3/4)*a^(1/3)*b^(2/3)*sqrt(a-b*x^3)*sqrt(a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[(e+f*x)/((-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))*sqrt(-a+b*x^3)),x,4,(a^(1/3)-b^(1/3)*x)*EllipticF((-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*(b^(1/3)*e+a^(1/3)*f*(1+sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2-sqrt(3))/(3^(3/4)*a^(1/3)*b^(2/3)*sqrt(-a+b*x^3)*sqrt(-a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))+arctan(a^(1/6)*(a^(1/3)-b^(1/3)*x)*sqrt(-3+2*sqrt(3))/sqrt(-a+b*x^3))*(b^(1/3)*e+a^(1/3)*f*(1-sqrt(3)))/(b^(2/3)*sqrt(a)*sqrt(3*(-3+2*sqrt(3))))],
[(e+f*x)/((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))*sqrt(-a-b*x^3)),x,4,-(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*(b^(1/3)*e-a^(1/3)*f*(1+sqrt(3)))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2-sqrt(3))/(3^(3/4)*a^(1/3)*b^(2/3)*sqrt(-a-b*x^3)*sqrt(-a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))-arctan(a^(1/6)*(a^(1/3)+b^(1/3)*x)*sqrt(-3+2*sqrt(3))/sqrt(-a-b*x^3))*(b^(1/3)*e-a^(1/3)*f*(1-sqrt(3)))/(b^(2/3)*sqrt(a)*sqrt(3*(-3+2*sqrt(3))))],

# e = 0
[x/((1+x+sqrt(3))*sqrt(1+x^3)),x,4,-arctan((1+x)*sqrt(3+2*sqrt(3))/sqrt(1+x^3))*sqrt(2)/3^(3/4)+(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(3/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[x/((1-x+sqrt(3))*sqrt(1-x^3)),x,4,-arctan((1-x)*sqrt(3+2*sqrt(3))/sqrt(1-x^3))*sqrt(2)/3^(3/4)+(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(3/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[x/((1-x+sqrt(3))*sqrt(-1+x^3)),x,4,-arctanh((1-x)*sqrt(3+2*sqrt(3))/sqrt(-1+x^3))*sqrt(2)/3^(3/4)+2*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(7/6+(-2)/sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[x/((1+x+sqrt(3))*sqrt(-1-x^3)),x,4,-arctanh((1+x)*sqrt(3+2*sqrt(3))/sqrt(-1-x^3))*sqrt(2)/3^(3/4)+2*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(7/6+(-2)/sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[x/((1+x-sqrt(3))*sqrt(1+x^3)),x,4,-arctanh((1+x)*sqrt(-3+2*sqrt(3))/sqrt(1+x^3))*sqrt(2)/3^(3/4)+2*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(7/6+2/sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[x/((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))*sqrt(a+b*x^3)),x,4,-arctanh(a^(1/6)*(a^(1/3)+b^(1/3)*x)*sqrt(-3+2*sqrt(3))/sqrt(a+b*x^3))*sqrt(2)/(3^(3/4)*a^(1/6)*b^(2/3))+2*(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(7/6+2/sqrt(3))*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(2/3)*sqrt(a+b*x^3)*sqrt(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x/((-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))*sqrt(a-b*x^3)),x,4,-arctanh(a^(1/6)*(a^(1/3)-b^(1/3)*x)*sqrt(-3+2*sqrt(3))/sqrt(a-b*x^3))*sqrt(2)/(3^(3/4)*a^(1/6)*b^(2/3))+2*(a^(1/3)-b^(1/3)*x)*EllipticF((-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(7/6+2/sqrt(3))*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^(2/3)*sqrt(a-b*x^3)*sqrt(a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))^2))],
[x/((-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))*sqrt(-a+b*x^3)),x,4,-arctan(a^(1/6)*(a^(1/3)-b^(1/3)*x)*sqrt(-3+2*sqrt(3))/sqrt(-a+b*x^3))*sqrt(2)/(3^(3/4)*a^(1/6)*b^(2/3))+(a^(1/3)-b^(1/3)*x)*EllipticF((-b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)/(3^(3/4)*b^(2/3)*sqrt(-a+b*x^3)*sqrt(-a^(1/3)*(a^(1/3)-b^(1/3)*x)/(-b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))],
[x/((b^(1/3)*x+a^(1/3)*(1-sqrt(3)))*sqrt(-a-b*x^3)),x,4,-arctan(a^(1/6)*(a^(1/3)+b^(1/3)*x)*sqrt(-3+2*sqrt(3))/sqrt(-a-b*x^3))*sqrt(2)/(3^(3/4)*a^(1/6)*b^(2/3))+(a^(1/3)+b^(1/3)*x)*EllipticF((b^(1/3)*x+a^(1/3)*(1+sqrt(3)))/(b^(1/3)*x+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2)/(3^(3/4)*b^(2/3)*sqrt(-a-b*x^3)*sqrt(-a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1-sqrt(3)))^2))],

# Integrands of the form (e+f x) / ((c+d x) Sqrt[a+b x^3]) with b^2 e^6-20 a b e^3 f^3-8 a^2 f^6=0
[(1+x+sqrt(3))/((c+d*x)*sqrt(1+x^3)),x,6,-(1+x)*arctan(sqrt(c^2+c*d+d^2)*sqrt((1+x)/(1+x+sqrt(3))^2)/(sqrt(c-d)*sqrt(d)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)))*(c-d*(1+sqrt(3)))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(c-d)*sqrt(d)*sqrt(c^2+c*d+d^2)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))-4*3^(1/4)*(1+x)*EllipticPi((-1-x+sqrt(3))/(1+x+sqrt(3)),(c-d*(1+sqrt(3)))^2/(c-d*(1-sqrt(3)))^2,sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/((c-d*(1-sqrt(3)))*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[(1-x+sqrt(3))/((c+d*x)*sqrt(1-x^3)),x,6,-(1-x)*arctanh(sqrt(c^2-c*d+d^2)*sqrt((1-x)/(1-x+sqrt(3))^2)/(sqrt(d)*sqrt(c+d)*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)))*(c+d+d*sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(d)*sqrt(c+d)*sqrt(c^2-c*d+d^2)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))+4*3^(1/4)*(1-x)*EllipticPi((-1+x+sqrt(3))/(1-x+sqrt(3)),(c+d+d*sqrt(3))^2/(c+d-d*sqrt(3))^2,sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/((c+d-d*sqrt(3))*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[(1-x+sqrt(3))/((c+d*x)*sqrt(-1+x^3)),x,6,-(1-x)*arctanh(sqrt(c^2-c*d+d^2)*sqrt((1-x)/(1-x+sqrt(3))^2)/(sqrt(d)*sqrt(c+d)*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)))*(c+d+d*sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(d)*sqrt(c+d)*sqrt(c^2-c*d+d^2)*sqrt(-1+x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))+4*3^(1/4)*(1-x)*EllipticPi((-1+x+sqrt(3))/(1-x+sqrt(3)),(c+d+d*sqrt(3))^2/(c+d-d*sqrt(3))^2,sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/((c+d-d*sqrt(3))*sqrt(-1+x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[(1+x+sqrt(3))/((c+d*x)*sqrt(-1-x^3)),x,6,-(1+x)*arctan(sqrt(c^2+c*d+d^2)*sqrt((1+x)/(1+x+sqrt(3))^2)/(sqrt(c-d)*sqrt(d)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)))*(c-d*(1+sqrt(3)))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(c-d)*sqrt(d)*sqrt(c^2+c*d+d^2)*sqrt(-1-x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))-4*3^(1/4)*(1+x)*EllipticPi((-1-x+sqrt(3))/(1+x+sqrt(3)),(c-d*(1+sqrt(3)))^2/(c-d*(1-sqrt(3)))^2,sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/((c-d*(1-sqrt(3)))*sqrt(-1-x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[(1+x-sqrt(3))/((c+d*x)*sqrt(1+x^3)),x,6,-(1+x)*arctanh(2*sqrt(c^2+c*d+d^2)*sqrt((-1-x)/(1+x-sqrt(3))^2)*sqrt(2+sqrt(3))/(sqrt(c-d)*sqrt(d)*sqrt(7+4*sqrt(3)+(1+x+sqrt(3))^2/(1+x-sqrt(3))^2)))*(c-d*(1-sqrt(3)))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(sqrt(c-d)*sqrt(d)*sqrt(c^2+c*d+d^2)*sqrt(1+x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))+4*3^(1/4)*(1+x)*EllipticPi((-1-x-sqrt(3))/(1+x-sqrt(3)),(c-d*(1-sqrt(3)))^2/(c-d*(1+sqrt(3)))^2,sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/((c-d-d*sqrt(3))*sqrt(1+x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[(1-x-sqrt(3))/((c+d*x)*sqrt(1-x^3)),x,6,-(1-x)*arctan(sqrt(c^2-c*d+d^2)*sqrt((-1+x)/(1-x-sqrt(3))^2)/(sqrt(d)*sqrt(c+d)*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)))*(c+d-d*sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(sqrt(d)*sqrt(c+d)*sqrt(c^2-c*d+d^2)*sqrt(1-x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))-4*3^(1/4)*(1-x)*EllipticPi((-1+x-sqrt(3))/(1-x-sqrt(3)),(c+d-d*sqrt(3))^2/(c+d+d*sqrt(3))^2,sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/((c+d+d*sqrt(3))*sqrt(1-x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[(1-x-sqrt(3))/((c+d*x)*sqrt(-1+x^3)),x,6,-(1-x)*arctan(sqrt(c^2-c*d+d^2)*sqrt((-1+x)/(1-x-sqrt(3))^2)/(sqrt(d)*sqrt(c+d)*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)))*(c+d-d*sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(sqrt(d)*sqrt(c+d)*sqrt(c^2-c*d+d^2)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))-4*3^(1/4)*(1-x)*EllipticPi((-1+x-sqrt(3))/(1-x-sqrt(3)),(c+d-d*sqrt(3))^2/(c+d+d*sqrt(3))^2,sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/((c+d+d*sqrt(3))*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[(1+x-sqrt(3))/((c+d*x)*sqrt(-1-x^3)),x,6,-(1+x)*arctanh(2*sqrt(c^2+c*d+d^2)*sqrt((-1-x)/(1+x-sqrt(3))^2)*sqrt(2+sqrt(3))/(sqrt(c-d)*sqrt(d)*sqrt(7+4*sqrt(3)+(1+x+sqrt(3))^2/(1+x-sqrt(3))^2)))*(c-d*(1-sqrt(3)))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(sqrt(c-d)*sqrt(d)*sqrt(c^2+c*d+d^2)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))+4*3^(1/4)*(1+x)*EllipticPi((-1-x-sqrt(3))/(1+x-sqrt(3)),(c-d*(1-sqrt(3)))^2/(c-d*(1+sqrt(3)))^2,sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/((c-d-d*sqrt(3))*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[(1+x+sqrt(3))/(x*sqrt(1+x^3)),x,5,-2/3*arctanh(sqrt(1+x^3))*(1+sqrt(3))+2*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[(1-x+sqrt(3))/(x*sqrt(1-x^3)),x,5,-2/3*arctanh(sqrt(1-x^3))*(1+sqrt(3))+2*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[(1-x+sqrt(3))/(x*sqrt(-1+x^3)),x,5,2/3*arctan(sqrt(-1+x^3))*(1+sqrt(3))+2*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[(1+x+sqrt(3))/(x*sqrt(-1-x^3)),x,5,2/3*arctan(sqrt(-1-x^3))*(1+sqrt(3))+2*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[(1+x-sqrt(3))/(x*sqrt(1+x^3)),x,5,-2/3*arctanh(sqrt(1+x^3))*(1-sqrt(3))+2*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[(1-x-sqrt(3))/(x*sqrt(1-x^3)),x,5,-2/3*arctanh(sqrt(1-x^3))*(1-sqrt(3))+2*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[(1-x-sqrt(3))/(x*sqrt(-1+x^3)),x,5,2/3*arctan(sqrt(-1+x^3))*(1-sqrt(3))+2*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[(1+x-sqrt(3))/(x*sqrt(-1-x^3)),x,5,2/3*arctan(sqrt(-1-x^3))*(1-sqrt(3))+2*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],

# Integrands of the form (e+f x) / ((c+d x) Sqrt[a+b x^3])
[x/((3+x)*sqrt(1+x^3)),x,8,-3*(1+x)*arctan(sqrt(13/2)*sqrt((1+x)/(1+x+sqrt(3))^2)/sqrt((1-x+x^2)/(1+x+sqrt(3))^2))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(26)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))-12*3^(1/4)*(1+x)*EllipticPi((-1-x+sqrt(3))/(1+x+sqrt(3)),97-56*sqrt(3),sqrt(-7-4*sqrt(3)))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(1+x^3)*sqrt(2-sqrt(3))*sqrt((1+x)/(1+x+sqrt(3))^2))-2*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)*sqrt(2*(97+56*sqrt(3)))/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[x/((3+x)*sqrt(1-x^3)),x,8,3/2*(1-x)*arctanh(1/2*sqrt(7)*sqrt((1-x)/(1-x+sqrt(3))^2)/sqrt((1+x+x^2)/(1-x+sqrt(3))^2))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(7)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))-12/13*3^(1/4)*(1-x)*EllipticPi((-1+x+sqrt(3))/(1-x+sqrt(3)),1/169*(553+304*sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))-2/13*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)*sqrt(2*(37+20*sqrt(3)))/(3^(1/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[x/((3+x)*sqrt(-1+x^3)),x,8,-2*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*(4+sqrt(3))*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))+3/2*(1-x)*arctanh(1/2*sqrt(7)*sqrt((1-x)/(1-x+sqrt(3))^2)/sqrt((1+x+x^2)/(1-x+sqrt(3))^2))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(7)*sqrt(-1+x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))-12/13*3^(1/4)*(1-x)*EllipticPi((-1+x+sqrt(3))/(1-x+sqrt(3)),1/169*(553+304*sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(-1+x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[x/((3+x)*sqrt(-1-x^3)),x,8,-3*(1+x)*arctan(sqrt(13/2)*sqrt((1+x)/(1+x+sqrt(3))^2)/sqrt((1-x+x^2)/(1+x+sqrt(3))^2))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(26)*sqrt(-1-x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))-12*3^(1/4)*(1+x)*EllipticPi((-1-x+sqrt(3))/(1+x+sqrt(3)),97-56*sqrt(3),sqrt(-7-4*sqrt(3)))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(-1-x^3)*sqrt(2-sqrt(3))*sqrt((1+x)/(1+x+sqrt(3))^2))-2*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)*sqrt(14+8*sqrt(3))/(3^(1/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],
[(e+f*x)/((c+d*x)*sqrt(1+x^3)),x,8,(d*e-c*f)*(1+x)*arctan(sqrt(c^2+c*d+d^2)*sqrt((1+x)/(1+x+sqrt(3))^2)/(sqrt(c-d)*sqrt(d)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(c-d)*sqrt(d)*sqrt(c^2+c*d+d^2)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))+4*3^(1/4)*(d*e-c*f)*(1+x)*EllipticPi((-1-x+sqrt(3))/(1+x+sqrt(3)),(c-d*(1+sqrt(3)))^2/(c-d*(1-sqrt(3)))^2,sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/((c^2-2*c*d-2*d^2)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))+2*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*(e-f-f*sqrt(3))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*(c-d-d*sqrt(3))*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[(e+f*x)/((c+d*x)*sqrt(1-x^3)),x,8,-(d*e-c*f)*(1-x)*arctanh(sqrt(c^2-c*d+d^2)*sqrt((1-x)/(1-x+sqrt(3))^2)/(sqrt(d)*sqrt(c+d)*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(d)*sqrt(c+d)*sqrt(c^2-c*d+d^2)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))+4*3^(1/4)*(d*e-c*f)*(1-x)*EllipticPi((-1+x+sqrt(3))/(1-x+sqrt(3)),(c+d+d*sqrt(3))^2/(c+d-d*sqrt(3))^2,sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/((c^2+2*c*d-2*d^2)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))-2*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*(e+f+f*sqrt(3))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*(c+d+d*sqrt(3))*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[(e+f*x)/((c+d*x)*sqrt(-1+x^3)),x,8,-2*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*(e+f+f*sqrt(3))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*(c+d+d*sqrt(3))*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))-(d*e-c*f)*(1-x)*arctanh(sqrt(c^2-c*d+d^2)*sqrt((1-x)/(1-x+sqrt(3))^2)/(sqrt(d)*sqrt(c+d)*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(sqrt(d)*sqrt(c+d)*sqrt(c^2-c*d+d^2)*sqrt(-1+x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))+4*3^(1/4)*(d*e-c*f)*(1-x)*EllipticPi((-1+x+sqrt(3))/(1-x+sqrt(3)),(c+d+d*sqrt(3))^2/(c+d-d*sqrt(3))^2,sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/((c^2+2*c*d-2*d^2)*sqrt(-1+x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[(e+f*x)/((c+d*x)*sqrt(-1-x^3)),x,8,2*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*(e-f-f*sqrt(3))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*(c-d-d*sqrt(3))*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))+(d*e-c*f)*(1+x)*arctan(sqrt(c^2+c*d+d^2)*sqrt((1+x)/(1+x+sqrt(3))^2)/(sqrt(c-d)*sqrt(d)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(c-d)*sqrt(d)*sqrt(c^2+c*d+d^2)*sqrt(-1-x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))+4*3^(1/4)*(d*e-c*f)*(1+x)*EllipticPi((-1-x+sqrt(3))/(1+x+sqrt(3)),(c-d*(1+sqrt(3)))^2/(c-d*(1-sqrt(3)))^2,sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/((c^2-2*c*d-2*d^2)*sqrt(-1-x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[(e+f*x)/(x*sqrt(1+x^3)),x,6,-2/3*e*arctanh(sqrt(1+x^3))+2*f*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[(e+f*x)/(x*sqrt(1-x^3)),x,6,-2/3*e*arctanh(sqrt(1-x^3))-2*f*(1-x)*EllipticF((1-x-sqrt(3))/(1-x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((1+x+x^2)/(1-x+sqrt(3))^2)/(3^(1/4)*sqrt(1-x^3)*sqrt((1-x)/(1-x+sqrt(3))^2))],
[(e+f*x)/(x*sqrt(-1+x^3)),x,6,2/3*e*arctan(sqrt(-1+x^3))-2*f*(1-x)*EllipticF((1-x+sqrt(3))/(1-x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1+x+x^2)/(1-x-sqrt(3))^2)/(3^(1/4)*sqrt(-1+x^3)*sqrt((-1+x)/(1-x-sqrt(3))^2))],
[(e+f*x)/(x*sqrt(-1-x^3)),x,6,2/3*e*arctan(sqrt(-1-x^3))+2*f*(1+x)*EllipticF((1+x+sqrt(3))/(1+x-sqrt(3)),sqrt(-7+4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x-sqrt(3))^2)/(3^(1/4)*sqrt(-1-x^3)*sqrt((-1-x)/(1+x-sqrt(3))^2))],

# Integrands of the form (e+f x) / ((c+d x) (a+b x^3)^(1/3)) when 2 b c^3-a d^3=0
[(c-d*x)/((c+d*x)*(2*c^3+d^3*x^3)^(1/3)),x,1,-log(c+d*x)/d+3/2*log(d*(2*c+d*x)-d*(2*c^3+d^3*x^3)^(1/3))/d-arctan((1+2*(2*c+d*x)/(2*c^3+d^3*x^3)^(1/3))/sqrt(3))*sqrt(3)/d],

# Integrands of the form (e+f x) / ((c+d x) (a+b x^3)^(1/3)) when b c^3+a d^3=0
[(e+f*x)/((c+d*x)*(-c^3+d^3*x^3)^(1/3)),x,3,1/4*(d*e-c*f)*log((c-d*x)*(c+d*x)^2)/(2^(1/3)*c*d^2)-1/2*f*log(-d*x+(-c^3+d^3*x^3)^(1/3))/d^2-3/4*(d*e-c*f)*log(d*(c-d*x)+2^(2/3)*d*(-c^3+d^3*x^3)^(1/3))/(2^(1/3)*c*d^2)+f*arctan((1+2*d*x/(-c^3+d^3*x^3)^(1/3))/sqrt(3))/(d^2*sqrt(3))+1/2*(d*e-c*f)*arctan((1-2^(1/3)*(c-d*x)/(-c^3+d^3*x^3)^(1/3))/sqrt(3))*sqrt(3)/(2^(1/3)*c*d^2)],

# Integrands of the form x^m (c+d x)^n (a+b x^3)^p

# Integrands of the form x^m (c+d x)^n (a+b x^3)^p with n symbolic

# p>0
[x^2*(a+b*x)^n*(c+d*x^3),x,2,a^2*(b^3*c-a^3*d)*(a+b*x)^(1+n)/(b^6*(1+n))-a*(2*b^3*c-5*a^3*d)*(a+b*x)^(2+n)/(b^6*(2+n))+(b^3*c-10*a^3*d)*(a+b*x)^(3+n)/(b^6*(3+n))+10*a^2*d*(a+b*x)^(4+n)/(b^6*(4+n))-5*a*d*(a+b*x)^(5+n)/(b^6*(5+n))+d*(a+b*x)^(6+n)/(b^6*(6+n))],
[x*(a+b*x)^n*(c+d*x^3),x,2,-a*(b^3*c-a^3*d)*(a+b*x)^(1+n)/(b^5*(1+n))+(b^3*c-4*a^3*d)*(a+b*x)^(2+n)/(b^5*(2+n))+6*a^2*d*(a+b*x)^(3+n)/(b^5*(3+n))-4*a*d*(a+b*x)^(4+n)/(b^5*(4+n))+d*(a+b*x)^(5+n)/(b^5*(5+n))],
[(a+b*x)^n*(c+d*x^3),x,2,(b^3*c-a^3*d)*(a+b*x)^(1+n)/(b^4*(1+n))+3*a^2*d*(a+b*x)^(2+n)/(b^4*(2+n))-3*a*d*(a+b*x)^(3+n)/(b^4*(3+n))+d*(a+b*x)^(4+n)/(b^4*(4+n))],
[(a+b*x)^n*(c+d*x^3)/x,x,3,a^2*d*(a+b*x)^(1+n)/(b^3*(1+n))-2*a*d*(a+b*x)^(2+n)/(b^3*(2+n))+d*(a+b*x)^(3+n)/(b^3*(3+n))-c*(a+b*x)^(1+n)*hypergeom([1,1+n],[2+n],1+b*x/a)/(a*(1+n))],
[x^2*(a+b*x)^n*(c+d*x^3)^2,x,2,a^2*(b^3*c-a^3*d)^2*(a+b*x)^(1+n)/(b^9*(1+n))-2*a*(b^3*c-4*a^3*d)*(b^3*c-a^3*d)*(a+b*x)^(2+n)/(b^9*(2+n))+(b^6*c^2-20*a^3*b^3*c*d+28*a^6*d^2)*(a+b*x)^(3+n)/(b^9*(3+n))+4*a^2*d*(5*b^3*c-14*a^3*d)*(a+b*x)^(4+n)/(b^9*(4+n))-10*a*d*(b^3*c-7*a^3*d)*(a+b*x)^(5+n)/(b^9*(5+n))+2*d*(b^3*c-28*a^3*d)*(a+b*x)^(6+n)/(b^9*(6+n))+28*a^2*d^2*(a+b*x)^(7+n)/(b^9*(7+n))-8*a*d^2*(a+b*x)^(8+n)/(b^9*(8+n))+d^2*(a+b*x)^(9+n)/(b^9*(9+n))],
[x*(a+b*x)^n*(c+d*x^3)^2,x,2,-a*(b^3*c-a^3*d)^2*(a+b*x)^(1+n)/(b^8*(1+n))+(b^3*c-7*a^3*d)*(b^3*c-a^3*d)*(a+b*x)^(2+n)/(b^8*(2+n))+3*a^2*d*(4*b^3*c-7*a^3*d)*(a+b*x)^(3+n)/(b^8*(3+n))-a*d*(8*b^3*c-35*a^3*d)*(a+b*x)^(4+n)/(b^8*(4+n))+d*(2*b^3*c-35*a^3*d)*(a+b*x)^(5+n)/(b^8*(5+n))+21*a^2*d^2*(a+b*x)^(6+n)/(b^8*(6+n))-7*a*d^2*(a+b*x)^(7+n)/(b^8*(7+n))+d^2*(a+b*x)^(8+n)/(b^8*(8+n))],
[(a+b*x)^n*(c+d*x^3)^2,x,2,(b^3*c-a^3*d)^2*(a+b*x)^(1+n)/(b^7*(1+n))+6*a^2*d*(b^3*c-a^3*d)*(a+b*x)^(2+n)/(b^7*(2+n))-3*a*d*(2*b^3*c-5*a^3*d)*(a+b*x)^(3+n)/(b^7*(3+n))+2*d*(b^3*c-10*a^3*d)*(a+b*x)^(4+n)/(b^7*(4+n))+15*a^2*d^2*(a+b*x)^(5+n)/(b^7*(5+n))-6*a*d^2*(a+b*x)^(6+n)/(b^7*(6+n))+d^2*(a+b*x)^(7+n)/(b^7*(7+n))],
[(a+b*x)^n*(c+d*x^3)^2/x,x,3,a^2*d*(2*b^3*c-a^3*d)*(a+b*x)^(1+n)/(b^6*(1+n))-a*d*(4*b^3*c-5*a^3*d)*(a+b*x)^(2+n)/(b^6*(2+n))+2*d*(b^3*c-5*a^3*d)*(a+b*x)^(3+n)/(b^6*(3+n))+10*a^2*d^2*(a+b*x)^(4+n)/(b^6*(4+n))-5*a*d^2*(a+b*x)^(5+n)/(b^6*(5+n))+d^2*(a+b*x)^(6+n)/(b^6*(6+n))-c^2*(a+b*x)^(1+n)*hypergeom([1,1+n],[2+n],1+b*x/a)/(a*(1+n))],
[x^2*(a+b*x)^n*(c+d*x^3)^3,x,2,a^2*(b^3*c-a^3*d)^3*(a+b*x)^(1+n)/(b^12*(1+n))-a*(2*b^3*c-11*a^3*d)*(b^3*c-a^3*d)^2*(a+b*x)^(2+n)/(b^12*(2+n))+(b^3*c-a^3*d)*(b^6*c^2-29*a^3*b^3*c*d+55*a^6*d^2)*(a+b*x)^(3+n)/(b^12*(3+n))+3*a^2*d*(10*b^6*c^2-56*a^3*b^3*c*d+55*a^6*d^2)*(a+b*x)^(4+n)/(b^12*(4+n))-15*a*d*(b^6*c^2-14*a^3*b^3*c*d+22*a^6*d^2)*(a+b*x)^(5+n)/(b^12*(5+n))+3*d*(b^6*c^2-56*a^3*b^3*c*d+154*a^6*d^2)*(a+b*x)^(6+n)/(b^12*(6+n))+42*a^2*d^2*(2*b^3*c-11*a^3*d)*(a+b*x)^(7+n)/(b^12*(7+n))-6*a*d^2*(4*b^3*c-55*a^3*d)*(a+b*x)^(8+n)/(b^12*(8+n))+3*d^2*(b^3*c-55*a^3*d)*(a+b*x)^(9+n)/(b^12*(9+n))+55*a^2*d^3*(a+b*x)^(10+n)/(b^12*(10+n))-11*a*d^3*(a+b*x)^(11+n)/(b^12*(11+n))+d^3*(a+b*x)^(12+n)/(b^12*(12+n))],
[x*(a+b*x)^n*(c+d*x^3)^3,x,2,-a*(b^3*c-a^3*d)^3*(a+b*x)^(1+n)/(b^11*(1+n))+(b^3*c-10*a^3*d)*(b^3*c-a^3*d)^2*(a+b*x)^(2+n)/(b^11*(2+n))+9*a^2*d*(2*b^3*c-5*a^3*d)*(b^3*c-a^3*d)*(a+b*x)^(3+n)/(b^11*(3+n))-3*a*d*(4*b^6*c^2-35*a^3*b^3*c*d+40*a^6*d^2)*(a+b*x)^(4+n)/(b^11*(4+n))+3*d*(b^6*c^2-35*a^3*b^3*c*d+70*a^6*d^2)*(a+b*x)^(5+n)/(b^11*(5+n))+63*a^2*d^2*(b^3*c-4*a^3*d)*(a+b*x)^(6+n)/(b^11*(6+n))-21*a*d^2*(b^3*c-10*a^3*d)*(a+b*x)^(7+n)/(b^11*(7+n))+3*d^2*(b^3*c-40*a^3*d)*(a+b*x)^(8+n)/(b^11*(8+n))+45*a^2*d^3*(a+b*x)^(9+n)/(b^11*(9+n))-10*a*d^3*(a+b*x)^(10+n)/(b^11*(10+n))+d^3*(a+b*x)^(11+n)/(b^11*(11+n))],
[(a+b*x)^n*(c+d*x^3)^3,x,2,(b^3*c-a^3*d)^3*(a+b*x)^(1+n)/(b^10*(1+n))+9*a^2*d*(b^3*c-a^3*d)^2*(a+b*x)^(2+n)/(b^10*(2+n))-9*a*d*(b^3*c-4*a^3*d)*(b^3*c-a^3*d)*(a+b*x)^(3+n)/(b^10*(3+n))+3*d*(b^6*c^2-20*a^3*b^3*c*d+28*a^6*d^2)*(a+b*x)^(4+n)/(b^10*(4+n))+9*a^2*d^2*(5*b^3*c-14*a^3*d)*(a+b*x)^(5+n)/(b^10*(5+n))-18*a*d^2*(b^3*c-7*a^3*d)*(a+b*x)^(6+n)/(b^10*(6+n))+3*d^2*(b^3*c-28*a^3*d)*(a+b*x)^(7+n)/(b^10*(7+n))+36*a^2*d^3*(a+b*x)^(8+n)/(b^10*(8+n))-9*a*d^3*(a+b*x)^(9+n)/(b^10*(9+n))+d^3*(a+b*x)^(10+n)/(b^10*(10+n))],
[(a+b*x)^n*(c+d*x^3)^3/x,x,3,a^2*d*(3*b^6*c^2-3*a^3*b^3*c*d+a^6*d^2)*(a+b*x)^(1+n)/(b^9*(1+n))-a*d*(6*b^6*c^2-15*a^3*b^3*c*d+8*a^6*d^2)*(a+b*x)^(2+n)/(b^9*(2+n))+d*(3*b^6*c^2-30*a^3*b^3*c*d+28*a^6*d^2)*(a+b*x)^(3+n)/(b^9*(3+n))+2*a^2*d^2*(15*b^3*c-28*a^3*d)*(a+b*x)^(4+n)/(b^9*(4+n))-5*a*d^2*(3*b^3*c-14*a^3*d)*(a+b*x)^(5+n)/(b^9*(5+n))+d^2*(3*b^3*c-56*a^3*d)*(a+b*x)^(6+n)/(b^9*(6+n))+28*a^2*d^3*(a+b*x)^(7+n)/(b^9*(7+n))-8*a*d^3*(a+b*x)^(8+n)/(b^9*(8+n))+d^3*(a+b*x)^(9+n)/(b^9*(9+n))-c^3*(a+b*x)^(1+n)*hypergeom([1,1+n],[2+n],1+b*x/a)/(a*(1+n))],

# p<0
[x^5*(e+f*x)^n/(a+b*x^3),x,7,e^2*(e+f*x)^(1+n)/(b*f^3*(1+n))-2*e*(e+f*x)^(2+n)/(b*f^3*(2+n))+(e+f*x)^(3+n)/(b*f^3*(3+n))+1/3*a*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],b^(1/3)*(e+f*x)/(b^(1/3)*e-a^(1/3)*f))/(b^(5/3)*(b^(1/3)*e-a^(1/3)*f)*(1+n))+1/3*a*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],b^(1/3)*(e+f*x)/(b^(1/3)*e+(-1)^(1/3)*a^(1/3)*f))/(b^(5/3)*(b^(1/3)*e+(-1)^(1/3)*a^(1/3)*f)*(1+n))+1/3*a*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],b^(1/3)*(e+f*x)/(b^(1/3)*e-(-1)^(2/3)*a^(1/3)*f))/(b^(5/3)*(b^(1/3)*e-(-1)^(2/3)*a^(1/3)*f)*(1+n))],
[x^4*(e+f*x)^n/(a+b*x^3),x,7,-e*(e+f*x)^(1+n)/(b*f^2*(1+n))+(e+f*x)^(2+n)/(b*f^2*(2+n))-1/3*a^(2/3)*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],b^(1/3)*(e+f*x)/(b^(1/3)*e-a^(1/3)*f))/(b^(4/3)*(b^(1/3)*e-a^(1/3)*f)*(1+n))+1/3*(-1)^(1/3)*a^(2/3)*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],(-1)^(2/3)*b^(1/3)*(e+f*x)/((-1)^(2/3)*b^(1/3)*e-a^(1/3)*f))/(b^(4/3)*((-1)^(2/3)*b^(1/3)*e-a^(1/3)*f)*(1+n))+1/3*(-1)^(2/3)*a^(2/3)*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],(-1)^(1/3)*b^(1/3)*(e+f*x)/((-1)^(1/3)*b^(1/3)*e+a^(1/3)*f))/(b^(4/3)*((-1)^(1/3)*b^(1/3)*e+a^(1/3)*f)*(1+n))],
[x^3*(e+f*x)^n/(a+b*x^3),x,7,(e+f*x)^(1+n)/(b*f*(1+n))+1/3*a^(1/3)*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],b^(1/3)*(e+f*x)/(b^(1/3)*e-a^(1/3)*f))/(b*(b^(1/3)*e-a^(1/3)*f)*(1+n))+1/3*a^(1/3)*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],(-1)^(2/3)*b^(1/3)*(e+f*x)/((-1)^(2/3)*b^(1/3)*e-a^(1/3)*f))/(b*((-1)^(2/3)*b^(1/3)*e-a^(1/3)*f)*(1+n))-1/3*a^(1/3)*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],(-1)^(1/3)*b^(1/3)*(e+f*x)/((-1)^(1/3)*b^(1/3)*e+a^(1/3)*f))/(b*((-1)^(1/3)*b^(1/3)*e+a^(1/3)*f)*(1+n))],
[x^2*(e+f*x)^n/(a+b*x^3),x,5,-1/3*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],b^(1/3)*(e+f*x)/(b^(1/3)*e-a^(1/3)*f))/(b^(2/3)*(b^(1/3)*e-a^(1/3)*f)*(1+n))-1/3*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],b^(1/3)*(e+f*x)/(b^(1/3)*e+(-1)^(1/3)*a^(1/3)*f))/(b^(2/3)*(b^(1/3)*e+(-1)^(1/3)*a^(1/3)*f)*(1+n))-1/3*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],b^(1/3)*(e+f*x)/(b^(1/3)*e-(-1)^(2/3)*a^(1/3)*f))/(b^(2/3)*(b^(1/3)*e-(-1)^(2/3)*a^(1/3)*f)*(1+n))],
[x*(e+f*x)^n/(a+b*x^3),x,5,1/3*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],b^(1/3)*(e+f*x)/(b^(1/3)*e-a^(1/3)*f))/(a^(1/3)*b^(1/3)*(b^(1/3)*e-a^(1/3)*f)*(1+n))-1/3*(-1)^(1/3)*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],(-1)^(2/3)*b^(1/3)*(e+f*x)/((-1)^(2/3)*b^(1/3)*e-a^(1/3)*f))/(a^(1/3)*b^(1/3)*((-1)^(2/3)*b^(1/3)*e-a^(1/3)*f)*(1+n))-1/3*(-1)^(2/3)*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],(-1)^(1/3)*b^(1/3)*(e+f*x)/((-1)^(1/3)*b^(1/3)*e+a^(1/3)*f))/(a^(1/3)*b^(1/3)*((-1)^(1/3)*b^(1/3)*e+a^(1/3)*f)*(1+n))],
[(e+f*x)^n/(a+b*x^3),x,5,-1/3*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],b^(1/3)*(e+f*x)/(b^(1/3)*e-a^(1/3)*f))/(a^(2/3)*(b^(1/3)*e-a^(1/3)*f)*(1+n))-1/3*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],(-1)^(2/3)*b^(1/3)*(e+f*x)/((-1)^(2/3)*b^(1/3)*e-a^(1/3)*f))/(a^(2/3)*((-1)^(2/3)*b^(1/3)*e-a^(1/3)*f)*(1+n))+1/3*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],(-1)^(1/3)*b^(1/3)*(e+f*x)/((-1)^(1/3)*b^(1/3)*e+a^(1/3)*f))/(a^(2/3)*((-1)^(1/3)*b^(1/3)*e+a^(1/3)*f)*(1+n))],
[(e+f*x)^n/(x*(a+b*x^3)),x,8,1/3*b^(1/3)*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],b^(1/3)*(e+f*x)/(b^(1/3)*e-a^(1/3)*f))/(a*(b^(1/3)*e-a^(1/3)*f)*(1+n))+1/3*b^(1/3)*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],b^(1/3)*(e+f*x)/(b^(1/3)*e+(-1)^(1/3)*a^(1/3)*f))/(a*(b^(1/3)*e+(-1)^(1/3)*a^(1/3)*f)*(1+n))+1/3*b^(1/3)*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],b^(1/3)*(e+f*x)/(b^(1/3)*e-(-1)^(2/3)*a^(1/3)*f))/(a*(b^(1/3)*e-(-1)^(2/3)*a^(1/3)*f)*(1+n))-(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],1+f*x/e)/(a*e*(1+n))],
[(e+f*x)^n/(x^2*(a+b*x^3)),x,8,-1/3*b^(2/3)*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],b^(1/3)*(e+f*x)/(b^(1/3)*e-a^(1/3)*f))/(a^(4/3)*(b^(1/3)*e-a^(1/3)*f)*(1+n))+1/3*(-1)^(1/3)*b^(2/3)*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],(-1)^(2/3)*b^(1/3)*(e+f*x)/((-1)^(2/3)*b^(1/3)*e-a^(1/3)*f))/(a^(4/3)*((-1)^(2/3)*b^(1/3)*e-a^(1/3)*f)*(1+n))+1/3*(-1)^(2/3)*b^(2/3)*(e+f*x)^(1+n)*hypergeom([1,1+n],[2+n],(-1)^(1/3)*b^(1/3)*(e+f*x)/((-1)^(1/3)*b^(1/3)*e+a^(1/3)*f))/(a^(4/3)*((-1)^(1/3)*b^(1/3)*e+a^(1/3)*f)*(1+n))+f*(e+f*x)^(1+n)*hypergeom([2,1+n],[2+n],1+f*x/e)/(a*e^2*(1+n))],
[x^2*(c+d*x)^(1+n)/(a+b*x^3),x,5,-1/3*(c+d*x)^(2+n)*hypergeom([1,2+n],[3+n],b^(1/3)*(c+d*x)/(b^(1/3)*c-a^(1/3)*d))/(b^(2/3)*(b^(1/3)*c-a^(1/3)*d)*(2+n))-1/3*(c+d*x)^(2+n)*hypergeom([1,2+n],[3+n],b^(1/3)*(c+d*x)/(b^(1/3)*c+(-1)^(1/3)*a^(1/3)*d))/(b^(2/3)*(b^(1/3)*c+(-1)^(1/3)*a^(1/3)*d)*(2+n))-1/3*(c+d*x)^(2+n)*hypergeom([1,2+n],[3+n],b^(1/3)*(c+d*x)/(b^(1/3)*c-(-1)^(2/3)*a^(1/3)*d))/(b^(2/3)*(b^(1/3)*c-(-1)^(2/3)*a^(1/3)*d)*(2+n))],
[x^m*(e+f*x)^n/(a+b*x^3),x,8,1/3*x^(1+m)*(e+f*x)^n*AppellF1(1+m,-n,1,2+m,-f*x/e,-b^(1/3)*x/a^(1/3))/(a*(1+m)*(1+f*x/e)^n)+1/3*x^(1+m)*(e+f*x)^n*AppellF1(1+m,-n,1,2+m,-f*x/e,(-1)^(1/3)*b^(1/3)*x/a^(1/3))/(a*(1+m)*(1+f*x/e)^n)+1/3*x^(1+m)*(e+f*x)^n*AppellF1(1+m,-n,1,2+m,-f*x/e,-(-1)^(2/3)*b^(1/3)*x/a^(1/3))/(a*(1+m)*(1+f*x/e)^n)],

# Integrands of the form x^m (c+d x)^n (a+b x^3)^(p/2)

# p>0
[sqrt(c+d*x^3)/(a+b*x),x,13,2/3*sqrt(c+d*x^3)/b-2*a*d^(1/3)*sqrt(c+d*x^3)/(b^2*(d^(1/3)*x+c^(1/3)*(1+sqrt(3))))+3^(1/4)*a*c^(1/3)*d^(1/3)*(c^(1/3)+d^(1/3)*x)*EllipticE((d^(1/3)*x+c^(1/3)*(1-sqrt(3)))/(d^(1/3)*x+c^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2-sqrt(3))*sqrt((c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/(d^(1/3)*x+c^(1/3)*(1+sqrt(3)))^2)/(b^2*sqrt(c+d*x^3)*sqrt(c^(1/3)*(c^(1/3)+d^(1/3)*x)/(d^(1/3)*x+c^(1/3)*(1+sqrt(3)))^2))+2*a*d^(1/3)*(c^(1/3)+d^(1/3)*x)*EllipticF((d^(1/3)*x+c^(1/3)*(1-sqrt(3)))/(d^(1/3)*x+c^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(a*d^(1/3)+b*c^(1/3)*(1-sqrt(3)))*sqrt(2+sqrt(3))*sqrt((c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/(d^(1/3)*x+c^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^3*sqrt(c+d*x^3)*sqrt(c^(1/3)*(c^(1/3)+d^(1/3)*x)/(d^(1/3)*x+c^(1/3)*(1+sqrt(3)))^2))-2*(b^3*c-a^3*d)*(c^(1/3)+d^(1/3)*x)*EllipticF((d^(1/3)*x+c^(1/3)*(1-sqrt(3)))/(d^(1/3)*x+c^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt((c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/(d^(1/3)*x+c^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*b^3*(-a*d^(1/3)+b*c^(1/3)*(1+sqrt(3)))*sqrt(c+d*x^3)*sqrt(c^(1/3)*(c^(1/3)+d^(1/3)*x)/(d^(1/3)*x+c^(1/3)*(1+sqrt(3)))^2))-c^(1/6)*(c^(1/3)+d^(1/3)*x)*arctanh(sqrt(b^2*c^(2/3)+a*b*c^(1/3)*d^(1/3)+a^2*d^(2/3))*sqrt(2-sqrt(3))*sqrt(1-(d^(1/3)*x+c^(1/3)*(1-sqrt(3)))^2/(d^(1/3)*x+c^(1/3)*(1+sqrt(3)))^2)/(3^(1/4)*c^(1/6)*sqrt(b)*sqrt(b*c^(1/3)-a*d^(1/3))*sqrt(7-4*sqrt(3)+(d^(1/3)*x+c^(1/3)*(1-sqrt(3)))^2/(d^(1/3)*x+c^(1/3)*(1+sqrt(3)))^2)))*sqrt(b*c^(1/3)-a*d^(1/3))*sqrt(b^2*c^(2/3)+a*b*c^(1/3)*d^(1/3)+a^2*d^(2/3))*sqrt(c^(2/3)*(1-d^(1/3)*x/c^(1/3)+d^(2/3)*x^2/c^(2/3))/(d^(1/3)*x+c^(1/3)*(1+sqrt(3)))^2)/(b^(5/2)*sqrt(c+d*x^3)*sqrt(c^(1/3)*(c^(1/3)+d^(1/3)*x)/(d^(1/3)*x+c^(1/3)*(1+sqrt(3)))^2))-4*3^(1/4)*c^(1/3)*(b^3*c-a^3*d)*(c^(1/3)+d^(1/3)*x)*EllipticPi((-d^(1/3)*x-c^(1/3)*(1-sqrt(3)))/(d^(1/3)*x+c^(1/3)*(1+sqrt(3))),(-a*d^(1/3)+b*c^(1/3)*(1+sqrt(3)))^2/(-a*d^(1/3)+b*c^(1/3)*(1-sqrt(3)))^2,sqrt(-7-4*sqrt(3)))*sqrt(2+sqrt(3))*sqrt(c^(2/3)*(1-d^(1/3)*x/c^(1/3)+d^(2/3)*x^2/c^(2/3))/(d^(1/3)*x+c^(1/3)*(1+sqrt(3)))^2)/(b^2*(2*b^2*c^(2/3)+2*a*b*c^(1/3)*d^(1/3)-a^2*d^(2/3))*sqrt(c+d*x^3)*sqrt(c^(1/3)*(c^(1/3)+d^(1/3)*x)/(d^(1/3)*x+c^(1/3)*(1+sqrt(3)))^2))],

# p<0

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

# Integrands of the form  (c+d x+e x^2)^m (f+g x+h x^2)^n (a+b x^3)^p

# Integrands of the form (f+g x+h x^2) / ((c+e x^2) Sqrt[a+b x^3]) with b g^3 - 8 a h^3 = 0 && g^2 + 2 f h = 0 && b c g - 4 a e h = 0
[(2-2*x-x^2)/((2+x^2)*sqrt(1+x^3)),x,2,2*arctan((1+x)/sqrt(1+x^3))],
[(2+2*x-x^2)/((2+x^2)*sqrt(1-x^3)),x,2,-2*arctan((1-x)/sqrt(1-x^3))],
[(2+2*x-x^2)/((2+x^2)*sqrt(-1+x^3)),x,2,-2*arctanh((1-x)/sqrt(-1+x^3))],
[(2-2*x-x^2)/((2+x^2)*sqrt(-1-x^3)),x,2,2*arctanh((1+x)/sqrt(-1-x^3))],

# Integrands of the form (f+g x+h x^2) / ((c+d x+e x^2) Sqrt[a+b x^3]) with b g^3 - 8 a h^3 = 0 && g^2 + 2 f h = 0 && b d f + b c g - 4 a e h = 0
[(2-2*x-x^2)/((2+d+d*x+x^2)*sqrt(1+x^3)),x,2,2*arctan((1+x)*sqrt(1+d)/sqrt(1+x^3))/sqrt(1+d)],
[(2+2*x-x^2)/((2-d+d*x+x^2)*sqrt(1-x^3)),x,2,-2*arctan((1-x)*sqrt(1-d)/sqrt(1-x^3))/sqrt(1-d)],
[(2+2*x-x^2)/((2-d+d*x+x^2)*sqrt(-1+x^3)),x,2,-2*arctanh((1-x)*sqrt(1-d)/sqrt(-1+x^3))/sqrt(1-d)],
[(2-2*x-x^2)/((2+d+d*x+x^2)*sqrt(-1-x^3)),x,2,2*arctanh((1+x)*sqrt(1+d)/sqrt(-1-x^3))/sqrt(1+d)],

# Algebraic Function Integration Problems

# Integrands of the form u (a+b x^2+c x^4)^p

# Integrands of the form (d+e x)^q (a+c x^4)^(p/2)

# p>0
[(d+e*x)^3*sqrt(a+c*x^4),x,11,1/6*e^3*(a+c*x^4)^(3/2)/c+3/4*a*d^2*e*arctanh(x^2*sqrt(c)/sqrt(a+c*x^4))/sqrt(c)+3/4*d^2*e*x^2*sqrt(a+c*x^4)+1/15*d*x*(5*d^2+9*e^2*x^2)*sqrt(a+c*x^4)+6/5*a*d*e^2*x*sqrt(a+c*x^4)/(sqrt(c)*(sqrt(a)+x^2*sqrt(c)))-6/5*a^(5/4)*d*e^2*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(a+c*x^4))+1/15*a^(3/4)*d*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(9*e^2*sqrt(a)+5*d^2*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(a+c*x^4))],
[(d+e*x)^2*sqrt(a+c*x^4),x,10,1/2*a*d*e*arctanh(x^2*sqrt(c)/sqrt(a+c*x^4))/sqrt(c)+1/2*d*e*x^2*sqrt(a+c*x^4)+1/15*x*(5*d^2+3*e^2*x^2)*sqrt(a+c*x^4)+2/5*a*e^2*x*sqrt(a+c*x^4)/(sqrt(c)*(sqrt(a)+x^2*sqrt(c)))-2/5*a^(5/4)*e^2*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(a+c*x^4))+1/15*a^(3/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(3*e^2*sqrt(a)+5*d^2*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(a+c*x^4))],
[(d+e*x)*sqrt(a+c*x^4),x,8,1/4*a*e*arctanh(x^2*sqrt(c)/sqrt(a+c*x^4))/sqrt(c)+1/3*d*x*sqrt(a+c*x^4)+1/4*e*x^2*sqrt(a+c*x^4)+1/3*a^(3/4)*d*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(1/4)*sqrt(a+c*x^4))],
[sqrt(a+c*x^4),x,2,1/3*x*sqrt(a+c*x^4)+1/3*a^(3/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(1/4)*sqrt(a+c*x^4))],
[sqrt(a+c*x^4)/(d+e*x),x,15,1/2*d^2*arctanh(x^2*sqrt(c)/sqrt(a+c*x^4))*sqrt(c)/e^3-1/2*arctan(x*sqrt(-c*d^4-a*e^4)/(d*e*sqrt(a+c*x^4)))*sqrt(-c*d^4-a*e^4)/e^3-1/2*arctanh((a*e^2+c*d^2*x^2)/(sqrt(c*d^4+a*e^4)*sqrt(a+c*x^4)))*sqrt(c*d^4+a*e^4)/e^3+1/2*sqrt(a+c*x^4)/e-d*x*sqrt(c)*sqrt(a+c*x^4)/(e^2*(sqrt(a)+x^2*sqrt(c)))+a^(1/4)*c^(1/4)*d*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(e^2*sqrt(a+c*x^4))+1/2*c^(1/4)*d*(c*d^4+a*e^4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*e^4*(e^2*sqrt(a)+d^2*sqrt(c))*sqrt(a+c*x^4))-1/4*(c*d^4+a*e^4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/4*(e^2*sqrt(a)+d^2*sqrt(c))^2/(d^2*e^2*sqrt(a)*sqrt(c)),sqrt(1/2))*(-e^2*sqrt(a)+d^2*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*c^(1/4)*d*e^4*(e^2*sqrt(a)+d^2*sqrt(c))*sqrt(a+c*x^4))-1/2*a^(1/4)*c^(1/4)*d*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*(e^2+d^2*sqrt(c)/sqrt(a))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(e^4*sqrt(a+c*x^4))],
[sqrt(a+c*x^4)/(d+e*x)^2,x,32,-d*arctanh(x^2*sqrt(c)/sqrt(a+c*x^4))*sqrt(c)/e^3-1/2*(c*d^4-a*e^4)*arctan(x*sqrt(-c*d^4-a*e^4)/(d*e*sqrt(a+c*x^4)))/(d*e^3*sqrt(-c*d^4-a*e^4))+1/2*arctan(x*sqrt(-c*d^4-a*e^4)/(d*e*sqrt(a+c*x^4)))*sqrt(-c*d^4-a*e^4)/(d*e^3)+c*d^3*arctanh((a*e^2+c*d^2*x^2)/(sqrt(c*d^4+a*e^4)*sqrt(a+c*x^4)))/(e^3*sqrt(c*d^4+a*e^4))-d*sqrt(a+c*x^4)/(e*(d^2-e^2*x^2))+x*sqrt(a+c*x^4)/(d^2-e^2*x^2)+2*x*sqrt(c)*sqrt(a+c*x^4)/(e^2*(sqrt(a)+x^2*sqrt(c)))-2*a^(1/4)*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(e^2*sqrt(a+c*x^4))-1/2*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(-e^2*sqrt(a)+d^2*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*e^4*sqrt(a+c*x^4))+1/4*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/4*(e^2*sqrt(a)+d^2*sqrt(c))^2/(d^2*e^2*sqrt(a)*sqrt(c)),sqrt(1/2))*(-e^2*sqrt(a)+d^2*sqrt(c))^2*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*c^(1/4)*d^2*e^4*sqrt(a+c*x^4))-1/2*c^(1/4)*(c*d^4+a*e^4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*e^4*(e^2*sqrt(a)+d^2*sqrt(c))*sqrt(a+c*x^4))+1/4*(c*d^4+a*e^4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/4*(e^2*sqrt(a)+d^2*sqrt(c))^2/(d^2*e^2*sqrt(a)*sqrt(c)),sqrt(1/2))*(-e^2*sqrt(a)+d^2*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*c^(1/4)*d^2*e^4*(e^2*sqrt(a)+d^2*sqrt(c))*sqrt(a+c*x^4))+1/4*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(e^2*sqrt(a)+d^2*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*e^4*sqrt(a+c*x^4))+3/4*a^(1/4)*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*(e^2+d^2*sqrt(c)/sqrt(a))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(e^4*sqrt(a+c*x^4))],

# p<0
[(d+e*x)^3/sqrt(a+c*x^4),x,9,3/2*d^2*e*arctanh(x^2*sqrt(c)/sqrt(a+c*x^4))/sqrt(c)+1/2*e^3*sqrt(a+c*x^4)/c+3*d*e^2*x*sqrt(a+c*x^4)/(sqrt(c)*(sqrt(a)+x^2*sqrt(c)))-3*a^(1/4)*d*e^2*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(a+c*x^4))+1/2*d*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(3*e^2*sqrt(a)+d^2*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*c^(3/4)*sqrt(a+c*x^4))],
[(d+e*x)^2/sqrt(a+c*x^4),x,8,d*e*arctanh(x^2*sqrt(c)/sqrt(a+c*x^4))/sqrt(c)+e^2*x*sqrt(a+c*x^4)/(sqrt(c)*(sqrt(a)+x^2*sqrt(c)))-a^(1/4)*e^2*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(a+c*x^4))+1/2*a^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*(e^2+d^2*sqrt(c)/sqrt(a))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(c^(3/4)*sqrt(a+c*x^4))],
[(d+e*x)/sqrt(a+c*x^4),x,6,1/2*e*arctanh(x^2*sqrt(c)/sqrt(a+c*x^4))/sqrt(c)+1/2*d*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*c^(1/4)*sqrt(a+c*x^4))],
[1/sqrt(a+c*x^4),x,1,1/2*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*c^(1/4)*sqrt(a+c*x^4))],
[1/((d+e*x)*sqrt(a+c*x^4)),x,7,1/2*e*arctan(x*sqrt(-c*d^4-a*e^4)/(d*e*sqrt(a+c*x^4)))/sqrt(-c*d^4-a*e^4)-1/2*e*arctanh((a*e^2+c*d^2*x^2)/(sqrt(c*d^4+a*e^4)*sqrt(a+c*x^4)))/sqrt(c*d^4+a*e^4)+1/2*c^(1/4)*d*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*(e^2*sqrt(a)+d^2*sqrt(c))*sqrt(a+c*x^4))-1/4*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/4*(e^2*sqrt(a)+d^2*sqrt(c))^2/(d^2*e^2*sqrt(a)*sqrt(c)),sqrt(1/2))*(-e^2*sqrt(a)+d^2*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*c^(1/4)*d*(e^2*sqrt(a)+d^2*sqrt(c))*sqrt(a+c*x^4))],
[1/((d+e*x)^2*sqrt(a+c*x^4)),x,11,-c*d^3*e*arctan(x*sqrt(-c*d^4-a*e^4)/(d*e*sqrt(a+c*x^4)))/(-c*d^4-a*e^4)^(3/2)-c*d^3*e*arctanh((a*e^2+c*d^2*x^2)/(sqrt(c*d^4+a*e^4)*sqrt(a+c*x^4)))/(c*d^4+a*e^4)^(3/2)-e^3*sqrt(a+c*x^4)/((c*d^4+a*e^4)*(d+e*x))+e^2*x*sqrt(c)*sqrt(a+c*x^4)/((c*d^4+a*e^4)*(sqrt(a)+x^2*sqrt(c)))-a^(1/4)*c^(1/4)*e^2*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/((c*d^4+a*e^4)*sqrt(a+c*x^4))+1/2*c^(1/4)*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*(e^2*sqrt(a)+d^2*sqrt(c))*sqrt(a+c*x^4))-1/2*c^(3/4)*d^2*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/4*(e^2*sqrt(a)+d^2*sqrt(c))^2/(d^2*e^2*sqrt(a)*sqrt(c)),sqrt(1/2))*(-e^2*sqrt(a)+d^2*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*(c*d^4+a*e^4)*(e^2*sqrt(a)+d^2*sqrt(c))*sqrt(a+c*x^4))],
[1/((d+e*x)^3*sqrt(a+c*x^4)),x,12,3/2*c*d^2*e*(c*d^4-a*e^4)*arctan(x*sqrt(-c*d^4-a*e^4)/(d*e*sqrt(a+c*x^4)))/(-c*d^4-a*e^4)^(5/2)-3/2*c*d^2*e*(c*d^4-a*e^4)*arctanh((a*e^2+c*d^2*x^2)/(sqrt(c*d^4+a*e^4)*sqrt(a+c*x^4)))/(c*d^4+a*e^4)^(5/2)-1/2*e^3*sqrt(a+c*x^4)/((c*d^4+a*e^4)*(d+e*x)^2)-3*c*d^3*e^3*sqrt(a+c*x^4)/((c*d^4+a*e^4)^2*(d+e*x))+3*c^(3/2)*d^3*e^2*x*sqrt(a+c*x^4)/((c*d^4+a*e^4)^2*(sqrt(a)+x^2*sqrt(c)))-3*a^(1/4)*c^(5/4)*d^3*e^2*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/((c*d^4+a*e^4)^2*sqrt(a+c*x^4))+1/2*c^(3/4)*d*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*(c*d^4+a*e^4)*sqrt(a+c*x^4))-3/4*c^(3/4)*d*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/4*(e^2*sqrt(a)+d^2*sqrt(c))^2/(d^2*e^2*sqrt(a)*sqrt(c)),sqrt(1/2))*(-e^2*sqrt(a)+d^2*sqrt(c))^2*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*(c*d^4+a*e^4)^2*sqrt(a+c*x^4))],
[(d+e*x)^3/(a+c*x^4)^(3/2),x,4,1/2*(-a*e^3+c*x*(d^3+3*d^2*e*x+3*d*e^2*x^2))/(a*c*sqrt(a+c*x^4))-3/2*d*e^2*x*sqrt(a+c*x^4)/(a*sqrt(c)*(sqrt(a)+x^2*sqrt(c)))+3/2*d*e^2*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(3/4)*c^(3/4)*sqrt(a+c*x^4))+1/4*d*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(-3*e^2*sqrt(a)+d^2*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(5/4)*c^(3/4)*sqrt(a+c*x^4))],
[(d+e*x)^2/(a+c*x^4)^(3/2),x,4,1/2*x*(d+e*x)^2/(a*sqrt(a+c*x^4))-1/2*e^2*x*sqrt(a+c*x^4)/(a*sqrt(c)*(sqrt(a)+x^2*sqrt(c)))+1/2*e^2*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(3/4)*c^(3/4)*sqrt(a+c*x^4))+1/4*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(-e^2*sqrt(a)+d^2*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(5/4)*c^(3/4)*sqrt(a+c*x^4))],
[(d+e*x)/(a+c*x^4)^(3/2),x,3,1/2*x*(d+e*x)/(a*sqrt(a+c*x^4))+1/4*d*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(5/4)*c^(1/4)*sqrt(a+c*x^4))],
[1/(a+c*x^4)^(3/2),x,2,1/2*x/(a*sqrt(a+c*x^4))+1/4*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(5/4)*c^(1/4)*sqrt(a+c*x^4))],
[1/((d+e*x)*(a+c*x^4)^(3/2)),x,14,-1/2*e^5*arctan(x*sqrt(-c*d^4-a*e^4)/(d*e*sqrt(a+c*x^4)))/(-c*d^4-a*e^4)^(3/2)-1/2*e^5*arctanh((a*e^2+c*d^2*x^2)/(sqrt(c*d^4+a*e^4)*sqrt(a+c*x^4)))/(c*d^4+a*e^4)^(3/2)+1/2*e*(a*e^2-c*d^2*x^2)/(a*(c*d^4+a*e^4)*sqrt(a+c*x^4))+1/2*c*d*x*(d^2+e^2*x^2)/(a*(c*d^4+a*e^4)*sqrt(a+c*x^4))-1/2*d*e^2*x*sqrt(c)*sqrt(a+c*x^4)/(a*(c*d^4+a*e^4)*(sqrt(a)+x^2*sqrt(c)))+1/2*c^(1/4)*d*e^2*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(3/4)*(c*d^4+a*e^4)*sqrt(a+c*x^4))+1/4*c^(1/4)*d*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(-e^2*sqrt(a)+d^2*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(5/4)*(c*d^4+a*e^4)*sqrt(a+c*x^4))+1/2*c^(1/4)*d*e^4*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*(c*d^4+a*e^4)*(e^2*sqrt(a)+d^2*sqrt(c))*sqrt(a+c*x^4))-1/4*e^4*sqrt(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/4*(e^2*sqrt(a)+d^2*sqrt(c))^2/(d^2*e^2*sqrt(a)*sqrt(c)),sqrt(1/2))*(-e^2*sqrt(a)+d^2*sqrt(c))*(sqrt(a)+x^2*sqrt(c))*sqrt((a+c*x^4)/(sqrt(a)+x^2*sqrt(c))^2)/(a^(1/4)*c^(1/4)*d*(c*d^4+a*e^4)*(e^2*sqrt(a)+d^2*sqrt(c))*sqrt(a+c*x^4))],

#  {1/((d + e*x)^2*(a + c*x^4)^(3/2)), x, 79, (d*e*(a*e^2 - c*d^2*x^2))/(a*(c*d^4 + a*e^4)*(d^2 - e^2*x^2)*Sqrt[a + c*x^4]) - (c*x*(d^2 + e^2*x^2))/(2*a*(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) + (c*d^2*x*(c*d^4 - a*e^4 + 2*c*d^2*e^2*x^2))/(a*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) + (e^7*Sqrt[a + c*x^4])/(2*(c*d^4 + a*e^4)^2*(d - e*x)) - (e^7*Sqrt[a + c*x^4])/(2*(c*d^4 + a*e^4)^2*(d + e*x)) - (2*c^(3/2)*d^4*e^2*x*Sqrt[a + c*x^4])/(a*(c*d^4 + a*e^4)^2*(Sqrt[a] + Sqrt[c]*x^2)) + (Sqrt[c]*e^6*x*Sqrt[a + c*x^4])/((c*d^4 + a*e^4)^2*(Sqrt[a] + Sqrt[c]*x^2)) + (Sqrt[c]*e^2*x*Sqrt[a + c*x^4])/(2*a*(c*d^4 + a*e^4)*(Sqrt[a] + Sqrt[c]*x^2)) + (d*e^3*(c*d^4 - 2*a*e^4)*Sqrt[a + c*x^4])/(a*(c*d^4 + a*e^4)^2*(d^2 - e^2*x^2)) + (c*d^3*e^5*ArcTan[(Sqrt[(-c)*d^4 - a*e^4]*x)/(d*e*Sqrt[a + c*x^4])])/((-c)*d^4 - a*e^4)^(5/2) + (e^5*ArcTan[(Sqrt[(-c)*d^4 - a*e^4]*x)/(d*e*Sqrt[a + c*x^4])])/(2*d*((-c)*d^4 - a*e^4)^(3/2)) + (e^5*(5*c*d^4 + a*e^4)*ArcTan[(Sqrt[(-c)*d^4 - a*e^4]*x)/(d*e*Sqrt[a + c*x^4])])/(2*d*((-c)*d^4 - a*e^4)^(5/2)) - (3*c*d^3*e^5*ArcTanh[(a*e^2 + c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])])/(c*d^4 + a*e^4)^(5/2) + (2*c^(5/4)*d^4*e^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(a^(3/4)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) - (a^(1/4)*c^(1/4)*e^6*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/((c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) - (c^(1/4)*e^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(3/4)*(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) + (c^(3/4)*d^2*(c*d^4 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2 - a*e^4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(5/4)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) - (c^(1/4)*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(5/4)*(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) + (c^(1/4)*e^4*(5*c*d^4 + a*e^4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) - (c^(3/4)*d^2*e^4*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[(Sqrt[c]*d^2 + Sqrt[a]*e^2)^2/(4*Sqrt[a]*Sqrt[c]*d^2*e^2), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) + (e^4*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[(Sqrt[c]*d^2 + Sqrt[a]*e^2)^2/(4*Sqrt[a]*Sqrt[c]*d^2*e^2), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*c^(1/4)*d^2*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*(c*d^4 + a*e^4)*Sqrt[a + c*x^4]) - (e^4*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*(5*c*d^4 + a*e^4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[(Sqrt[c]*d^2 + Sqrt[a]*e^2)^2/(4*Sqrt[a]*Sqrt[c]*d^2*e^2), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*c^(1/4)*d^2*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*(c*d^4 + a*e^4)^2*Sqrt[a + c*x^4])} 

# Integrands of the form x^m (c+d x)^n (a+b x^4)^p with n symbolic

# p>0

# p<0
[x^3*(c+d*x)^n/(a+b*x^4),x,10,-1/4*(c+d*x)^(1+n)*hypergeom([1,1+n],[2+n],b^(1/4)*(c+d*x)/(b^(1/4)*c-(-a)^(1/4)*d))/(b^(3/4)*(b^(1/4)*c-(-a)^(1/4)*d)*(1+n))-1/4*(c+d*x)^(1+n)*hypergeom([1,1+n],[2+n],b^(1/4)*(c+d*x)/(b^(1/4)*c+(-a)^(1/4)*d))/(b^(3/4)*(b^(1/4)*c+(-a)^(1/4)*d)*(1+n))-1/4*(c+d*x)^(1+n)*hypergeom([1,1+n],[2+n],b^(1/4)*(c+d*x)/(b^(1/4)*c-d*sqrt(-sqrt(-a))))/(b^(3/4)*(1+n)*(b^(1/4)*c-d*sqrt(-sqrt(-a))))-1/4*(c+d*x)^(1+n)*hypergeom([1,1+n],[2+n],b^(1/4)*(c+d*x)/(b^(1/4)*c+d*sqrt(-sqrt(-a))))/(b^(3/4)*(1+n)*(b^(1/4)*c+d*sqrt(-sqrt(-a))))],
[x^3*(c+d*x)^(1+n)/(a+b*x^4),x,10,-1/4*(c+d*x)^(2+n)*hypergeom([1,2+n],[3+n],b^(1/4)*(c+d*x)/(b^(1/4)*c-(-a)^(1/4)*d))/(b^(3/4)*(b^(1/4)*c-(-a)^(1/4)*d)*(2+n))-1/4*(c+d*x)^(2+n)*hypergeom([1,2+n],[3+n],b^(1/4)*(c+d*x)/(b^(1/4)*c+(-a)^(1/4)*d))/(b^(3/4)*(b^(1/4)*c+(-a)^(1/4)*d)*(2+n))-1/4*(c+d*x)^(2+n)*hypergeom([1,2+n],[3+n],b^(1/4)*(c+d*x)/(b^(1/4)*c-d*sqrt(-sqrt(-a))))/(b^(3/4)*(2+n)*(b^(1/4)*c-d*sqrt(-sqrt(-a))))-1/4*(c+d*x)^(2+n)*hypergeom([1,2+n],[3+n],b^(1/4)*(c+d*x)/(b^(1/4)*c+d*sqrt(-sqrt(-a))))/(b^(3/4)*(2+n)*(b^(1/4)*c+d*sqrt(-sqrt(-a))))],

# Integrands of the form (c+d x+e x^2)^n (a+b x^4)^(p/2)

# p>0

# p<0
[1/((c+d*x+e*x^2)*sqrt(a+b*x^4)),x,16,1/2*b^(1/4)*e*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*(d-sqrt(d^2-4*c*e))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(1/4)*sqrt(d^2-4*c*e)*(2*e^2*sqrt(a)+sqrt(b)*(d^2-2*c*e-d*sqrt(d^2-4*c*e)))*sqrt(a+b*x^4))+1/2*e*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/4*(2*e^2*sqrt(a)+sqrt(b)*(d^2-2*c*e-d*sqrt(d^2-4*c*e)))^2/(e^2*sqrt(a)*sqrt(b)*(d-sqrt(d^2-4*c*e))^2),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*(2*e^2*sqrt(a)-sqrt(b)*(d^2-2*c*e-d*sqrt(d^2-4*c*e)))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(1/4)*b^(1/4)*(d-sqrt(d^2-4*c*e))*sqrt(d^2-4*c*e)*(2*e^2*sqrt(a)+sqrt(b)*(d^2-2*c*e-d*sqrt(d^2-4*c*e)))*sqrt(a+b*x^4))-1/2*b^(1/4)*e*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*(d+sqrt(d^2-4*c*e))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(1/4)*sqrt(d^2-4*c*e)*(2*e^2*sqrt(a)+sqrt(b)*(d^2-2*c*e+d*sqrt(d^2-4*c*e)))*sqrt(a+b*x^4))-1/2*e*sqrt(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/4*(2*e^2*sqrt(a)+sqrt(b)*(d^2-2*c*e+d*sqrt(d^2-4*c*e)))^2/(e^2*sqrt(a)*sqrt(b)*(d+sqrt(d^2-4*c*e))^2),sqrt(1/2))*(sqrt(a)+x^2*sqrt(b))*(2*e^2*sqrt(a)-sqrt(b)*(d^2-2*c*e+d*sqrt(d^2-4*c*e)))*sqrt((a+b*x^4)/(sqrt(a)+x^2*sqrt(b))^2)/(a^(1/4)*b^(1/4)*sqrt(d^2-4*c*e)*(d+sqrt(d^2-4*c*e))*(2*e^2*sqrt(a)+sqrt(b)*(d^2-2*c*e+d*sqrt(d^2-4*c*e)))*sqrt(a+b*x^4))-e^2*arctanh(1/2*(4*a*e^2+b*x^2*(d-sqrt(d^2-4*c*e))^2)/(sqrt(2)*sqrt(a+b*x^4)*sqrt(b*d^4-4*b*c*d^2*e+2*b*c^2*e^2+2*a*e^4-b*d*(d^2-2*c*e)*sqrt(d^2-4*c*e))))/(sqrt(2)*sqrt(d^2-4*c*e)*sqrt(b*d^4-4*b*c*d^2*e+2*b*c^2*e^2+2*a*e^4-b*d*(d^2-2*c*e)*sqrt(d^2-4*c*e)))+e^2*arctanh(1/2*(4*a*e^2+b*x^2*(d+sqrt(d^2-4*c*e))^2)/(sqrt(2)*sqrt(a+b*x^4)*sqrt(b*d^4-4*b*c*d^2*e+2*b*c^2*e^2+2*a*e^4+b*d*(d^2-2*c*e)*sqrt(d^2-4*c*e))))/(sqrt(2)*sqrt(d^2-4*c*e)*sqrt(b*d^4-4*b*c*d^2*e+2*b*c^2*e^2+2*a*e^4+b*d*(d^2-2*c*e)*sqrt(d^2-4*c*e)))-e^2*arctan(x*sqrt(2)*sqrt(-b*d^4+4*b*c*d^2*e-2*b*c^2*e^2-2*a*e^4-b*d*(d^2-2*c*e)*sqrt(d^2-4*c*e))/(e*(d+sqrt(d^2-4*c*e))*sqrt(a+b*x^4)))/(sqrt(2)*sqrt(d^2-4*c*e)*sqrt(-2*a*e^4-b*(d^4-4*c*d^2*e+2*c^2*e^2+d^3*sqrt(d^2-4*c*e)-2*c*d*e*sqrt(d^2-4*c*e))))+e^2*arctan(x*sqrt(2)*sqrt(-b*d^4+4*b*c*d^2*e-2*b*c^2*e^2-2*a*e^4+b*d*(d^2-2*c*e)*sqrt(d^2-4*c*e))/(e*(d-sqrt(d^2-4*c*e))*sqrt(a+b*x^4)))/(sqrt(2)*sqrt(d^2-4*c*e)*sqrt(-2*a*e^4-b*(d^4-4*c*d^2*e+2*c^2*e^2-d^3*sqrt(d^2-4*c*e)+2*c*d*e*sqrt(d^2-4*c*e))))],

# Integrands of the form u (c (a+b x^n)^q)^p

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

# q>0
[x^m*(c*(a+b*x^2)^2)^(3/2),x,3,a^3*c*x^(1+m)*sqrt(c*(a+b*x^2)^2)/((1+m)*(a+b*x^2))+3*a^2*b*c*x^(3+m)*sqrt(c*(a+b*x^2)^2)/((3+m)*(a+b*x^2))+3*a*b^2*c*x^(5+m)*sqrt(c*(a+b*x^2)^2)/((5+m)*(a+b*x^2))+b^3*c*x^(7+m)*sqrt(c*(a+b*x^2)^2)/((7+m)*(a+b*x^2))],
[x^5*(c*(a+b*x^2)^2)^(3/2),x,4,1/6*a^3*c*x^6*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)+3/8*a^2*b*c*x^8*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)+3/10*a*b^2*c*x^10*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)+1/12*b^3*c*x^12*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)],
[x^4*(c*(a+b*x^2)^2)^(3/2),x,3,1/5*a^3*c*x^5*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)+3/7*a^2*b*c*x^7*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)+1/3*a*b^2*c*x^9*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)+1/11*b^3*c*x^11*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)],
[x^3*(c*(a+b*x^2)^2)^(3/2),x,4,-1/8*a*c*(a+b*x^2)^3*sqrt(c*(a+b*x^2)^2)/b^2+1/10*c*(a+b*x^2)^4*sqrt(c*(a+b*x^2)^2)/b^2],
[x^2*(c*(a+b*x^2)^2)^(3/2),x,3,1/3*a^3*c*x^3*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)+3/5*a^2*b*c*x^5*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)+3/7*a*b^2*c*x^7*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)+1/9*b^3*c*x^9*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)],
[x*(c*(a+b*x^2)^2)^(3/2),x,3,1/8*c*(a+b*x^2)^3*sqrt(c*(a+b*x^2)^2)/b],
[(c*(a+b*x^2)^2)^(3/2),x,3,a^3*c*x*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)+a^2*b*c*x^3*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)+3/5*a*b^2*c*x^5*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)+1/7*b^3*c*x^7*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)],
[(c*(a+b*x^2)^2)^(3/2)/x,x,4,3/2*a^2*b*c*x^2*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)+3/4*a*b^2*c*x^4*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)+1/6*b^3*c*x^6*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)+a^3*c*log(x)*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)],
[(c*(a+b*x^2)^2)^(3/2)/x^2,x,3,-a^3*c*sqrt(c*(a+b*x^2)^2)/(x*(a+b*x^2))+3*a^2*b*c*x*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)+a*b^2*c*x^3*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)+1/5*b^3*c*x^5*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)],
[(c*(a+b*x^2)^2)^(3/2)/x^3,x,4,-1/2*a^3*c*sqrt(c*(a+b*x^2)^2)/(x^2*(a+b*x^2))+3/2*a*b^2*c*x^2*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)+1/4*b^3*c*x^4*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)+3*a^2*b*c*log(x)*sqrt(c*(a+b*x^2)^2)/(a+b*x^2)],
[x^2*(c*(a+b*x^2)^3)^(3/2),x,8,7/128*a^3*c*x^3*sqrt(c*(a+b*x^2)^3)+21/1024*a^5*c*x*sqrt(c*(a+b*x^2)^3)/(b*(a+b*x^2))+21/512*a^4*c*x^3*sqrt(c*(a+b*x^2)^3)/(a+b*x^2)+21/320*a^2*c*x^3*(a+b*x^2)*sqrt(c*(a+b*x^2)^3)+3/40*a*c*x^3*(a+b*x^2)^2*sqrt(c*(a+b*x^2)^3)+1/12*c*x^3*(a+b*x^2)^3*sqrt(c*(a+b*x^2)^3)-21/1024*a^(9/2)*c*arcsinh(x*sqrt(b)/sqrt(a))*sqrt(c*(a+b*x^2)^3)/(b^(3/2)*(1+b*x^2/a)^(3/2))],
[x*(c*(a+b*x^2)^3)^(3/2),x,3,1/11*c*(a+b*x^2)^4*sqrt(c*(a+b*x^2)^3)/b],
[(c*(a+b*x^2)^3)^(3/2),x,7,21/128*a^3*c*x*sqrt(c*(a+b*x^2)^3)+63/256*a^4*c*x*sqrt(c*(a+b*x^2)^3)/(a+b*x^2)+21/160*a^2*c*x*(a+b*x^2)*sqrt(c*(a+b*x^2)^3)+9/80*a*c*x*(a+b*x^2)^2*sqrt(c*(a+b*x^2)^3)+1/10*c*x*(a+b*x^2)^3*sqrt(c*(a+b*x^2)^3)+63/256*a^(7/2)*c*arcsinh(x*sqrt(b)/sqrt(a))*sqrt(c*(a+b*x^2)^3)/((1+b*x^2/a)^(3/2)*sqrt(b))],
[(c*(a+b*x^2)^3)^(3/2)/x,x,9,1/3*a^3*c*sqrt(c*(a+b*x^2)^3)+a^4*c*sqrt(c*(a+b*x^2)^3)/(a+b*x^2)+1/5*a^2*c*(a+b*x^2)*sqrt(c*(a+b*x^2)^3)+1/7*a*c*(a+b*x^2)^2*sqrt(c*(a+b*x^2)^3)+1/9*c*(a+b*x^2)^3*sqrt(c*(a+b*x^2)^3)-a^3*c*arctanh(sqrt(1+b*x^2/a))*sqrt(c*(a+b*x^2)^3)/(1+b*x^2/a)^(3/2)],
[(c*(a+b*x^2)^3)^(3/2)/x^2,x,7,105/64*a^2*b*c*x*sqrt(c*(a+b*x^2)^3)+315/128*a^3*b*c*x*sqrt(c*(a+b*x^2)^3)/(a+b*x^2)+21/16*a*b*c*x*(a+b*x^2)*sqrt(c*(a+b*x^2)^3)+9/8*b*c*x*(a+b*x^2)^2*sqrt(c*(a+b*x^2)^3)-c*(a+b*x^2)^3*sqrt(c*(a+b*x^2)^3)/x+315/128*a^(5/2)*c*arcsinh(x*sqrt(b)/sqrt(a))*sqrt(b)*sqrt(c*(a+b*x^2)^3)/(1+b*x^2/a)^(3/2)],
[(c*(a+b*x^2)^3)^(3/2)/x^3,x,9,3/2*a^2*b*c*sqrt(c*(a+b*x^2)^3)+9/2*a^3*b*c*sqrt(c*(a+b*x^2)^3)/(a+b*x^2)+9/10*a*b*c*(a+b*x^2)*sqrt(c*(a+b*x^2)^3)+9/14*b*c*(a+b*x^2)^2*sqrt(c*(a+b*x^2)^3)-1/2*c*(a+b*x^2)^3*sqrt(c*(a+b*x^2)^3)/x^2-9/2*a^2*b*c*arctanh(sqrt(1+b*x^2/a))*sqrt(c*(a+b*x^2)^3)/(1+b*x^2/a)^(3/2)],

# q<0
[x^2*(c/(a+b*x^2))^(3/2),x,3,-c*x*sqrt(c/(a+b*x^2))/b+c*arcsinh(x*sqrt(b)/sqrt(a))*sqrt(a)*sqrt(c/(a+b*x^2))*sqrt(1+b*x^2/a)/b^(3/2)],
[x*(c/(a+b*x^2))^(3/2),x,3,-c*sqrt(c/(a+b*x^2))/b],
[(c/(a+b*x^2))^(3/2),x,2,c*x*sqrt(c/(a+b*x^2))/a],
[(c/(a+b*x^2))^(3/2)/x,x,5,c*sqrt(c/(a+b*x^2))/a-c*arctanh(sqrt(1+b*x^2/a))*sqrt(c/(a+b*x^2))*sqrt(1+b*x^2/a)/a],
[(c/(a+b*x^2))^(3/2)/x^2,x,3,-c*sqrt(c/(a+b*x^2))/(a*x)-2*b*c*x*sqrt(c/(a+b*x^2))/a^2],
[(c/(a+b*x^2))^(3/2)/x^3,x,6,-3/2*b*c*sqrt(c/(a+b*x^2))/a^2-1/2*c*sqrt(c/(a+b*x^2))/(a*x^2)+3/2*b*c*arctanh(sqrt(1+b*x^2/a))*sqrt(c/(a+b*x^2))*sqrt(1+b*x^2/a)/a^2],

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

# q>0
[x^7*(c*(a+b*x^2)^(1/2))^(3/2),x,4,-2/7*a^3*(a+b*x^2)*(c*sqrt(a+b*x^2))^(3/2)/b^4+6/11*a^2*(a+b*x^2)^2*(c*sqrt(a+b*x^2))^(3/2)/b^4-2/5*a*(a+b*x^2)^3*(c*sqrt(a+b*x^2))^(3/2)/b^4+2/19*(a+b*x^2)^4*(c*sqrt(a+b*x^2))^(3/2)/b^4],
[x^5*(c*(a+b*x^2)^(1/2))^(3/2),x,4,2/7*a^2*(a+b*x^2)*(c*sqrt(a+b*x^2))^(3/2)/b^3-4/11*a*(a+b*x^2)^2*(c*sqrt(a+b*x^2))^(3/2)/b^3+2/15*(a+b*x^2)^3*(c*sqrt(a+b*x^2))^(3/2)/b^3],
[x^3*(c*(a+b*x^2)^(1/2))^(3/2),x,4,-2/7*a*(a+b*x^2)*(c*sqrt(a+b*x^2))^(3/2)/b^2+2/11*(a+b*x^2)^2*(c*sqrt(a+b*x^2))^(3/2)/b^2],
[x*(c*(a+b*x^2)^(1/2))^(3/2),x,3,2/7*c*(a+b*x^2)^(3/2)*sqrt(c*sqrt(a+b*x^2))/b],
[(c*(a+b*x^2)^(1/2))^(3/2)/x,x,7,2/3*(c*sqrt(a+b*x^2))^(3/2)+arctan((1+b*x^2/a)^(1/4))*(c*sqrt(a+b*x^2))^(3/2)/(1+b*x^2/a)^(3/4)-arctanh((1+b*x^2/a)^(1/4))*(c*sqrt(a+b*x^2))^(3/2)/(1+b*x^2/a)^(3/4)],
[(c*(a+b*x^2)^(1/2))^(3/2)/x^3,x,7,-1/2*(c*sqrt(a+b*x^2))^(3/2)/x^2+3/4*b*arctan((1+b*x^2/a)^(1/4))*(c*sqrt(a+b*x^2))^(3/2)/(a*(1+b*x^2/a)^(3/4))-3/4*b*arctanh((1+b*x^2/a)^(1/4))*(c*sqrt(a+b*x^2))^(3/2)/(a*(1+b*x^2/a)^(3/4))],
[x^2*(c*(a+b*x^2)^(1/2))^(3/2),x,5,2/15*a*x*(c*sqrt(a+b*x^2))^(3/2)/b+2/9*x^3*(c*sqrt(a+b*x^2))^(3/2)-4/15*a^2*x*(c*sqrt(a+b*x^2))^(3/2)/(b*(a+b*x^2))+4/15*a^(3/2)*sqrt(cos(1/2*arctan(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x*sqrt(b)/sqrt(a))),sqrt(2))*(c*sqrt(a+b*x^2))^(3/2)/(b^(3/2)*(1+b*x^2/a)^(3/4))],
[(c*(a+b*x^2)^(1/2))^(3/2),x,4,2/5*x*(c*sqrt(a+b*x^2))^(3/2)+6/5*a*x*(c*sqrt(a+b*x^2))^(3/2)/(a+b*x^2)-6/5*sqrt(cos(1/2*arctan(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)*(c*sqrt(a+b*x^2))^(3/2)/((1+b*x^2/a)^(3/4)*sqrt(b))],
[(c*(a+b*x^2)^(1/2))^(3/2)/x^2,x,4,-(c*sqrt(a+b*x^2))^(3/2)/x+3*b*x*(c*sqrt(a+b*x^2))^(3/2)/(a+b*x^2)-3*sqrt(cos(1/2*arctan(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)*(c*sqrt(a+b*x^2))^(3/2)/((1+b*x^2/a)^(3/4)*sqrt(a))],
[(c*(a+b*x^2)^(1/2))^(3/2)/x^4,x,5,-1/3*(c*sqrt(a+b*x^2))^(3/2)/x^3-1/2*b*(c*sqrt(a+b*x^2))^(3/2)/(a*x)+1/2*b^2*x*(c*sqrt(a+b*x^2))^(3/2)/(a*(a+b*x^2))-1/2*b^(3/2)*sqrt(cos(1/2*arctan(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x*sqrt(b)/sqrt(a)))*EllipticE(sin(1/2*arctan(x*sqrt(b)/sqrt(a))),sqrt(2))*(c*sqrt(a+b*x^2))^(3/2)/(a^(3/2)*(1+b*x^2/a)^(3/4))],

# q<0

# Integrands of the form u (e (a+b x^n)^q (c+d x^n)^r)^p

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

# p>0
[sqrt((b-x)*(-a+x)),x,4,-1/8*(a-b)^2*arctan(1/2*(a+b-2*x)/sqrt(-a*b+(a+b)*x-x^2))-1/4*(a+b-2*x)*sqrt(-a*b+(a+b)*x-x^2)],
[sqrt((1-x^2)*(3+x^2)),x,6,-2*EllipticE(x,sqrt(-1/3))/sqrt(3)+4*EllipticF(x,sqrt(-1/3))/sqrt(3)+1/3*x*sqrt(3-2*x^2-x^4)],

# p<0
[1/sqrt((b-x)*(-a+x)),x,3,-arctan(1/2*(a+b-2*x)/sqrt(-a*b+(a+b)*x-x^2))],
[1/sqrt((1-x^2)*(3+x^2)),x,3,EllipticF(x,sqrt(-1/3))/sqrt(3)],

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

# p>0
[x^5*sqrt(e*(a+b*x^2)/(c+d*x^2)),x,6,1/6*(e*(a+b*x^2)/(c+d*x^2))^(3/2)*(c+d*x^2)^3/(b*d^2*e)-1/16*(b*c-a*d)*(5*b^2*c^2+2*a*b*c*d+a^2*d^2)*arctanh(sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(b)*sqrt(e)))*sqrt(e)/(b^(5/2)*d^(7/2))+1/16*(11*b^2*c^2-2*a*b*c*d-a^2*d^2)*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b^2*d^3)-1/8*(3*b*c+a*d)*(c+d*x^2)^2*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b*d^3)],
[x^3*sqrt(e*(a+b*x^2)/(c+d*x^2)),x,5,1/8*(b*c-a*d)*(3*b*c+a*d)*arctanh(sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(b)*sqrt(e)))*sqrt(e)/(b^(3/2)*d^(5/2))-1/8*(5*b*c-a*d)*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b*d^2)+1/4*(c+d*x^2)^2*sqrt(e*(a+b*x^2)/(c+d*x^2))/d^2],
[x*sqrt(e*(a+b*x^2)/(c+d*x^2)),x,4,-1/2*(b*c-a*d)*arctanh(sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(b)*sqrt(e)))*sqrt(e)/(d^(3/2)*sqrt(b))+1/2*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/d],
[sqrt(e*(a+b*x^2)/(c+d*x^2))/x,x,5,-arctanh(sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(a)*sqrt(e)))*sqrt(a)*sqrt(e)/sqrt(c)+arctanh(sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(b)*sqrt(e)))*sqrt(b)*sqrt(e)/sqrt(d)],
[sqrt(e*(a+b*x^2)/(c+d*x^2))/x^3,x,4,-1/2*(b*c-a*d)*arctanh(sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(a)*sqrt(e)))*sqrt(e)/(c^(3/2)*sqrt(a))+1/2*(b*c-a*d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c*(a-c*(a+b*x^2)/(c+d*x^2)))],
[sqrt(e*(a+b*x^2)/(c+d*x^2))/x^5,x,5,1/8*(b*c-a*d)*(b*c+3*a*d)*arctanh(sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(a)*sqrt(e)))*sqrt(e)/(a^(3/2)*c^(5/2))-1/4*(b*c-a*d)^2*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c^2*(a-c*(a+b*x^2)/(c+d*x^2))^2)+1/8*(b*c-5*a*d)*(b*c-a*d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a*c^2*(a-c*(a+b*x^2)/(c+d*x^2)))],
[sqrt(e*(a+b*x^2)/(c+d*x^2))/x^7,x,6,1/6*(b*c-a*d)^3*e^2*(e*(a+b*x^2)/(c+d*x^2))^(3/2)/(a*c^2*(a*e-c*e*(a+b*x^2)/(c+d*x^2))^3)-1/16*(b*c-a*d)*(b^2*c^2+2*a*b*c*d+5*a^2*d^2)*arctanh(sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(a)*sqrt(e)))*sqrt(e)/(a^(5/2)*c^(7/2))+1/8*(b*c-a*d)^2*(b*c+3*a*d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a*c^3*(a-c*(a+b*x^2)/(c+d*x^2))^2)-1/16*(b*c-a*d)*(b^2*c^2+2*a*b*c*d-11*a^2*d^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a^2*c^3*(a-c*(a+b*x^2)/(c+d*x^2)))],
[x^4*sqrt(e*(a+b*x^2)/(c+d*x^2)),x,7,1/15*(8*b^2*c^2-3*a*b*c*d-2*a^2*d^2)*x*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b^2*d^2)-1/15*(4*b*c-a*d)*x*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b*d^2)+1/5*x^3*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/d+1/15*c^(3/2)*(4*b*c-a*d)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b*d^(5/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))-1/15*(8*b^2*c^2-3*a*b*c*d-2*a^2*d^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b^2*d^(5/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))],
[x^2*sqrt(e*(a+b*x^2)/(c+d*x^2)),x,6,-1/3*(2*b*c-a*d)*x*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b*d)+1/3*x*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/d-1/3*c^(3/2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(e*(a+b*x^2)/(c+d*x^2))/(d^(3/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))+1/3*(2*b*c-a*d)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b*d^(3/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))],
[sqrt(e*(a+b*x^2)/(c+d*x^2)),x,5,x*sqrt(e*(a+b*x^2)/(c+d*x^2))-sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))+sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))],
[sqrt(e*(a+b*x^2)/(c+d*x^2))/x^2,x,7,d*x*sqrt(e*(a+b*x^2)/(c+d*x^2))/c-(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c*x)+b*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))-sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(c)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))],
[sqrt(e*(a+b*x^2)/(c+d*x^2))/x^4,x,7,1/3*d*(b*c-2*a*d)*x*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a*c^2)-1/3*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c*x^3)-1/3*(b*c-2*a*d)*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a*c^2*x)-1/3*(b*c-2*a*d)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a*c^(3/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))-1/3*b*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a*sqrt(c)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))],
[sqrt(e*(a+b*x^2)/(c+d*x^2))/x^6,x,8,-1/15*d*(2*b^2*c^2+3*a*b*c*d-8*a^2*d^2)*x*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a^2*c^3)-1/5*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c*x^5)-1/15*(b*c-4*a*d)*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a*c^2*x^3)+1/15*(2*b^2*c^2+3*a*b*c*d-8*a^2*d^2)*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a^2*c^3*x)+1/15*(2*b^2*c^2+3*a*b*c*d-8*a^2*d^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a^2*c^(5/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))-1/15*b*(b*c-4*a*d)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a^2*c^(3/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))],
[x^5*(e*(a+b*x^2)/(c+d*x^2))^(3/2),x,7,1/6*(e*(a+b*x^2)/(c+d*x^2))^(5/2)*(c+d*x^2)^3/(b*d^2*e)-1/16*(b*c-a*d)*(35*b^2*c^2-10*a*b*c*d-a^2*d^2)*e^(3/2)*arctanh(sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(b)*sqrt(e)))/(b^(3/2)*d^(9/2))+c^2*(b*c-a*d)*e*sqrt(e*(a+b*x^2)/(c+d*x^2))/d^4+1/48*(79*b^2*c^2-50*a*b*c*d-5*a^2*d^2)*e*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b*d^4)-1/24*(11*b*c+a*d)*e*(c+d*x^2)^2*sqrt(e*(a+b*x^2)/(c+d*x^2))/d^4],
[x^3*(e*(a+b*x^2)/(c+d*x^2))^(3/2),x,6,3/8*(b*c-a*d)*(5*b*c-a*d)*e^(3/2)*arctanh(sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(b)*sqrt(e)))/(d^(7/2)*sqrt(b))-c*(b*c-a*d)*e*sqrt(e*(a+b*x^2)/(c+d*x^2))/d^3-1/8*(9*b*c-5*a*d)*e*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/d^3+1/4*b*e*(c+d*x^2)^2*sqrt(e*(a+b*x^2)/(c+d*x^2))/d^3],
[x*(e*(a+b*x^2)/(c+d*x^2))^(3/2),x,5,1/2*(e*(a+b*x^2)/(c+d*x^2))^(3/2)*(c+d*x^2)/d-3/2*(b*c-a*d)*e^(3/2)*arctanh(sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(b)*sqrt(e)))*sqrt(b)/d^(5/2)+3/2*(b*c-a*d)*e*sqrt(e*(a+b*x^2)/(c+d*x^2))/d^2],
[(e*(a+b*x^2)/(c+d*x^2))^(3/2)/x,x,6,-a^(3/2)*e^(3/2)*arctanh(sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(a)*sqrt(e)))/c^(3/2)+b^(3/2)*e^(3/2)*arctanh(sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(b)*sqrt(e)))/d^(3/2)-(b*c-a*d)*e*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c*d)],
[(e*(a+b*x^2)/(c+d*x^2))^(3/2)/x^3,x,5,1/2*(b*c-a*d)*(e*(a+b*x^2)/(c+d*x^2))^(3/2)/(c*(a-c*(a+b*x^2)/(c+d*x^2)))-3/2*(b*c-a*d)*e^(3/2)*arctanh(sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(a)*sqrt(e)))*sqrt(a)/c^(5/2)+3/2*(b*c-a*d)*e*sqrt(e*(a+b*x^2)/(c+d*x^2))/c^2],
[(e*(a+b*x^2)/(c+d*x^2))^(3/2)/x^5,x,6,-3/8*(b*c-5*a*d)*(b*c-a*d)*e^(3/2)*arctanh(sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(a)*sqrt(e)))/(c^(7/2)*sqrt(a))-d*(b*c-a*d)*e*sqrt(e*(a+b*x^2)/(c+d*x^2))/c^3-1/4*a*(b*c-a*d)^2*e^3*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c^3*(a*e-c*e*(a+b*x^2)/(c+d*x^2))^2)+1/8*(5*b*c-9*a*d)*(b*c-a*d)*e^2*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c^3*(a*e-c*e*(a+b*x^2)/(c+d*x^2)))],
[(e*(a+b*x^2)/(c+d*x^2))^(3/2)/x^7,x,7,1/6*(b*c-a*d)^3*e^2*(e*(a+b*x^2)/(c+d*x^2))^(5/2)/(a*c^2*(a*e-c*e*(a+b*x^2)/(c+d*x^2))^3)+1/16*(b*c-a*d)*(b^2*c^2+10*a*b*c*d-35*a^2*d^2)*e^(3/2)*arctanh(sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(a)*sqrt(e)))/(a^(3/2)*c^(9/2))+d^2*(b*c-a*d)*e*sqrt(e*(a+b*x^2)/(c+d*x^2))/c^4+1/24*(b*c-a*d)^2*(b*c+11*a*d)*e^3*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c^4*(a*e-c*e*(a+b*x^2)/(c+d*x^2))^2)-1/48*(b*c-a*d)*(5*b^2*c^2+50*a*b*c*d-79*a^2*d^2)*e^2*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a*c^4*(a*e-c*e*(a+b*x^2)/(c+d*x^2)))],
[x^4*(e*(a+b*x^2)/(c+d*x^2))^(3/2),x,8,-1/5*(16*a*c-16*b*c^2/d-a^2*d/b)*e*x*sqrt(e*(a+b*x^2)/(c+d*x^2))/d^2-e*x^3*(a+b*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/d-1/5*(8*b*c-7*a*d)*e*x*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/d^3+6/5*b*e*x^3*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/d^2+1/5*c^(3/2)*(8*b*c-7*a*d)*e*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(e*(a+b*x^2)/(c+d*x^2))/(d^(7/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))-1/5*(16*b^2*c^2-16*a*b*c*d+a^2*d^2)*e*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b*d^(7/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))],
[x^2*(e*(a+b*x^2)/(c+d*x^2))^(3/2),x,7,-1/3*(8*b*c-7*a*d)*e*x*sqrt(e*(a+b*x^2)/(c+d*x^2))/d^2-e*x*(a+b*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/d+4/3*b*e*x*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/d^2+1/3*(8*b*c-7*a*d)*e*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(d^(5/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))-1/3*(4*b*c-3*a*d)*e*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(d^(5/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))],
[(e*(a+b*x^2)/(c+d*x^2))^(3/2),x,6,-(b*c-a*d)*e*x*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c*d)+(2*b*c-a*d)*e*x*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c*d)-(2*b*c-a*d)*e*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(e*(a+b*x^2)/(c+d*x^2))/(d^(3/2)*sqrt(c)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))+b*e*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(d^(3/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))],
[(e*(a+b*x^2)/(c+d*x^2))^(3/2)/x^2,x,7,-(b*c-a*d)*e*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c*d*x)-(b*c-2*a*d)*e*x*sqrt(e*(a+b*x^2)/(c+d*x^2))/c^2+(b*c-2*a*d)*e*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c^2*d*x)+(b*c-2*a*d)*e*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c^(3/2)*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))+b*e*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(c)*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))],
[(e*(a+b*x^2)/(c+d*x^2))^(3/2)/x^4,x,8,-(b*c-a*d)*e*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c*d*x^3)+1/3*d*(7*b*c-8*a*d)*e*x*sqrt(e*(a+b*x^2)/(c+d*x^2))/c^3+1/3*(3*b*c-4*a*d)*e*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c^2*d*x^3)-1/3*(7*b*c-8*a*d)*e*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c^3*x)+1/3*b*(3*b*c-4*a*d)*e*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a*c^(3/2)*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))-1/3*(7*b*c-8*a*d)*e*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c^(5/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))],
[(e*(a+b*x^2)/(c+d*x^2))^(3/2)/x^6,x,9,-(b*c-a*d)*e*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c*d*x^5)+1/5*d*(b^2*c^2-16*a*b*c*d+16*a^2*d^2)*e*x*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a*c^4)+1/5*(5*b*c-6*a*d)*e*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c^2*d*x^5)-1/5*(7*b*c-8*a*d)*e*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(c^3*x^3)-1/5*(b^2*c^2-16*a*b*c*d+16*a^2*d^2)*e*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a*c^4*x)-1/5*(b^2*c^2-16*a*b*c*d+16*a^2*d^2)*e*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a*c^(7/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))-1/5*b*(7*b*c-8*a*d)*e*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a*c^(5/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))],
[x*sqrt((1-x^2)/(1+x^2)),x,4,-arctan(sqrt((1-x^2)/(1+x^2)))+1/2*(1+x^2)*sqrt((1-x^2)/(1+x^2))],
[x*sqrt((5-7*x^2)/(7+5*x^2)),x,4,-37/5*arctan(sqrt(5/7)*sqrt((5-7*x^2)/(7+5*x^2)))/sqrt(35)+1/10*(7+5*x^2)*sqrt((5-7*x^2)/(7+5*x^2))],
[x^2*sqrt((1-x^3)/(1+x^3)),x,4,-2/3*arctan(sqrt((1-x^3)/(1+x^3)))+1/3*(1+x^3)*sqrt((1-x^3)/(1+x^3))],
[x^8*sqrt((1-x^3)/(1+x^3)),x,6,-1/9*((1-x^3)/(1+x^3))^(3/2)*(1+x^3)^3-1/3*arctan(sqrt((1-x^3)/(1+x^3)))+1/2*(1+x^3)*sqrt((1-x^3)/(1+x^3))-1/6*(1+x^3)^2*sqrt((1-x^3)/(1+x^3))],
[x^9*sqrt((5-7*x^5)/(7+5*x^5)),x,5,2257/875*arctan(sqrt(5/7)*sqrt((5-7*x^5)/(7+5*x^5)))/sqrt(35)-27/350*(7+5*x^5)*sqrt((5-7*x^5)/(7+5*x^5))+1/250*(7+5*x^5)^2*sqrt((5-7*x^5)/(7+5*x^5))],
[sqrt(x^2/(-1+x^2))/(1+x^2),x,5,arctan(sqrt(-1+x^2)/sqrt(2))*sqrt(-x^2/(1-x^2))*sqrt(-1+x^2)/(x*sqrt(2))],
[sqrt(x^2/(-1+a+(1+a)*x^2))/(1+x^2),x,5,arctan(sqrt(-1+a+(1+a)*x^2)/sqrt(2))*sqrt(-x^2/(1-a-(1+a)*x^2))*sqrt(-1+a+(1+a)*x^2)/(x*sqrt(2))],

# p<0
[x^5/sqrt(e*(a+b*x^2)/(c+d*x^2)),x,6,1/16*(b*c-a*d)*(b^2*c^2+2*a*b*c*d+5*a^2*d^2)*arctanh(sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(b)*sqrt(e)))/(b^(7/2)*d^(5/2)*sqrt(e))+1/16*(b^2*c^2+2*a*b*c*d+5*a^2*d^2)*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b^3*d^2*e)-1/24*(3*b*c+5*a*d)*(c+d*x^2)^2*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b^2*d^2*e)-1/6*(c+d*x^2)^3*(a-c*(a+b*x^2)/(c+d*x^2))*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b*d*(b*c-a*d)*e)],
[x^3/sqrt(e*(a+b*x^2)/(c+d*x^2)),x,5,-1/8*(b*c-a*d)*(b*c+3*a*d)*arctanh(sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(b)*sqrt(e)))/(b^(5/2)*d^(3/2)*sqrt(e))-1/8*(b*c+3*a*d)*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b^2*d*e)+1/4*(c+d*x^2)^2*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b*d*e)],
[x/sqrt(e*(a+b*x^2)/(c+d*x^2)),x,4,1/2*(b*c-a*d)*arctanh(sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(b)*sqrt(e)))/(b^(3/2)*sqrt(d)*sqrt(e))+1/2*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b*e)],
[1/(x*sqrt(e*(a+b*x^2)/(c+d*x^2))),x,5,-arctanh(sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(a)*sqrt(e)))*sqrt(c)/(sqrt(a)*sqrt(e))+arctanh(sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(b)*sqrt(e)))*sqrt(d)/(sqrt(b)*sqrt(e))],
[1/(x^3*sqrt(e*(a+b*x^2)/(c+d*x^2))),x,4,1/2*(b*c-a*d)*arctanh(sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(a)*sqrt(e)))/(a^(3/2)*sqrt(c)*sqrt(e))+1/2*(b*c-a*d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a*(a*e-c*e*(a+b*x^2)/(c+d*x^2)))],
[1/(x^5*sqrt(e*(a+b*x^2)/(c+d*x^2))),x,5,-1/8*(b*c-a*d)*(3*b*c+a*d)*arctanh(sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(a)*sqrt(e)))/(a^(5/2)*c^(3/2)*sqrt(e))-1/4*(b*c-a*d)^2*e*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a*c*(a*e-c*e*(a+b*x^2)/(c+d*x^2))^2)-1/8*(b*c-a*d)*(3*b*c+a*d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a^2*c*(a*e-c*e*(a+b*x^2)/(c+d*x^2)))],
[x^4/sqrt(e*(a+b*x^2)/(c+d*x^2)),x,7,1/15*(b*c-4*a*d)*x*(a+b*x^2)/(b^2*d*sqrt(e*(a+b*x^2)/(c+d*x^2)))+1/5*x^3*(a+b*x^2)/(b*sqrt(e*(a+b*x^2)/(c+d*x^2)))-1/15*(2*b^2*c^2+3*a*b*c*d-8*a^2*d^2)*x*(a+b*x^2)/(b^3*d*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2)))-1/15*c^(3/2)*(b*c-4*a*d)*(a+b*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))/(b^2*d^(3/2)*(c+d*x^2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e*(a+b*x^2)/(c+d*x^2)))+1/15*(2*b^2*c^2+3*a*b*c*d-8*a^2*d^2)*(a+b*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)/(b^3*d^(3/2)*(c+d*x^2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e*(a+b*x^2)/(c+d*x^2)))],
[x^2/sqrt(e*(a+b*x^2)/(c+d*x^2)),x,6,1/3*x*(a+b*x^2)/(b*sqrt(e*(a+b*x^2)/(c+d*x^2)))+1/3*(b*c-2*a*d)*x*(a+b*x^2)/(b^2*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2)))-1/3*c^(3/2)*(a+b*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))/(b*(c+d*x^2)*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e*(a+b*x^2)/(c+d*x^2)))-1/3*(b*c-2*a*d)*(a+b*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)/(b^2*(c+d*x^2)*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e*(a+b*x^2)/(c+d*x^2)))],
[1/sqrt(e*(a+b*x^2)/(c+d*x^2)),x,5,d*x*(a+b*x^2)/(b*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2)))+c^(3/2)*(a+b*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))/(a*(c+d*x^2)*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e*(a+b*x^2)/(c+d*x^2)))-(a+b*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(d)/(b*(c+d*x^2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e*(a+b*x^2)/(c+d*x^2)))],
[1/(x^2*sqrt(e*(a+b*x^2)/(c+d*x^2))),x,7,(-a-b*x^2)/(a*x*sqrt(e*(a+b*x^2)/(c+d*x^2)))+d*x*(a+b*x^2)/(a*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2)))-(a+b*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(d)/(a*(c+d*x^2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e*(a+b*x^2)/(c+d*x^2)))+(a+b*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(d)/(a*(c+d*x^2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e*(a+b*x^2)/(c+d*x^2)))],
[1/(x^4*sqrt(e*(a+b*x^2)/(c+d*x^2))),x,7,1/3*(-a-b*x^2)/(a*x^3*sqrt(e*(a+b*x^2)/(c+d*x^2)))+1/3*(2*b*c-a*d)*(a+b*x^2)/(a^2*c*x*sqrt(e*(a+b*x^2)/(c+d*x^2)))-1/3*d*(2*b*c-a*d)*x*(a+b*x^2)/(a^2*c*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2)))+1/3*(2*b*c-a*d)*(a+b*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(d)/(a^2*(c+d*x^2)*sqrt(c)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e*(a+b*x^2)/(c+d*x^2)))-1/3*b*(a+b*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(d)/(a^2*(c+d*x^2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e*(a+b*x^2)/(c+d*x^2)))],
[x^5/(e*(a+b*x^2)/(c+d*x^2))^(3/2),x,7,-1/16*(b*c-a*d)*(b^2*c^2+5*a*d*(2*b*c-7*a*d))*arctanh(sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(b)*sqrt(e)))/(b^(9/2)*d^(3/2)*e^(3/2))-a^2*(c+d*x^2)^3/(b*(b*c-a*d)^2*e*sqrt(e*(a+b*x^2)/(c+d*x^2)))-1/16*(b^2*c^2+5*a*d*(2*b*c-7*a*d))*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b^4*d*e^2)-1/24*(b^2*c^2+5*a*d*(2*b*c-7*a*d))*(c+d*x^2)^2*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b^3*d*(b*c-a*d)*e^2)+1/6*(b^2*c^2-2*a*b*c*d+7*a^2*d^2)*(c+d*x^2)^3*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b^2*d*(b*c-a*d)^2*e^2)],
[x^3/(e*(a+b*x^2)/(c+d*x^2))^(3/2),x,6,3/8*(b*c-5*a*d)*(b*c-a*d)*arctanh(sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(b)*sqrt(e)))/(b^(7/2)*e^(3/2)*sqrt(d))+a*(b*c-a*d)/(b^3*e*sqrt(e*(a+b*x^2)/(c+d*x^2)))+1/8*(3*b*c-7*a*d)*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b^3*e^2)+1/4*(c+d*x^2)^2*sqrt(e*(a+b*x^2)/(c+d*x^2))/(b^2*e^2)],
[x/(e*(a+b*x^2)/(c+d*x^2))^(3/2),x,5,3/2*(b*c-a*d)*arctanh(sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(b)*sqrt(e)))*sqrt(d)/(b^(5/2)*e^(3/2))-3/2*(b*c-a*d)/(b^2*e*sqrt(e*(a+b*x^2)/(c+d*x^2)))+1/2*(c+d*x^2)/(b*e*sqrt(e*(a+b*x^2)/(c+d*x^2)))],
[1/(x*(e*(a+b*x^2)/(c+d*x^2))^(3/2)),x,6,-c^(3/2)*arctanh(sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(a)*sqrt(e)))/(a^(3/2)*e^(3/2))+d^(3/2)*arctanh(sqrt(d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(b)*sqrt(e)))/(b^(3/2)*e^(3/2))+(b*c-a*d)/(a*b*e*sqrt(e*(a+b*x^2)/(c+d*x^2)))],
[1/(x^3*(e*(a+b*x^2)/(c+d*x^2))^(3/2)),x,5,3/2*(b*c-a*d)*arctanh(sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(a)*sqrt(e)))*sqrt(c)/(a^(5/2)*e^(3/2))-3/2*(b*c-a*d)/(a^2*e*sqrt(e*(a+b*x^2)/(c+d*x^2)))+1/2*(b*c-a*d)/(a*(a*e-c*e*(a+b*x^2)/(c+d*x^2))*sqrt(e*(a+b*x^2)/(c+d*x^2)))],
[1/(x^5*(e*(a+b*x^2)/(c+d*x^2))^(3/2)),x,6,-3/8*(b*c-a*d)*(5*b*c-a*d)*arctanh(sqrt(c)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(sqrt(a)*sqrt(e)))/(a^(7/2)*e^(3/2)*sqrt(c))+b*(b*c-a*d)/(a^3*e*sqrt(e*(a+b*x^2)/(c+d*x^2)))-1/4*(b*c-a*d)^2*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a^2*(a*e-c*e*(a+b*x^2)/(c+d*x^2))^2)-1/8*(7*b*c-3*a*d)*(b*c-a*d)*sqrt(e*(a+b*x^2)/(c+d*x^2))/(a^3*(a*e^2-c*e^2*(a+b*x^2)/(c+d*x^2)))],
[x^4/(e*(a+b*x^2)/(c+d*x^2))^(3/2),x,8,1/5*(7*b*c-8*a*d)*x*(a+b*x^2)/(b^3*e*sqrt(e*(a+b*x^2)/(c+d*x^2)))+6/5*d*x^3*(a+b*x^2)/(b^2*e*sqrt(e*(a+b*x^2)/(c+d*x^2)))+1/5*(b^2*c^2-16*a*b*c*d+16*a^2*d^2)*x*(a+b*x^2)/(b^4*e*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2)))-x^3*(c+d*x^2)/(b*e*sqrt(e*(a+b*x^2)/(c+d*x^2)))-1/5*c^(3/2)*(7*b*c-8*a*d)*(a+b*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))/(b^3*e*(c+d*x^2)*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e*(a+b*x^2)/(c+d*x^2)))-1/5*(b^2*c^2-16*a*b*c*d+16*a^2*d^2)*(a+b*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)/(b^4*e*(c+d*x^2)*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e*(a+b*x^2)/(c+d*x^2)))],
[x^2/(e*(a+b*x^2)/(c+d*x^2))^(3/2),x,7,4/3*d*x*(a+b*x^2)/(b^2*e*sqrt(e*(a+b*x^2)/(c+d*x^2)))+1/3*d*(7*b*c-8*a*d)*x*(a+b*x^2)/(b^3*e*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2)))-x*(c+d*x^2)/(b*e*sqrt(e*(a+b*x^2)/(c+d*x^2)))+1/3*c^(3/2)*(3*b*c-4*a*d)*(a+b*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))/(a*b^2*e*(c+d*x^2)*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e*(a+b*x^2)/(c+d*x^2)))-1/3*(7*b*c-8*a*d)*(a+b*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(d)/(b^3*e*(c+d*x^2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e*(a+b*x^2)/(c+d*x^2)))],
[1/(e*(a+b*x^2)/(c+d*x^2))^(3/2),x,6,(b*c-a*d)*x/(a*b*e*sqrt(e*(a+b*x^2)/(c+d*x^2)))-d*(b*c-2*a*d)*x*(a+b*x^2)/(a*b^2*e*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2)))+c^(3/2)*(a+b*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(d)/(a*b*e*(c+d*x^2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e*(a+b*x^2)/(c+d*x^2)))+(b*c-2*a*d)*(a+b*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(d)/(a*b^2*e*(c+d*x^2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e*(a+b*x^2)/(c+d*x^2)))],
[1/(x^2*(e*(a+b*x^2)/(c+d*x^2))^(3/2)),x,7,(b*c-a*d)/(a*b*e*x*sqrt(e*(a+b*x^2)/(c+d*x^2)))-(2*b*c-a*d)*(a+b*x^2)/(a^2*b*e*x*sqrt(e*(a+b*x^2)/(c+d*x^2)))+d*(2*b*c-a*d)*x*(a+b*x^2)/(a^2*b*e*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2)))+c^(3/2)*(a+b*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(d)/(a^2*e*(c+d*x^2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e*(a+b*x^2)/(c+d*x^2)))-(2*b*c-a*d)*(a+b*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(d)/(a^2*b*e*(c+d*x^2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e*(a+b*x^2)/(c+d*x^2)))],
[1/(x^4*(e*(a+b*x^2)/(c+d*x^2))^(3/2)),x,8,(b*c-a*d)/(a*b*e*x^3*sqrt(e*(a+b*x^2)/(c+d*x^2)))-1/3*(4*b*c-3*a*d)*(a+b*x^2)/(a^2*b*e*x^3*sqrt(e*(a+b*x^2)/(c+d*x^2)))+1/3*(8*b*c-7*a*d)*(a+b*x^2)/(a^3*e*x*sqrt(e*(a+b*x^2)/(c+d*x^2)))-1/3*d*(8*b*c-7*a*d)*x*(a+b*x^2)/(a^3*e*(c+d*x^2)*sqrt(e*(a+b*x^2)/(c+d*x^2)))+1/3*(8*b*c-7*a*d)*(a+b*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(d)/(a^3*e*(c+d*x^2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e*(a+b*x^2)/(c+d*x^2)))-1/3*(4*b*c-3*a*d)*(a+b*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-b*c/(a*d)))*sqrt(c)*sqrt(d)/(a^3*e*(c+d*x^2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e*(a+b*x^2)/(c+d*x^2)))],

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

# p>0
[x^5*sqrt(a+b/(c+d*x^2)),x,7,1/6*(c+d*x^2)^3*((b+a*c+a*d*x^2)/(c+d*x^2))^(3/2)/(a*d^3)+1/16*b*(b^2+4*a*b*c+8*a^2*c^2)*arctanh(sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(a))/(a^(5/2)*d^3)-1/16*(b^2+4*a*b*c-8*a^2*c^2)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a^2*d^3)-1/8*(b+4*a*c)*(c+d*x^2)^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a*d^3)],
[x^3*sqrt(a+b/(c+d*x^2)),x,6,-1/8*b*(b+4*a*c)*arctanh(sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(a))/(a^(3/2)*d^2)+1/8*(b-4*a*c)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a*d^2)+1/4*(c+d*x^2)^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/d^2],
[x*sqrt(a+b/(c+d*x^2)),x,5,1/2*b*arctanh(sqrt(a+b/(c+d*x^2))/sqrt(a))/(d*sqrt(a))+1/2*(c+d*x^2)*sqrt(a+b/(c+d*x^2))/d],
[sqrt(a+b/(c+d*x^2))/x,x,6,arctanh(sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(a))*sqrt(a)-arctanh(sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(b+a*c))*sqrt(b+a*c)/sqrt(c)],
[sqrt(a+b/(c+d*x^2))/x^3,x,5,1/2*b*d*arctanh(sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(b+a*c))/(c^(3/2)*sqrt(b+a*c))-1/2*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c*x^2)],
[sqrt(a+b/(c+d*x^2))/x^5,x,6,-1/8*b*(3*b+4*a*c)*d^2*arctanh(sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(b+a*c))/(c^(5/2)*(b+a*c)^(3/2))+1/8*(5*b+4*a*c)*d*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^2*(b+a*c)*x^2)-1/4*(c+d*x^2)^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^2*x^4)],
[sqrt(a+b/(c+d*x^2))/x^7,x,7,-1/6*(c+d*x^2)^3*((b+a*c+a*d*x^2)/(c+d*x^2))^(3/2)/(c^2*(b+a*c)*x^6)+1/16*b*(5*b^2+12*a*b*c+8*a^2*c^2)*d^3*arctanh(sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(b+a*c))/(c^(7/2)*(b+a*c)^(5/2))-1/16*(11*b^2+20*a*b*c+8*a^2*c^2)*d^2*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^3*(b+a*c)^2*x^2)+1/8*(3*b+4*a*c)*d*(c+d*x^2)^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^3*(b+a*c)*x^4)],
[x^4*sqrt(a+b/(c+d*x^2)),x,8,-1/15*(2*b^2+7*a*b*c-3*a^2*c^2)*x*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a^2*d^2)+1/15*(b-3*a*c)*x*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a*d^2)+1/5*x^3*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/d-1/15*c^(3/2)*(b-3*a*c)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a*d^(5/2)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))+1/15*(2*b^2+7*a*b*c-3*a^2*c^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a^2*d^(5/2)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],
[x^2*sqrt(a+b/(c+d*x^2)),x,7,1/3*(b-a*c)*x*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a*d)+1/3*x*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/d-1/3*c^(3/2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(d^(3/2)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))-1/3*(b-a*c)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a*d^(3/2)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],
[sqrt(a+b/(c+d*x^2)),x,6,x*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))-sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(sqrt(d)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))+sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(sqrt(d)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],
[sqrt(a+b/(c+d*x^2))/x^2,x,8,d*x*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/c-(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c*x)-sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(d)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(sqrt(c)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))+a*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)*sqrt(d)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/((b+a*c)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],
[sqrt(a+b/(c+d*x^2))/x^4,x,8,-1/3*(2*b+a*c)*d^2*x*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^2*(b+a*c))-1/3*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c*x^3)+1/3*(2*b+a*c)*d*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^2*(b+a*c)*x)+1/3*(2*b+a*c)*d^(3/2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^(3/2)*(b+a*c)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))-1/3*a*d^(3/2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/((b+a*c)*sqrt(c)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],
[sqrt(a+b/(c+d*x^2))/x^6,x,9,1/15*(8*b^2+13*a*b*c+3*a^2*c^2)*d^3*x*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^3*(b+a*c)^2)-1/5*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c*x^5)+1/15*(4*b+3*a*c)*d*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^2*(b+a*c)*x^3)-1/15*(8*b^2+13*a*b*c+3*a^2*c^2)*d^2*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^3*(b+a*c)^2*x)-1/15*(8*b^2+13*a*b*c+3*a^2*c^2)*d^(5/2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^(5/2)*(b+a*c)^2*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))+1/15*a*(4*b+3*a*c)*d^(5/2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^(3/2)*(b+a*c)^2*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],
[x^5*(a+b/(c+d*x^2))^(3/2),x,8,1/6*(c+d*x^2)^3*((b+a*c+a*d*x^2)/(c+d*x^2))^(5/2)/(a*d^3)-1/16*b*(b^2+12*a*b*c-24*a^2*c^2)*arctanh(sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(a))/(a^(3/2)*d^3)-b*c^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/d^3-1/48*(5*b^2+60*a*b*c-24*a^2*c^2)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a*d^3)-1/24*(b+12*a*c)*(c+d*x^2)^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/d^3],
[x^3*(a+b/(c+d*x^2))^(3/2),x,7,3/8*b*(b-4*a*c)*arctanh(sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(a))/(d^2*sqrt(a))+b*c*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/d^2+1/8*(5*b-4*a*c)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/d^2+1/4*a*(c+d*x^2)^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/d^2],
[x*(a+b/(c+d*x^2))^(3/2),x,6,1/2*(c+d*x^2)*(a+b/(c+d*x^2))^(3/2)/d+3/2*b*arctanh(sqrt(a+b/(c+d*x^2))/sqrt(a))*sqrt(a)/d-3/2*b*sqrt(a+b/(c+d*x^2))/d],
[(a+b/(c+d*x^2))^(3/2)/x,x,7,a^(3/2)*arctanh(sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(a))-(b+a*c)^(3/2)*arctanh(sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(b+a*c))/c^(3/2)+b*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/c],
[(a+b/(c+d*x^2))^(3/2)/x^3,x,6,-1/2*(c+d*x^2)*((b+a*c+a*d*x^2)/(c+d*x^2))^(3/2)/(c*x^2)+3/2*b*d*arctanh(sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(b+a*c))*sqrt(b+a*c)/c^(5/2)-3/2*b*d*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/c^2],
[(a+b/(c+d*x^2))^(3/2)/x^5,x,7,-3/8*b*(5*b+4*a*c)*d^2*arctanh(sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(b+a*c))/(c^(7/2)*sqrt(b+a*c))+b*d^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/c^3+1/8*(9*b+4*a*c)*d*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^3*x^2)-1/4*(b+a*c)*(c+d*x^2)^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^3*x^4)],
[(a+b/(c+d*x^2))^(3/2)/x^7,x,8,-1/6*(c+d*x^2)^3*((b+a*c+a*d*x^2)/(c+d*x^2))^(5/2)/(c^2*(b+a*c)*x^6)+1/16*b*(35*b^2+60*a*b*c+24*a^2*c^2)*d^3*arctanh(sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(b+a*c))/(c^(9/2)*(b+a*c)^(3/2))-b*d^3*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/c^4-1/48*(79*b^2+108*a*b*c+24*a^2*c^2)*d^2*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^4*(b+a*c)*x^2)+1/24*(11*b+12*a*c)*d*(c+d*x^2)^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^4*x^4)],
[x^4*(a+b/(c+d*x^2))^(3/2),x,9,1/5*(b^2-14*a*b*c+a^2*c^2)*x*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a*d^2)+1/5*(7*b-a*c)*x*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/d^2+6/5*a*x^3*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/d-x^3*(b+a*c+a*d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/d-1/5*c^(3/2)*(7*b-a*c)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(d^(5/2)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))-1/5*(b^2-14*a*b*c+a^2*c^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a*d^(5/2)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],
[x^2*(a+b/(c+d*x^2))^(3/2),x,8,1/3*(7*b-a*c)*x*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/d+4/3*a*x*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/d-x*(b+a*c+a*d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/d-1/3*(7*b-a*c)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(d^(3/2)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))+1/3*(3*b-a*c)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(d^(3/2)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],
[(a+b/(c+d*x^2))^(3/2),x,7,b*x*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/c-(b-a*c)*x*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/c+(b-a*c)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(sqrt(c)*sqrt(d)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))+a*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(sqrt(d)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],
[(a+b/(c+d*x^2))^(3/2)/x^2,x,8,b*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c*x)+(2*b+a*c)*d*x*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/c^2-(2*b+a*c)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^2*x)-(2*b+a*c)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(d)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^(3/2)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))+a*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(d)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(sqrt(c)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],
[(a+b/(c+d*x^2))^(3/2)/x^4,x,9,b*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c*x^3)-1/3*(8*b+a*c)*d^2*x*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/c^3-1/3*(4*b+a*c)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^2*x^3)+1/3*(8*b+a*c)*d*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^3*x)+1/3*(8*b+a*c)*d^(3/2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^(5/2)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))-1/3*a*(4*b+a*c)*d^(3/2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^(3/2)*(b+a*c)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],
[(a+b/(c+d*x^2))^(3/2)/x^6,x,10,b*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c*x^5)+1/5*(16*b^2+16*a*b*c+a^2*c^2)*d^3*x*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^4*(b+a*c))-1/5*(6*b+a*c)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^2*x^5)+1/5*(8*b+a*c)*d*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^3*x^3)-1/5*(16*b^2+16*a*b*c+a^2*c^2)*d^2*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^4*(b+a*c)*x)-1/5*(16*b^2+16*a*b*c+a^2*c^2)*d^(5/2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^(7/2)*(b+a*c)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))+1/5*a*(8*b+a*c)*d^(5/2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c^(5/2)*(b+a*c)*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],

# p<0
[x^5/sqrt(a+b/(c+d*x^2)),x,7,-1/16*b*(5*b^2+12*a*b*c+8*a^2*c^2)*arctanh(sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(a))/(a^(7/2)*d^3)+1/16*(5*b^2+12*a*b*c+8*a^2*c^2)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a^3*d^3)-1/24*(5*b+8*a*c)*(c+d*x^2)^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a^2*d^3)+1/6*x^2*(c+d*x^2)^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a*d^2)],
[x^3/sqrt(a+b/(c+d*x^2)),x,6,1/8*b*(3*b+4*a*c)*arctanh(sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(a))/(a^(5/2)*d^2)-1/8*(3*b+4*a*c)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a^2*d^2)+1/4*(c+d*x^2)^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a*d^2)],
[x/sqrt(a+b/(c+d*x^2)),x,5,-1/2*b*arctanh(sqrt(a+b/(c+d*x^2))/sqrt(a))/(a^(3/2)*d)+1/2*(c+d*x^2)*sqrt(a+b/(c+d*x^2))/(a*d)],
[1/(x*sqrt(a+b/(c+d*x^2))),x,6,arctanh(sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(a))/sqrt(a)-arctanh(sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(b+a*c))*sqrt(c)/sqrt(b+a*c)],
[1/(x^3*sqrt(a+b/(c+d*x^2))),x,5,-1/2*b*d*arctanh(sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(b+a*c))/((b+a*c)^(3/2)*sqrt(c))-1/2*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/((b+a*c)*x^2)],
[1/(x^5*sqrt(a+b/(c+d*x^2))),x,6,1/8*b*(b+4*a*c)*d^2*arctanh(sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(b+a*c))/(c^(3/2)*(b+a*c)^(5/2))+1/8*(b+4*a*c)*d*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c*(b+a*c)^2*x^2)-1/4*(c+d*x^2)^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(c*(b+a*c)*x^4)],
[x^4/sqrt(a+b/(c+d*x^2)),x,8,-1/15*(4*b+3*a*c)*x*(b+a*c+a*d*x^2)/(a^2*d^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))+1/5*x^3*(b+a*c+a*d*x^2)/(a*d*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))+1/15*(8*b^2+13*a*b*c+3*a^2*c^2)*x*(b+a*c+a*d*x^2)/(a^3*d^2*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))+1/15*c^(3/2)*(4*b+3*a*c)*(b+a*c+a*d*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))/(a^2*d^(5/2)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))-1/15*(8*b^2+13*a*b*c+3*a^2*c^2)*(b+a*c+a*d*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)/(a^3*d^(5/2)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],
[x^2/sqrt(a+b/(c+d*x^2)),x,7,1/3*x*(b+a*c+a*d*x^2)/(a*d*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))-1/3*(2*b+a*c)*x*(b+a*c+a*d*x^2)/(a^2*d*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))-1/3*c^(3/2)*(b+a*c+a*d*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))/(a*d^(3/2)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))+1/3*(2*b+a*c)*(b+a*c+a*d*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)/(a^2*d^(3/2)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],
[1/sqrt(a+b/(c+d*x^2)),x,6,x*(b+a*c+a*d*x^2)/(a*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))+c^(3/2)*(b+a*c+a*d*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))/((b+a*c)*(c+d*x^2)*sqrt(d)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))-(b+a*c+a*d*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)/(a*(c+d*x^2)*sqrt(d)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],
[1/(x^2*sqrt(a+b/(c+d*x^2))),x,8,(-b-a*c-a*d*x^2)/((b+a*c)*x*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))+d*x*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))-(b+a*c+a*d*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)*sqrt(d)/((b+a*c)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))+(b+a*c+a*d*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)*sqrt(d)/((b+a*c)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],
[1/(x^4*sqrt(a+b/(c+d*x^2))),x,8,1/3*(-b-a*c-a*d*x^2)/((b+a*c)*x^3*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))-1/3*(b-a*c)*d*(b+a*c+a*d*x^2)/(c*(b+a*c)^2*x*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))+1/3*(b-a*c)*d^2*x*(b+a*c+a*d*x^2)/(c*(b+a*c)^2*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))-1/3*(b-a*c)*d^(3/2)*(b+a*c+a*d*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))/((b+a*c)^2*(c+d*x^2)*sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))-1/3*a*d^(3/2)*(b+a*c+a*d*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)/((b+a*c)^2*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],
[x^5/(a+b/(c+d*x^2))^(3/2),x,8,-1/16*b*(35*b^2+60*a*b*c+24*a^2*c^2)*arctanh(sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(a))/(a^(9/2)*d^3)-(b+a*c)^2*(c+d*x^2)^3/(a*b^2*d^3*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))+1/16*(35*b^2+60*a*b*c+24*a^2*c^2)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a^4*d^3)-1/24*(35*b^2+60*a*b*c+24*a^2*c^2)*(c+d*x^2)^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a^3*b*d^3)+1/6*(7*b^2+12*a*b*c+6*a^2*c^2)*(c+d*x^2)^3*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a^2*b^2*d^3)],
[x^3/(a+b/(c+d*x^2))^(3/2),x,7,3/8*b*(5*b+4*a*c)*arctanh(sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(a))/(a^(7/2)*d^2)-b*(b+a*c)/(a^3*d^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))-1/8*(7*b+4*a*c)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a^3*d^2)+1/4*(c+d*x^2)^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/(a^2*d^2)],
[x/(a+b/(c+d*x^2))^(3/2),x,6,-3/2*b*arctanh(sqrt(a+b/(c+d*x^2))/sqrt(a))/(a^(5/2)*d)+3/2*b/(a^2*d*sqrt(a+b/(c+d*x^2)))+1/2*(c+d*x^2)/(a*d*sqrt(a+b/(c+d*x^2)))],
[1/(x*(a+b/(c+d*x^2))^(3/2)),x,7,arctanh(sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(a))/a^(3/2)-c^(3/2)*arctanh(sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(b+a*c))/(b+a*c)^(3/2)-b/(a*(b+a*c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))],
[1/(x^3*(a+b/(c+d*x^2))^(3/2)),x,6,-3/2*b*d*arctanh(sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(b+a*c))*sqrt(c)/(b+a*c)^(5/2)+3/2*b*d/((b+a*c)^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))+1/2*(-c-d*x^2)/((b+a*c)*x^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))],
[1/(x^5*(a+b/(c+d*x^2))^(3/2)),x,7,-3/8*b*(b-4*a*c)*d^2*arctanh(sqrt(c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/sqrt(b+a*c))/((b+a*c)^(7/2)*sqrt(c))-a*b*d^2/((b+a*c)^3*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))-1/8*(3*b-4*a*c)*d*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/((b+a*c)^3*x^2)-1/4*(c+d*x^2)^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))/((b+a*c)^2*x^4)],
[x^4/(a+b/(c+d*x^2))^(3/2),x,9,-x^3*(c+d*x^2)/(a*d*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))-1/5*(8*b+a*c)*x*(b+a*c+a*d*x^2)/(a^3*d^2*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))+6/5*x^3*(b+a*c+a*d*x^2)/(a^2*d*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))+1/5*(16*b^2+16*a*b*c+a^2*c^2)*x*(b+a*c+a*d*x^2)/(a^4*d^2*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))+1/5*c^(3/2)*(8*b+a*c)*(b+a*c+a*d*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))/(a^3*d^(5/2)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))-1/5*(16*b^2+16*a*b*c+a^2*c^2)*(b+a*c+a*d*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)/(a^4*d^(5/2)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],
[x^2/(a+b/(c+d*x^2))^(3/2),x,8,-x*(c+d*x^2)/(a*d*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))+4/3*x*(b+a*c+a*d*x^2)/(a^2*d*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))-1/3*(8*b+a*c)*x*(b+a*c+a*d*x^2)/(a^3*d*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))-1/3*c^(3/2)*(4*b+a*c)*(b+a*c+a*d*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))/(a^2*(b+a*c)*d^(3/2)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))+1/3*(8*b+a*c)*(b+a*c+a*d*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)/(a^3*d^(3/2)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],
[1/(a+b/(c+d*x^2))^(3/2),x,7,-b*x/(a*(b+a*c)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))+(2*b+a*c)*x*(b+a*c+a*d*x^2)/(a^2*(b+a*c)*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))+c^(3/2)*(b+a*c+a*d*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))/(a*(b+a*c)*(c+d*x^2)*sqrt(d)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))-(2*b+a*c)*(b+a*c+a*d*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)/(a^2*(b+a*c)*(c+d*x^2)*sqrt(d)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],
[1/(x^2*(a+b/(c+d*x^2))^(3/2)),x,8,-b/(a*(b+a*c)*x*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))+(b-a*c)*(b+a*c+a*d*x^2)/(a*(b+a*c)^2*x*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))-(b-a*c)*d*x*(b+a*c+a*d*x^2)/(a*(b+a*c)^2*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))+c^(3/2)*(b+a*c+a*d*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(d)/((b+a*c)^2*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))+(b-a*c)*(b+a*c+a*d*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)*sqrt(d)/(a*(b+a*c)^2*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],
[1/(x^4*(a+b/(c+d*x^2))^(3/2)),x,9,-b/(a*(b+a*c)*x^3*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))+1/3*(3*b-a*c)*(b+a*c+a*d*x^2)/(a*(b+a*c)^2*x^3*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))-1/3*(7*b-a*c)*d*(b+a*c+a*d*x^2)/((b+a*c)^3*x*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))+1/3*(7*b-a*c)*d^2*x*(b+a*c+a*d*x^2)/((b+a*c)^3*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2)))-1/3*(7*b-a*c)*d^(3/2)*(b+a*c+a*d*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)/((b+a*c)^3*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))+1/3*(3*b-a*c)*d^(3/2)*(b+a*c+a*d*x^2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(b/(b+a*c)))*sqrt(c)/((b+a*c)^3*(c+d*x^2)*sqrt((b+a*c+a*d*x^2)/(c+d*x^2))*sqrt(c*(b+a*c+a*d*x^2)/((b+a*c)*(c+d*x^2))))],

# Integrands of the form u (a x^n)^p

# Integrands of the form u (a x^n)^p

#  Integrands of the form Sqrt[a*x^n]/Sqrt[1+x^5] where n mod 10 = 3 
[sqrt(a*x^23)/sqrt(1+x^5),x,6,3/20*arcsinh(x^(5/2))*sqrt(a*x^23)/x^(23/2)-3/20*sqrt(a*x^23)*sqrt(1+x^5)/x^9+1/10*sqrt(a*x^23)*sqrt(1+x^5)/x^4],
[sqrt(a*x^13)/sqrt(1+x^5),x,5,-1/5*arcsinh(x^(5/2))*sqrt(a*x^13)/x^(13/2)+1/5*sqrt(a*x^13)*sqrt(1+x^5)/x^4],
[sqrt(a*x^3)/sqrt(1+x^5),x,4,2/5*arcsinh(x^(5/2))*sqrt(a*x^3)/x^(3/2)],
[sqrt(a/x^7)/sqrt(1+x^5),x,2,-2/5*x*sqrt(a/x^7)*sqrt(1+x^5)],
[sqrt(a/x^17)/sqrt(1+x^5),x,3,-2/15*x*sqrt(a/x^17)*sqrt(1+x^5)+4/15*x^6*sqrt(a/x^17)*sqrt(1+x^5)],
[sqrt(a*x^6)/(x*(1-x^4)),x,4,-1/2*arctan(x)*sqrt(a*x^6)/x^3+1/2*arctanh(x)*sqrt(a*x^6)/x^3],
[sqrt(a*x^6)/(x-x^5),x,5,-1/2*arctan(x)*sqrt(a*x^6)/x^3+1/2*arctanh(x)*sqrt(a*x^6)/x^3],
[(a*x^6)^(3/2)/(x*(1-x^4)),x,6,-a*sqrt(a*x^6)/x^2-1/5*a*x^2*sqrt(a*x^6)+1/2*a*arctan(x)*sqrt(a*x^6)/x^3+1/2*a*arctanh(x)*sqrt(a*x^6)/x^3],
[1/(1-x^4)-sqrt(a*x^6)/(x*(1-x^4)),x,8,1/2*arctan(x)+1/2*arctanh(x)+1/2*arctan(x)*sqrt(a*x^6)/x^3-1/2*arctanh(x)*sqrt(a*x^6)/x^3],
[1/(1-x^4)-sqrt(a*x^6)/(x-x^5),x,9,1/2*arctan(x)+1/2*arctanh(x)+1/2*arctan(x)*sqrt(a*x^6)/x^3-1/2*arctanh(x)*sqrt(a*x^6)/x^3],
[sqrt(a*x^3)/(x-x^3),x,6,-arctan(sqrt(x))*sqrt(a*x^3)/x^(3/2)+arctanh(sqrt(x))*sqrt(a*x^3)/x^(3/2)],
[sqrt(a*x^4)/sqrt(1+x^2),x,3,-1/2*arcsinh(x)*sqrt(a*x^4)/x^2+1/2*sqrt(a*x^4)*sqrt(1+x^2)/x],
[sqrt(a*x^3)/sqrt(1+x^2),x,4,2/3*sqrt(a*x^3)*sqrt(1+x^2)/x-1/3*(1+x)*sqrt(cos(2*arctan(sqrt(x)))^2)/cos(2*arctan(sqrt(x)))*EllipticF(sin(2*arctan(sqrt(x))),sqrt(1/2))*sqrt(a*x^3)*sqrt((1+x^2)/(1+x)^2)/(x^(3/2)*sqrt(1+x^2))],
[sqrt(a*x^2)/sqrt(1+x^2),x,2,sqrt(a*x^2)*sqrt(1+x^2)/x],
[sqrt(a*x)/sqrt(1+x^2),x,4,2*sqrt(a*x)*sqrt(1+x^2)/(1+x)-2*(1+x)*sqrt(cos(2*arctan(sqrt(a*x)/sqrt(a)))^2)/cos(2*arctan(sqrt(a*x)/sqrt(a)))*EllipticE(sin(2*arctan(sqrt(a*x)/sqrt(a))),sqrt(1/2))*sqrt(a)*sqrt((1+x^2)/(1+x)^2)/sqrt(1+x^2)+(1+x)*sqrt(cos(2*arctan(sqrt(a*x)/sqrt(a)))^2)/cos(2*arctan(sqrt(a*x)/sqrt(a)))*EllipticF(sin(2*arctan(sqrt(a*x)/sqrt(a))),sqrt(1/2))*sqrt(a)*sqrt((1+x^2)/(1+x)^2)/sqrt(1+x^2)],
[sqrt(a/x)/sqrt(1+x^2),x,3,(1+x)*sqrt(cos(2*arctan(sqrt(x)))^2)/cos(2*arctan(sqrt(x)))*EllipticF(sin(2*arctan(sqrt(x))),sqrt(1/2))*sqrt(a/x)*sqrt(x)*sqrt((1+x^2)/(1+x)^2)/sqrt(1+x^2)],
[sqrt(a/x^2)/sqrt(1+x^2),x,4,-x*arctanh(sqrt(1+x^2))*sqrt(a/x^2)],
[sqrt(a/x^3)/sqrt(1+x^2),x,6,-2*x*sqrt(a/x^3)*sqrt(1+x^2)+2*x^2*sqrt(a/x^3)*sqrt(1+x^2)/(1+x)-2*x^(3/2)*(1+x)*sqrt(cos(2*arctan(sqrt(x)))^2)/cos(2*arctan(sqrt(x)))*EllipticE(sin(2*arctan(sqrt(x))),sqrt(1/2))*sqrt(a/x^3)*sqrt((1+x^2)/(1+x)^2)/sqrt(1+x^2)+x^(3/2)*(1+x)*sqrt(cos(2*arctan(sqrt(x)))^2)/cos(2*arctan(sqrt(x)))*EllipticF(sin(2*arctan(sqrt(x))),sqrt(1/2))*sqrt(a/x^3)*sqrt((1+x^2)/(1+x)^2)/sqrt(1+x^2)],
[sqrt(a/x^4)/sqrt(1+x^2),x,2,-x*sqrt(a/x^4)*sqrt(1+x^2)],
[sqrt(a*x^4)/sqrt(1+x^3),x,2,2/3*sqrt(a*x^4)*sqrt(1+x^3)/x^2],
[sqrt(a*x^3)/sqrt(1+x^3),x,5,(1+sqrt(3))*sqrt(a*x^3)*sqrt(1+x^3)/(x*(1+x*(1+sqrt(3))))-3^(1/4)*(1+x)*sqrt(cos(arccos((1+x*(1-sqrt(3)))/(1+x*(1+sqrt(3)))))^2)/cos(arccos((1+x*(1-sqrt(3)))/(1+x*(1+sqrt(3)))))*EllipticE(sin(arccos((1+x*(1-sqrt(3)))/(1+x*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*sqrt(a*x^3)*sqrt((1-x+x^2)/(1+x*(1+sqrt(3)))^2)/(x*sqrt(1+x^3)*sqrt(x*(1+x)/(1+x*(1+sqrt(3)))^2))-1/2*(1+x)*sqrt(cos(arccos((1+x*(1-sqrt(3)))/(1+x*(1+sqrt(3)))))^2)/cos(arccos((1+x*(1-sqrt(3)))/(1+x*(1+sqrt(3)))))*EllipticF(sin(arccos((1+x*(1-sqrt(3)))/(1+x*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(1-sqrt(3))*sqrt(a*x^3)*sqrt((1-x+x^2)/(1+x*(1+sqrt(3)))^2)/(3^(1/4)*x*sqrt(1+x^3)*sqrt(x*(1+x)/(1+x*(1+sqrt(3)))^2))],
[sqrt(a*x^2)/sqrt(1+x^3),x,4,2*sqrt(a*x^2)*sqrt(1+x^3)/(x*(1+x+sqrt(3)))+2*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt(a*x^2)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*x*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))-3^(1/4)*(1+x)*EllipticE((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(a*x^2)*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(x*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[sqrt(a*x)/sqrt(1+x^3),x,3,2/3*arcsinh((a*x)^(3/2)/a^(3/2))*sqrt(a)],
[sqrt(a/x)/sqrt(1+x^3),x,3,x*(1+x)*sqrt(cos(arccos((1+x*(1-sqrt(3)))/(1+x*(1+sqrt(3)))))^2)/cos(arccos((1+x*(1-sqrt(3)))/(1+x*(1+sqrt(3)))))*EllipticF(sin(arccos((1+x*(1-sqrt(3)))/(1+x*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*sqrt(a/x)*sqrt((1-x+x^2)/(1+x*(1+sqrt(3)))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt(x*(1+x)/(1+x*(1+sqrt(3)))^2))],
[sqrt(a/x^2)/sqrt(1+x^3),x,4,-2/3*x*arctanh(sqrt(1+x^3))*sqrt(a/x^2)],
[sqrt(a/x^3)/sqrt(1+x^3),x,6,-2*x*sqrt(a/x^3)*sqrt(1+x^3)+2*x^2*(1+sqrt(3))*sqrt(a/x^3)*sqrt(1+x^3)/(1+x*(1+sqrt(3)))-2*3^(1/4)*x^2*(1+x)*sqrt(cos(arccos((1+x*(1-sqrt(3)))/(1+x*(1+sqrt(3)))))^2)/cos(arccos((1+x*(1-sqrt(3)))/(1+x*(1+sqrt(3)))))*EllipticE(sin(arccos((1+x*(1-sqrt(3)))/(1+x*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*sqrt(a/x^3)*sqrt((1-x+x^2)/(1+x*(1+sqrt(3)))^2)/(sqrt(1+x^3)*sqrt(x*(1+x)/(1+x*(1+sqrt(3)))^2))-x^2*(1+x)*sqrt(cos(arccos((1+x*(1-sqrt(3)))/(1+x*(1+sqrt(3)))))^2)/cos(arccos((1+x*(1-sqrt(3)))/(1+x*(1+sqrt(3)))))*EllipticF(sin(arccos((1+x*(1-sqrt(3)))/(1+x*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(1-sqrt(3))*sqrt(a/x^3)*sqrt((1-x+x^2)/(1+x*(1+sqrt(3)))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt(x*(1+x)/(1+x*(1+sqrt(3)))^2))],
[sqrt(a/x^4)/sqrt(1+x^3),x,5,-x*sqrt(a/x^4)*sqrt(1+x^3)+x^2*sqrt(a/x^4)*sqrt(1+x^3)/(1+x+sqrt(3))+x^2*(1+x)*EllipticF((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(2)*sqrt(a/x^4)*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(3^(1/4)*sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))-1/2*3^(1/4)*x^2*(1+x)*EllipticE((1+x-sqrt(3))/(1+x+sqrt(3)),sqrt(-7-4*sqrt(3)))*sqrt(a/x^4)*sqrt(2-sqrt(3))*sqrt((1-x+x^2)/(1+x+sqrt(3))^2)/(sqrt(1+x^3)*sqrt((1+x)/(1+x+sqrt(3))^2))],
[sqrt(a*x^(2*n))/sqrt(1+x^n),x,2,x*hypergeom([1/2,1+1/n],[2+1/n],-x^n)*sqrt(a*x^(2*n))/(1+n)],
[sqrt(a*x^n)/sqrt(1+x^n),x,2,2*x*hypergeom([1/2,1/2*(1+2/n)],[1/2*(3+2/n)],-x^n)*sqrt(a*x^n)/(2+n)],
[sqrt(a*x^(1/2*n))/sqrt(1+x^n),x,2,4*x*hypergeom([1/2,1/4*(1+4/n)],[1/4*(5+4/n)],-x^n)*sqrt(a*x^(1/2*n))/(4+n)],
[sqrt(a*x^(2*n))/sqrt(1+x^n)+2*sqrt(a*x^(2*n))/((2+n)*x^n*sqrt(1+x^n)),x,-5,2*x^(1-n)*sqrt(a*x^(2*n))*sqrt(1+x^n)/(2+n)],
[sqrt(a*x)/(sqrt(d+e*x)*sqrt(e+f*x)),x,2,2*EllipticE(sqrt(f)*sqrt(d+e*x)/sqrt(-e^2+d*f),sqrt(1-e^2/(d*f)))*sqrt(-e^2+d*f)*sqrt(a*x)*sqrt(e*(e+f*x)/(e^2-d*f))/(e*sqrt(f)*sqrt(-e*x/d)*sqrt(e+f*x))],

# Integrands of the form (a x^n)^p (b x^m)^q ...
[(a*x^m)^r,x,2,x*(a*x^m)^r/(1+m*r)],
[(a*x^m)^r*(b*x^n)^s,x,3,x*(a*x^m)^r*(b*x^n)^s/(1+m*r+n*s)],
[(a*x^m)^r*(b*x^n)^s*(c*x^p)^t,x,4,x*(a*x^m)^r*(b*x^n)^s*(c*x^p)^t/(1+m*r+n*s+p*t)],

# Integrands of the form u (Sqrt[a+b x] + Sqrt[c+d x])^p

# Integrands of the form x^m (Sqrt[a+b x] + Sqrt[c+b x])^p

# p>0

# p<0
[x^2/(sqrt(a+b*x)+sqrt(c+b*x)),x,5,2/3*a^2*(a+b*x)^(3/2)/(b^3*(a-c))-4/5*a*(a+b*x)^(5/2)/(b^3*(a-c))+2/7*(a+b*x)^(7/2)/(b^3*(a-c))-2/3*c^2*(c+b*x)^(3/2)/(b^3*(a-c))+4/5*c*(c+b*x)^(5/2)/(b^3*(a-c))-2/7*(c+b*x)^(7/2)/(b^3*(a-c))],
[x/(sqrt(a+b*x)+sqrt(c+b*x)),x,5,-2/3*a*(a+b*x)^(3/2)/(b^2*(a-c))+2/5*(a+b*x)^(5/2)/(b^2*(a-c))+2/3*c*(c+b*x)^(3/2)/(b^2*(a-c))-2/5*(c+b*x)^(5/2)/(b^2*(a-c))],
[1/(sqrt(a+b*x)+sqrt(c+b*x)),x,2,2/3*(a+b*x)^(3/2)/(b*(a-c))-2/3*(c+b*x)^(3/2)/(b*(a-c))],
[1/(x*(sqrt(a+b*x)+sqrt(c+b*x))),x,7,-2*arctanh(sqrt(a+b*x)/sqrt(a))*sqrt(a)/(a-c)+2*arctanh(sqrt(c+b*x)/sqrt(c))*sqrt(c)/(a-c)+2*sqrt(a+b*x)/(a-c)-2*sqrt(c+b*x)/(a-c)],
[1/(x^2*(sqrt(a+b*x)+sqrt(c+b*x))),x,7,-b*arctanh(sqrt(a+b*x)/sqrt(a))/((a-c)*sqrt(a))+b*arctanh(sqrt(c+b*x)/sqrt(c))/((a-c)*sqrt(c))-sqrt(a+b*x)/((a-c)*x)+sqrt(c+b*x)/((a-c)*x)],
[x^2/(sqrt(a+b*x)+sqrt(c+b*x))^2,x,9,1/3*(a+c)*x^3/(a-c)^2+1/2*b*x^4/(a-c)^2+5/12*(a+c)*(a+b*x)^(3/2)*(c+b*x)^(3/2)/(b^3*(a-c)^2)-1/2*x*(a+b*x)^(3/2)*(c+b*x)^(3/2)/(b^2*(a-c)^2)-1/32*(4*a*c-5*(a+c)^2)*arctanh(sqrt(a+b*x)/sqrt(c+b*x))/b^3+1/16*(4*a*c-5*(a+c)^2)*(a+b*x)^(3/2)*sqrt(c+b*x)/(b^3*(a-c)^2)-1/32*(4*a*c-5*(a+c)^2)*sqrt(a+b*x)*sqrt(c+b*x)/(b^3*(a-c))],
[x/(sqrt(a+b*x)+sqrt(c+b*x))^2,x,8,1/2*(a+c)*x^2/(a-c)^2+2/3*b*x^3/(a-c)^2-2/3*(a+b*x)^(3/2)*(c+b*x)^(3/2)/(b^2*(a-c)^2)-1/4*(a+c)*arctanh(sqrt(a+b*x)/sqrt(c+b*x))/b^2+1/2*(a+c)*(a+b*x)^(3/2)*sqrt(c+b*x)/(b^2*(a-c)^2)-1/4*(a+c)*sqrt(a+b*x)*sqrt(c+b*x)/(b^2*(a-c))],
[1/(sqrt(a+b*x)+sqrt(c+b*x))^2,x,7,1/2*arctanh(sqrt(a+b*x)/sqrt(c+b*x))/b+1/8*(a-c)^2/(b*(sqrt(a+b*x)+sqrt(c+b*x))^4),(a+c)*x/(a-c)^2+b*x^2/(a-c)^2+1/2*arctanh(sqrt(a+b*x)/sqrt(c+b*x))/b-(a+b*x)^(3/2)*sqrt(c+b*x)/(b*(a-c)^2)+1/2*sqrt(a+b*x)*sqrt(c+b*x)/(b*(a-c))],
[1/(x*(sqrt(a+b*x)+sqrt(c+b*x))^2),x,9,2*b*x/(a-c)^2-2*(a+c)*arctanh(sqrt(a+b*x)/sqrt(c+b*x))/(a-c)^2+(a+c)*log(x)/(a-c)^2+4*arctanh(sqrt(c)*sqrt(a+b*x)/(sqrt(a)*sqrt(c+b*x)))*sqrt(a)*sqrt(c)/(a-c)^2-2*sqrt(a+b*x)*sqrt(c+b*x)/(a-c)^2],
[1/(x^2*(sqrt(a+b*x)+sqrt(c+b*x))^2),x,9,(-a-c)/((a-c)^2*x)-4*b*arctanh(sqrt(a+b*x)/sqrt(c+b*x))/(a-c)^2+2*b*log(x)/(a-c)^2+2*b*(a+c)*arctanh(sqrt(c)*sqrt(a+b*x)/(sqrt(a)*sqrt(c+b*x)))/((a-c)^2*sqrt(a)*sqrt(c))+2*sqrt(a+b*x)*sqrt(c+b*x)/((a-c)^2*x)],
[x^2/(sqrt(a+b*x)+sqrt(c+b*x))^3,x,10,-8/3*a^3*(a+b*x)^(3/2)/(b^3*(a-c)^3)+2/3*a^2*(a+3*c)*(a+b*x)^(3/2)/(b^3*(a-c)^3)+24/5*a^2*(a+b*x)^(5/2)/(b^3*(a-c)^3)-4/5*a*(a+3*c)*(a+b*x)^(5/2)/(b^3*(a-c)^3)-24/7*a*(a+b*x)^(7/2)/(b^3*(a-c)^3)+2/7*(a+3*c)*(a+b*x)^(7/2)/(b^3*(a-c)^3)+8/9*(a+b*x)^(9/2)/(b^3*(a-c)^3)+8/3*c^3*(c+b*x)^(3/2)/(b^3*(a-c)^3)-2/3*c^2*(3*a+c)*(c+b*x)^(3/2)/(b^3*(a-c)^3)-24/5*c^2*(c+b*x)^(5/2)/(b^3*(a-c)^3)+4/5*c*(3*a+c)*(c+b*x)^(5/2)/(b^3*(a-c)^3)+24/7*c*(c+b*x)^(7/2)/(b^3*(a-c)^3)-2/7*(3*a+c)*(c+b*x)^(7/2)/(b^3*(a-c)^3)-8/9*(c+b*x)^(9/2)/(b^3*(a-c)^3)],
[x/(sqrt(a+b*x)+sqrt(c+b*x))^3,x,10,8/3*a^2*(a+b*x)^(3/2)/(b^2*(a-c)^3)-2/3*a*(a+3*c)*(a+b*x)^(3/2)/(b^2*(a-c)^3)-16/5*a*(a+b*x)^(5/2)/(b^2*(a-c)^3)+2/5*(a+3*c)*(a+b*x)^(5/2)/(b^2*(a-c)^3)+8/7*(a+b*x)^(7/2)/(b^2*(a-c)^3)-8/3*c^2*(c+b*x)^(3/2)/(b^2*(a-c)^3)+2/3*c*(3*a+c)*(c+b*x)^(3/2)/(b^2*(a-c)^3)+16/5*c*(c+b*x)^(5/2)/(b^2*(a-c)^3)-2/5*(3*a+c)*(c+b*x)^(5/2)/(b^2*(a-c)^3)-8/7*(c+b*x)^(7/2)/(b^2*(a-c)^3)],
[1/(sqrt(a+b*x)+sqrt(c+b*x))^3,x,6,1/10*(a-c)^2/(b*(sqrt(a+b*x)+sqrt(c+b*x))^5)+(-1/2)/(b*(sqrt(a+b*x)+sqrt(c+b*x))),-8/3*a*(a+b*x)^(3/2)/(b*(a-c)^3)+2/3*(a+3*c)*(a+b*x)^(3/2)/(b*(a-c)^3)+8/5*(a+b*x)^(5/2)/(b*(a-c)^3)+8/3*c*(c+b*x)^(3/2)/(b*(a-c)^3)-2/3*(3*a+c)*(c+b*x)^(3/2)/(b*(a-c)^3)-8/5*(c+b*x)^(5/2)/(b*(a-c)^3)],
[1/(x*(sqrt(a+b*x)+sqrt(c+b*x))^3),x,8,8/3*(a+b*x)^(3/2)/(a-c)^3-8/3*(c+b*x)^(3/2)/(a-c)^3-2*(a+3*c)*arctanh(sqrt(a+b*x)/sqrt(a))*sqrt(a)/(a-c)^3+2*(3*a+c)*arctanh(sqrt(c+b*x)/sqrt(c))*sqrt(c)/(a-c)^3+2*(a+3*c)*sqrt(a+b*x)/(a-c)^3-2*(3*a+c)*sqrt(c+b*x)/(a-c)^3],
[1/(x^2*(sqrt(a+b*x)+sqrt(c+b*x))^3),x,14,-3*b*(3*a+c)*arctanh(sqrt(a+b*x)/sqrt(a))/((a-c)^3*sqrt(a))-3*b*(a+3*c)*arctanh(sqrt(c+b*x)/sqrt(c))/((-a+c)^3*sqrt(c))+8*b*sqrt(a+b*x)/(a-c)^3-(a+3*c)*sqrt(a+b*x)/((a-c)^3*x)-8*b*sqrt(c+b*x)/(a-c)^3+(3*a+c)*sqrt(c+b*x)/((a-c)^3*x),-b*(a+3*c)*arctanh(sqrt(a+b*x)/sqrt(a))/((a-c)^3*sqrt(a))-8*b*arctanh(sqrt(a+b*x)/sqrt(a))*sqrt(a)/(a-c)^3+b*(3*a+c)*arctanh(sqrt(c+b*x)/sqrt(c))/((a-c)^3*sqrt(c))+8*b*arctanh(sqrt(c+b*x)/sqrt(c))*sqrt(c)/(a-c)^3+8*b*sqrt(a+b*x)/(a-c)^3-(a+3*c)*sqrt(a+b*x)/((a-c)^3*x)-8*b*sqrt(c+b*x)/(a-c)^3+(3*a+c)*sqrt(c+b*x)/((a-c)^3*x)],
[1/(sqrt(x)+sqrt(1+x)),x,3,-2/3*x^(3/2)+2/3*(1+x)^(3/2)],
[1/(sqrt(-1+x)+sqrt(x)),x,3,-2/3*(-1+x)^(3/2)+2/3*x^(3/2)],
[1/(sqrt(-1+x)+sqrt(1+x)),x,2,-1/3*(-1+x)^(3/2)+1/3*(1+x)^(3/2)],

# Integrands of the form x^m (Sqrt[a+b x] + Sqrt[a+c x])^p

# p>0
[x^3*(sqrt(1-x)+sqrt(1+x))^2,x,5,1/2*x^4-2/3*(1-x^2)^(3/2)+2/5*(1-x^2)^(5/2)],
[x^2*(sqrt(1-x)+sqrt(1+x))^2,x,5,2/3*x^3+1/4*arcsin(x)-1/4*x*sqrt(1-x^2)+1/2*x^3*sqrt(1-x^2)],
[x*(sqrt(1-x)+sqrt(1+x))^2,x,3,x^2-2/3*(1-x^2)^(3/2)],
[(sqrt(1-x)+sqrt(1+x))^2,x,4,2*x+arcsin(x)+x*sqrt(1-x^2)],
[(sqrt(1-x)+sqrt(1+x))^2/x,x,6,-2*arctanh(sqrt(1-x^2))+2*log(x)+2*sqrt(1-x^2)],
[(sqrt(1-x)+sqrt(1+x))^2/x^2,x,4,(-2)/x-2*arcsin(x)-2*sqrt(1-x^2)/x],
[(sqrt(1-x)+sqrt(1+x))^2/x^3,x,6,(-1)/x^2+arctanh(sqrt(1-x^2))-sqrt(1-x^2)/x^2],

# p<0
[x^3/(sqrt(a+b*x)+sqrt(a+c*x)),x,5,2/3*a^2*(a+b*x)^(3/2)/(b^3*(b-c))-4/5*a*(a+b*x)^(5/2)/(b^3*(b-c))+2/7*(a+b*x)^(7/2)/(b^3*(b-c))-2/3*a^2*(a+c*x)^(3/2)/((b-c)*c^3)+4/5*a*(a+c*x)^(5/2)/((b-c)*c^3)-2/7*(a+c*x)^(7/2)/((b-c)*c^3)],
[x^2/(sqrt(a+b*x)+sqrt(a+c*x)),x,5,-2/3*a*(a+b*x)^(3/2)/(b^2*(b-c))+2/5*(a+b*x)^(5/2)/(b^2*(b-c))+2/3*a*(a+c*x)^(3/2)/((b-c)*c^2)-2/5*(a+c*x)^(5/2)/((b-c)*c^2)],
[x/(sqrt(a+b*x)+sqrt(a+c*x)),x,3,2/3*(a+b*x)^(3/2)/(b*(b-c))-2/3*(a+c*x)^(3/2)/((b-c)*c)],
[1/(sqrt(a+b*x)+sqrt(a+c*x)),x,8,-2*arctanh(sqrt(a+b*x)/sqrt(a))*sqrt(a)/(b-c)+2*arctanh(sqrt(a+c*x)/sqrt(a))*sqrt(a)/(b-c)+2*sqrt(a+b*x)/(b-c)-2*sqrt(a+c*x)/(b-c)],
[1/(x*(sqrt(a+b*x)+sqrt(a+c*x))),x,7,-b*arctanh(sqrt(a+b*x)/sqrt(a))/((b-c)*sqrt(a))+c*arctanh(sqrt(a+c*x)/sqrt(a))/((b-c)*sqrt(a))-sqrt(a+b*x)/((b-c)*x)+sqrt(a+c*x)/((b-c)*x)],
[1/(x^2*(sqrt(a+b*x)+sqrt(a+c*x))),x,9,1/4*b^2*arctanh(sqrt(a+b*x)/sqrt(a))/(a^(3/2)*(b-c))-1/4*c^2*arctanh(sqrt(a+c*x)/sqrt(a))/(a^(3/2)*(b-c))-1/2*sqrt(a+b*x)/((b-c)*x^2)-1/4*b*sqrt(a+b*x)/(a*(b-c)*x)+1/2*sqrt(a+c*x)/((b-c)*x^2)+1/4*c*sqrt(a+c*x)/(a*(b-c)*x)],
[x^3/(sqrt(a+b*x)+sqrt(a+c*x))^2,x,8,a*x^2/(b-c)^2+1/3*(b+c)*x^3/(b-c)^2-2/3*(a+b*x)^(3/2)*(a+c*x)^(3/2)/(b*(b-c)^2*c)-1/4*a^3*(b+c)*arctanh(sqrt(c)*sqrt(a+b*x)/(sqrt(b)*sqrt(a+c*x)))/(b^(5/2)*c^(5/2))+1/2*a*(b+c)*(a+b*x)^(3/2)*sqrt(a+c*x)/(b^2*(b-c)^2*c)+1/4*a^2*(b+c)*sqrt(a+b*x)*sqrt(a+c*x)/(b^2*(b-c)*c^2)],
[x^2/(sqrt(a+b*x)+sqrt(a+c*x))^2,x,7,2*a*x/(b-c)^2+1/2*(b+c)*x^2/(b-c)^2+1/2*a^2*arctanh(sqrt(c)*sqrt(a+b*x)/(sqrt(b)*sqrt(a+c*x)))/(b^(3/2)*c^(3/2))-(a+b*x)^(3/2)*sqrt(a+c*x)/(b*(b-c)^2)-1/2*a*sqrt(a+b*x)*sqrt(a+c*x)/(b*(b-c)*c)],
[x/(sqrt(a+b*x)+sqrt(a+c*x))^2,x,9,(b+c)*x/(b-c)^2+4*a*arctanh(sqrt(a+b*x)/sqrt(a+c*x))/(b-c)^2+2*a*log(x)/(b-c)^2-2*a*(b+c)*arctanh(sqrt(c)*sqrt(a+b*x)/(sqrt(b)*sqrt(a+c*x)))/((b-c)^2*sqrt(b)*sqrt(c))-2*sqrt(a+b*x)*sqrt(a+c*x)/(b-c)^2],
[1/(sqrt(a+b*x)+sqrt(a+c*x))^2,x,9,-2*a/((b-c)^2*x)+2*(b+c)*arctanh(sqrt(a+b*x)/sqrt(a+c*x))/(b-c)^2+(b+c)*log(x)/(b-c)^2-4*arctanh(sqrt(c)*sqrt(a+b*x)/(sqrt(b)*sqrt(a+c*x)))*sqrt(b)*sqrt(c)/(b-c)^2+2*sqrt(a+b*x)*sqrt(a+c*x)/((b-c)^2*x)],
[1/(x*(sqrt(a+b*x)+sqrt(a+c*x))^2),x,6,-a/((b-c)^2*x^2)+(-b-c)/((b-c)^2*x)-1/2*arctanh(sqrt(a+b*x)/sqrt(a+c*x))/a+(a+c*x)^(3/2)*sqrt(a+b*x)/(a*(b-c)^2*x^2)+1/2*sqrt(a+b*x)*sqrt(a+c*x)/(a*(b-c)*x)],
[1/(x^2*(sqrt(a+b*x)+sqrt(a+c*x))^2),x,7,-2/3*a/((b-c)^2*x^3)+1/2*(-b-c)/((b-c)^2*x^2)+2/3*(a+b*x)^(3/2)*(a+c*x)^(3/2)/(a^2*(b-c)^2*x^3)+1/4*(b+c)*arctanh(sqrt(a+b*x)/sqrt(a+c*x))/a^2-1/2*(b+c)*(a+c*x)^(3/2)*sqrt(a+b*x)/(a^2*(b-c)^2*x^2)-1/4*(b+c)*sqrt(a+b*x)*sqrt(a+c*x)/(a^2*(b-c)*x)],
[x^4/(sqrt(a+b*x)+sqrt(a+c*x))^3,x,10,-8/3*a^2*(a+b*x)^(3/2)/(b^2*(b-c)^3)+2/3*a^2*(b+3*c)*(a+b*x)^(3/2)/(b^3*(b-c)^3)+8/5*a*(a+b*x)^(5/2)/(b^2*(b-c)^3)-4/5*a*(b+3*c)*(a+b*x)^(5/2)/(b^3*(b-c)^3)+2/7*(b+3*c)*(a+b*x)^(7/2)/(b^3*(b-c)^3)+8/3*a^2*(a+c*x)^(3/2)/((b-c)^3*c^2)-2/3*a^2*(3*b+c)*(a+c*x)^(3/2)/((b-c)^3*c^3)-8/5*a*(a+c*x)^(5/2)/((b-c)^3*c^2)+4/5*a*(3*b+c)*(a+c*x)^(5/2)/((b-c)^3*c^3)-2/7*(3*b+c)*(a+c*x)^(7/2)/((b-c)^3*c^3)],
[x^3/(sqrt(a+b*x)+sqrt(a+c*x))^3,x,6,8/3*a*(a+b*x)^(3/2)/(b*(b-c)^3)-2/3*a*(b+3*c)*(a+b*x)^(3/2)/(b^2*(b-c)^3)+2/5*(b+3*c)*(a+b*x)^(5/2)/(b^2*(b-c)^3)-8/3*a*(a+c*x)^(3/2)/((b-c)^3*c)+2/3*a*(3*b+c)*(a+c*x)^(3/2)/((b-c)^3*c^2)-2/5*(3*b+c)*(a+c*x)^(5/2)/((b-c)^3*c^2)],
[x^2/(sqrt(a+b*x)+sqrt(a+c*x))^3,x,8,2/3*(b+3*c)*(a+b*x)^(3/2)/(b*(b-c)^3)-2/3*(3*b+c)*(a+c*x)^(3/2)/((b-c)^3*c)-8*a^(3/2)*arctanh(sqrt(a+b*x)/sqrt(a))/(b-c)^3+8*a^(3/2)*arctanh(sqrt(a+c*x)/sqrt(a))/(b-c)^3+8*a*sqrt(a+b*x)/(b-c)^3-8*a*sqrt(a+c*x)/(b-c)^3],
[x/(sqrt(a+b*x)+sqrt(a+c*x))^3,x,14,-6*(b+c)*arctanh(sqrt(a+b*x)/sqrt(a))*sqrt(a)/(b-c)^3+6*(b+c)*arctanh(sqrt(a+c*x)/sqrt(a))*sqrt(a)/(b-c)^3+2*(b+3*c)*sqrt(a+b*x)/(b-c)^3-4*a*sqrt(a+b*x)/((b-c)^3*x)-2*(3*b+c)*sqrt(a+c*x)/(b-c)^3+4*a*sqrt(a+c*x)/((b-c)^3*x),-4*b*arctanh(sqrt(a+b*x)/sqrt(a))*sqrt(a)/(b-c)^3-2*(b+3*c)*arctanh(sqrt(a+b*x)/sqrt(a))*sqrt(a)/(b-c)^3+4*c*arctanh(sqrt(a+c*x)/sqrt(a))*sqrt(a)/(b-c)^3+2*(3*b+c)*arctanh(sqrt(a+c*x)/sqrt(a))*sqrt(a)/(b-c)^3+2*(b+3*c)*sqrt(a+b*x)/(b-c)^3-4*a*sqrt(a+b*x)/((b-c)^3*x)-2*(3*b+c)*sqrt(a+c*x)/(b-c)^3+4*a*sqrt(a+c*x)/((b-c)^3*x)],
[1/(sqrt(a+b*x)+sqrt(a+c*x))^3,x,16,-3*b*c*arctanh(sqrt(a+b*x)/sqrt(a))/((b-c)^3*sqrt(a))+3*b*c*arctanh(sqrt(a+c*x)/sqrt(a))/((b-c)^3*sqrt(a))-2*a*sqrt(a+b*x)/((b-c)^3*x^2)-(2*b+3*c)*sqrt(a+b*x)/((b-c)^3*x)+2*a*sqrt(a+c*x)/((b-c)^3*x^2)+(3*b+2*c)*sqrt(a+c*x)/((b-c)^3*x),b^2*arctanh(sqrt(a+b*x)/sqrt(a))/((b-c)^3*sqrt(a))-b*(b+3*c)*arctanh(sqrt(a+b*x)/sqrt(a))/((b-c)^3*sqrt(a))-c^2*arctanh(sqrt(a+c*x)/sqrt(a))/((b-c)^3*sqrt(a))+c*(3*b+c)*arctanh(sqrt(a+c*x)/sqrt(a))/((b-c)^3*sqrt(a))-2*a*sqrt(a+b*x)/((b-c)^3*x^2)-b*sqrt(a+b*x)/((b-c)^3*x)-(b+3*c)*sqrt(a+b*x)/((b-c)^3*x)+2*a*sqrt(a+c*x)/((b-c)^3*x^2)+c*sqrt(a+c*x)/((b-c)^3*x)+(3*b+c)*sqrt(a+c*x)/((b-c)^3*x)],

# Integrands of the form (e+f x)^m (Sqrt[a+b x] + Sqrt[a+c x])^p

# p>0
[sqrt(1-x)*(sqrt(1-x)+sqrt(1+x)),x,4,x-1/2*x^2+1/2*arcsin(x)+1/2*x*sqrt(1-x^2)],

# p<0

# Integrands of the form x^m (-Sqrt[a+b x] - Sqrt[a+c x])^n (Sqrt[a+b x] + Sqrt[a+c x])^p

# p>0
[x^3*(-sqrt(1-x)-sqrt(1+x))*(sqrt(1-x)+sqrt(1+x)),x,6,-1/2*x^4+2/3*(1-x^2)^(3/2)-2/5*(1-x^2)^(5/2)],
[x^2*(-sqrt(1-x)-sqrt(1+x))*(sqrt(1-x)+sqrt(1+x)),x,6,-2/3*x^3-1/4*arcsin(x)+1/4*x*sqrt(1-x^2)-1/2*x^3*sqrt(1-x^2)],
[x*(-sqrt(1-x)-sqrt(1+x))*(sqrt(1-x)+sqrt(1+x)),x,4,-x^2+2/3*(1-x^2)^(3/2)],
[(-sqrt(1-x)-sqrt(1+x))*(sqrt(1-x)+sqrt(1+x)),x,5,-2*x-arcsin(x)-x*sqrt(1-x^2)],
[(-sqrt(1-x)-sqrt(1+x))*(sqrt(1-x)+sqrt(1+x))/x,x,7,2*arctanh(sqrt(1-x^2))-2*log(x)-2*sqrt(1-x^2)],
[(-sqrt(1-x)-sqrt(1+x))*(sqrt(1-x)+sqrt(1+x))/x^2,x,5,2/x+2*arcsin(x)+2*sqrt(1-x^2)/x],
[(-sqrt(1-x)-sqrt(1+x))*(sqrt(1-x)+sqrt(1+x))/x^3,x,7,1/x^2-arctanh(sqrt(1-x^2))+sqrt(1-x^2)/x^2],

# p<0

# Integrands of the form (Sqrt[a+b x] - Sqrt[a+c x])^n (Sqrt[a+b x] + Sqrt[a+c x])^p

# p>0
[(sqrt(1-x)+sqrt(1+x))/(-sqrt(1-x)+sqrt(1+x)),x,15,-arctanh(sqrt(1-x^2))+log(x)+sqrt(1-x^2)],

# p<0
[(-sqrt(-1+x)+sqrt(1+x))/(sqrt(-1+x)+sqrt(1+x)),x,9,1/2*x^2+1/2*arccosh(x)-1/2*x*sqrt(-1+x)*sqrt(1+x)],

# Integrands of the form (g+h x+i x^2)^m (d+e x+f Sqrt[a+b x+c x^2])^n

# Integrands of the form (d+e x+f Sqrt[a+c x^2])^n when e^2-c f^2=0
[(d+e*x+f*sqrt(a+e^2*x^2/f^2))^n,x,4,1/2*(d+e*x+f*sqrt(a+e^2*x^2/f^2))^(1+n)/(e*(1+n))+1/2*a*f^2*hypergeom([2,1+n],[2+n],(d+e*x+f*sqrt(a+e^2*x^2/f^2))/d)*(d+e*x+f*sqrt(a+e^2*x^2/f^2))^(1+n)/(d^2*e*(1+n))],
[(d+e*x+f*sqrt(a+e^2*x^2/f^2))^3,x,3,3/2*a*d^2*f^2*log(e*x+f*sqrt(a+e^2*x^2/f^2))/e-1/2*a*d^3*f^2/(e*(e*x+f*sqrt(a+e^2*x^2/f^2)))+a*d*f^2*(e*x+f*sqrt(a+e^2*x^2/f^2))/e+1/4*a*f^2*(d+e*x+f*sqrt(a+e^2*x^2/f^2))^2/e+1/8*(d+e*x+f*sqrt(a+e^2*x^2/f^2))^4/e],
[(d+e*x+f*sqrt(a+e^2*x^2/f^2))^2,x,3,a*d*f^2*log(e*x+f*sqrt(a+e^2*x^2/f^2))/e-1/2*a*d^2*f^2/(e*(e*x+f*sqrt(a+e^2*x^2/f^2)))+1/2*a*f^2*(e*x+f*sqrt(a+e^2*x^2/f^2))/e+1/6*(d+e*x+f*sqrt(a+e^2*x^2/f^2))^3/e],
[d+e*x+f*sqrt(a+e^2*x^2/f^2),x,4,d*x+1/2*e*x^2+1/2*a*f^2*arctanh(e*x/(f*sqrt(a+e^2*x^2/f^2)))/e+1/2*f*x*sqrt(a+e^2*x^2/f^2)],
[1/(d+e*x+f*sqrt(a+e^2*x^2/f^2)),x,3,-1/2*a*f^2*log(e*x+f*sqrt(a+e^2*x^2/f^2))/(d^2*e)+1/2*(1+a*f^2/d^2)*log(d+e*x+f*sqrt(a+e^2*x^2/f^2))/e-1/2*a*f^2/(d*e*(e*x+f*sqrt(a+e^2*x^2/f^2)))],
[1/(d+e*x+f*sqrt(a+e^2*x^2/f^2))^2,x,3,-a*f^2*log(e*x+f*sqrt(a+e^2*x^2/f^2))/(d^3*e)+a*f^2*log(d+e*x+f*sqrt(a+e^2*x^2/f^2))/(d^3*e)-1/2*a*f^2/(d^2*e*(e*x+f*sqrt(a+e^2*x^2/f^2)))+1/2*(-1-a*f^2/d^2)/(e*(d+e*x+f*sqrt(a+e^2*x^2/f^2)))],
[1/(d+e*x+f*sqrt(a+e^2*x^2/f^2))^3,x,3,-3/2*a*f^2*log(e*x+f*sqrt(a+e^2*x^2/f^2))/(d^4*e)+3/2*a*f^2*log(d+e*x+f*sqrt(a+e^2*x^2/f^2))/(d^4*e)-1/2*a*f^2/(d^3*e*(e*x+f*sqrt(a+e^2*x^2/f^2)))+1/4*(-1-a*f^2/d^2)/(e*(d+e*x+f*sqrt(a+e^2*x^2/f^2))^2)-a*f^2/(d^3*e*(d+e*x+f*sqrt(a+e^2*x^2/f^2)))],
[(d+e*x+f*sqrt(a+e^2*x^2/f^2))^(5/2),x,6,-5/2*a*d^(3/2)*f^2*arctanh(sqrt(d+e*x+f*sqrt(a+e^2*x^2/f^2))/sqrt(d))/e+1/3*a*f^2*(d+e*x+f*sqrt(a+e^2*x^2/f^2))^(3/2)/e+1/7*(d+e*x+f*sqrt(a+e^2*x^2/f^2))^(7/2)/e+2*a*d*f^2*sqrt(d+e*x+f*sqrt(a+e^2*x^2/f^2))/e-1/2*a*d^2*f^2*sqrt(d+e*x+f*sqrt(a+e^2*x^2/f^2))/(e*(e*x+f*sqrt(a+e^2*x^2/f^2)))],
[(d+e*x+f*sqrt(a+e^2*x^2/f^2))^(3/2),x,6,-3/2*a*f^2*arctanh(sqrt(d+e*x+f*sqrt(a+e^2*x^2/f^2))/sqrt(d))*sqrt(d)/e+1/5*(d+e*x+f*sqrt(a+e^2*x^2/f^2))^(5/2)/e+a*f^2*sqrt(d+e*x+f*sqrt(a+e^2*x^2/f^2))/e-1/2*a*d*f^2*sqrt(d+e*x+f*sqrt(a+e^2*x^2/f^2))/(e*(e*x+f*sqrt(a+e^2*x^2/f^2)))],
[(d+e*x+f*sqrt(a+e^2*x^2/f^2))^(1/2),x,6,-1/2*a*f^2*arctanh(sqrt(d+e*x+f*sqrt(a+e^2*x^2/f^2))/sqrt(d))/(e*sqrt(d))+1/3*(d+e*x+f*sqrt(a+e^2*x^2/f^2))^(3/2)/e-1/2*a*f^2*sqrt(d+e*x+f*sqrt(a+e^2*x^2/f^2))/(e*(e*x+f*sqrt(a+e^2*x^2/f^2)))],
[1/(d+e*x+f*sqrt(a+e^2*x^2/f^2))^(1/2),x,5,1/2*a*f^2*arctanh(sqrt(d+e*x+f*sqrt(a+e^2*x^2/f^2))/sqrt(d))/(d^(3/2)*e)+sqrt(d+e*x+f*sqrt(a+e^2*x^2/f^2))/e-1/2*a*f^2*sqrt(d+e*x+f*sqrt(a+e^2*x^2/f^2))/(d*e*(e*x+f*sqrt(a+e^2*x^2/f^2)))],
[1/(d+e*x+f*sqrt(a+e^2*x^2/f^2))^(3/2),x,5,3/2*a*f^2*arctanh(sqrt(d+e*x+f*sqrt(a+e^2*x^2/f^2))/sqrt(d))/(d^(5/2)*e)+(-1-a*f^2/d^2)/(e*sqrt(d+e*x+f*sqrt(a+e^2*x^2/f^2)))-1/2*a*f^2*sqrt(d+e*x+f*sqrt(a+e^2*x^2/f^2))/(d^2*e*(e*x+f*sqrt(a+e^2*x^2/f^2)))],
[1/(d+e*x+f*sqrt(a+e^2*x^2/f^2))^(5/2),x,6,5/2*a*f^2*arctanh(sqrt(d+e*x+f*sqrt(a+e^2*x^2/f^2))/sqrt(d))/(d^(7/2)*e)+1/3*(-1-a*f^2/d^2)/(e*(d+e*x+f*sqrt(a+e^2*x^2/f^2))^(3/2))-2*a*f^2/(d^3*e*sqrt(d+e*x+f*sqrt(a+e^2*x^2/f^2)))-1/2*a*f^2*sqrt(d+e*x+f*sqrt(a+e^2*x^2/f^2))/(d^3*e*(e*x+f*sqrt(a+e^2*x^2/f^2)))],
[sqrt(x-sqrt(-4+x^2)),x,3,1/3*(x-sqrt(-4+x^2))^(3/2)+4/sqrt(x-sqrt(-4+x^2))],
[sqrt(a*x+b*sqrt(c+a^2*x^2/b^2)),x,3,1/3*(a*x+b*sqrt(c+a^2*x^2/b^2))^(3/2)/a-b^2*c/(a*sqrt(a*x+b*sqrt(c+a^2*x^2/b^2)))],
[sqrt(1+sqrt(1-x^2)),x,1,-2/3*x^3/(1+sqrt(1-x^2))^(3/2)+2*x/sqrt(1+sqrt(1-x^2))],
[sqrt(1+sqrt(1+x^2)),x,1,2/3*x^3/(1+sqrt(1+x^2))^(3/2)+2*x/sqrt(1+sqrt(1+x^2))],
[sqrt(5+sqrt(25+x^2)),x,1,2/3*x^3/(5+sqrt(25+x^2))^(3/2)+10*x/sqrt(5+sqrt(25+x^2))],
[sqrt(a+b*sqrt(a^2/b^2+c*x^2)),x,1,2/3*b^2*c*x^3/(a+b*sqrt(a^2/b^2+c*x^2))^(3/2)+2*a*x/sqrt(a+b*sqrt(a^2/b^2+c*x^2))],

# Integrands of the form (d+e x+f Sqrt[a+b x+c x^2])^n when e^2-c f^2=0
[(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^n,x,4,1/2*(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^(1+n)/(e*(1+n))+1/2*f^2*(4*a*e^2-b^2*f^2)*hypergeom([2,1+n],[2+n],2*e*(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/(2*d*e-b*f^2))*(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^(1+n)/(e*(2*d*e-b*f^2)^2*(1+n))],
[(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^3,x,3,3/32*f^2*(2*d*e-b*f^2)^2*(4*a*e^2-b^2*f^2)*log(b*f^2+2*e*(e*x+f*sqrt(a+x*(b*f^2+e^2*x)/f^2)))/e^5+1/8*f^2*(2*d*e-b*f^2)*(4*a*e^2-b^2*f^2)*(e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/e^4+1/16*f^2*(4*a*e^2-b^2*f^2)*(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^2/e^3+1/8*(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^4/e-1/32*f^2*(2*d*e-b*f^2)^3*(4*a*e^2-b^2*f^2)/(e^5*(b*f^2+2*e*(e*x+f*sqrt(a+x*(b*f^2+e^2*x)/f^2))))],
[(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^2,x,3,1/8*f^2*(2*d*e-b*f^2)*(4*a*e^2-b^2*f^2)*log(b*f^2+2*e*(e*x+f*sqrt(a+x*(b*f^2+e^2*x)/f^2)))/e^4+1/8*f^2*(4*a*e^2-b^2*f^2)*(e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/e^3+1/6*(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^3/e-1/16*f^2*(2*d*e-b*f^2)^2*(4*a*e^2-b^2*f^2)/(e^4*(b*f^2+2*e*(e*x+f*sqrt(a+x*(b*f^2+e^2*x)/f^2))))],
[d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2),x,4,d*x+1/2*e*x^2+1/8*f^2*(4*a*e^2-b^2*f^2)*arctanh(1/2*(b*f^2+2*e^2*x)/(e*f*sqrt(a+b*x+e^2*x^2/f^2)))/e^3+1/4*f*(b*f^2+2*e^2*x)*sqrt(a+b*x+e^2*x^2/f^2)/e^2],
[1/(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2)),x,3,2*(d^2*e-b*d*f^2+a*e*f^2)*log(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/(2*d*e-b*f^2)^2-1/2*f^2*(4*a*e^2-b^2*f^2)*log(b*f^2+2*e*(e*x+f*sqrt(a+x*(b*f^2+e^2*x)/f^2)))/(e*(2*d*e-b*f^2)^2)-1/2*f^2*(4*a*e^2-b^2*f^2)/(e*(2*d*e-b*f^2)*(b*f^2+2*e*(e*x+f*sqrt(a+x*(b*f^2+e^2*x)/f^2))))],
[1/(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^2,x,3,2*f^2*(4*a*e^2-b^2*f^2)*log(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/(2*d*e-b*f^2)^3-2*f^2*(4*a*e^2-b^2*f^2)*log(b*f^2+2*e*(e*x+f*sqrt(a+x*(b*f^2+e^2*x)/f^2)))/(2*d*e-b*f^2)^3-2*(d^2*e-b*d*f^2+a*e*f^2)/((2*d*e-b*f^2)^2*(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2)))-f^2*(4*a*e^2-b^2*f^2)/((2*d*e-b*f^2)^2*(b*f^2+2*e*(e*x+f*sqrt(a+x*(b*f^2+e^2*x)/f^2))))],
[1/(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^3,x,3,6*e*f^2*(4*a*e^2-b^2*f^2)*log(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/(2*d*e-b*f^2)^4-6*e*f^2*(4*a*e^2-b^2*f^2)*log(b*f^2+2*e*(e*x+f*sqrt(a+x*(b*f^2+e^2*x)/f^2)))/(2*d*e-b*f^2)^4+(-d^2*e+b*d*f^2-a*e*f^2)/((2*d*e-b*f^2)^2*(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^2)-2*f^2*(4*a*e^2-b^2*f^2)/((2*d*e-b*f^2)^3*(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2)))-2*e*f^2*(4*a*e^2-b^2*f^2)/((2*d*e-b*f^2)^3*(b*f^2+2*e*(e*x+f*sqrt(a+x*(b*f^2+e^2*x)/f^2))))],
[(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^(5/2),x,6,-5/16*f^2*(2*d*e-b*f^2)^(3/2)*(4*a*e^2-b^2*f^2)*arctanh(sqrt(2)*sqrt(e)*sqrt(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/sqrt(2*d*e-b*f^2))/(e^(9/2)*sqrt(2))+1/12*f^2*(4*a*e^2-b^2*f^2)*(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^(3/2)/e^3+1/7*(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^(7/2)/e+1/4*f^2*(2*d*e-b*f^2)*(4*a*e^2-b^2*f^2)*sqrt(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/e^4-1/16*f^2*(2*d*e-b*f^2)^2*(4*a*e^2-b^2*f^2)*sqrt(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/(e^4*(b*f^2+2*e*(e*x+f*sqrt(a+x*(b*f^2+e^2*x)/f^2))))],
[(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^(3/2),x,6,-3/8*f^2*(4*a*e^2-b^2*f^2)*arctanh(sqrt(2)*sqrt(e)*sqrt(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/sqrt(2*d*e-b*f^2))*sqrt(2*d*e-b*f^2)/(e^(7/2)*sqrt(2))+1/5*(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^(5/2)/e+1/4*f^2*(4*a*e^2-b^2*f^2)*sqrt(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/e^3-1/8*f^2*(2*d*e-b*f^2)*(4*a*e^2-b^2*f^2)*sqrt(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/(e^3*(b*f^2+2*e*(e*x+f*sqrt(a+x*(b*f^2+e^2*x)/f^2))))],
[(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^(1/2),x,6,-1/4*f^2*(4*a*e^2-b^2*f^2)*arctanh(sqrt(2)*sqrt(e)*sqrt(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/sqrt(2*d*e-b*f^2))/(e^(5/2)*sqrt(2)*sqrt(2*d*e-b*f^2))+1/3*(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^(3/2)/e-1/4*f^2*(4*a-b^2*f^2/e^2)*sqrt(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/(b*f^2+2*e*(e*x+f*sqrt(a+x*(b*f^2+e^2*x)/f^2)))],
[1/(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^(1/2),x,5,1/2*f^2*(4*a*e^2-b^2*f^2)*arctanh(sqrt(2)*sqrt(e)*sqrt(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/sqrt(2*d*e-b*f^2))/(e^(3/2)*(2*d*e-b*f^2)^(3/2)*sqrt(2))+sqrt(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/e-1/2*f^2*(4*a*e-b^2*f^2/e)*sqrt(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/((2*d*e-b*f^2)*(b*f^2+2*e*(e*x+f*sqrt(a+x*(b*f^2+e^2*x)/f^2))))],
[1/(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^(3/2),x,5,3*f^2*(4*a*e^2-b^2*f^2)*arctanh(sqrt(2)*sqrt(e)*sqrt(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/sqrt(2*d*e-b*f^2))/((2*d*e-b*f^2)^(5/2)*sqrt(2)*sqrt(e))-4*(d^2*e-b*d*f^2+a*e*f^2)/((2*d*e-b*f^2)^2*sqrt(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2)))-f^2*(4*a*e^2-b^2*f^2)*sqrt(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/((2*d*e-b*f^2)^2*(b*f^2+2*e*(e*x+f*sqrt(a+x*(b*f^2+e^2*x)/f^2))))],
[1/(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^(5/2),x,6,5*f^2*(4*a*e^2-b^2*f^2)*arctanh(sqrt(2)*sqrt(e)*sqrt(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/sqrt(2*d*e-b*f^2))*sqrt(2)*sqrt(e)/(2*d*e-b*f^2)^(7/2)-4/3*(d^2*e-b*d*f^2+a*e*f^2)/((2*d*e-b*f^2)^2*(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))^(3/2))-4*f^2*(4*a*e^2-b^2*f^2)/((2*d*e-b*f^2)^3*sqrt(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2)))-2*e*f^2*(4*a*e^2-b^2*f^2)*sqrt(d+e*x+f*sqrt(a+b*x+e^2*x^2/f^2))/((2*d*e-b*f^2)^3*(b*f^2+2*e*(e*x+f*sqrt(a+x*(b*f^2+e^2*x)/f^2))))],

# Integrands of the form (a+c x^2)^m (d x+e Sqrt[a+c x^2])^n
[(a+x^2)^2*(x+sqrt(a+x^2))^n,x,3,-1/32*a^5*(x+sqrt(a+x^2))^(-5+n)/(5-n)-5/32*a^4*(x+sqrt(a+x^2))^(-3+n)/(3-n)-5/16*a^3*(x+sqrt(a+x^2))^(-1+n)/(1-n)+5/16*a^2*(x+sqrt(a+x^2))^(1+n)/(1+n)+5/32*a*(x+sqrt(a+x^2))^(3+n)/(3+n)+1/32*(x+sqrt(a+x^2))^(5+n)/(5+n)],
[(a+x^2)*(x+sqrt(a+x^2))^n,x,3,-1/8*a^3*(x+sqrt(a+x^2))^(-3+n)/(3-n)-3/8*a^2*(x+sqrt(a+x^2))^(-1+n)/(1-n)+3/8*a*(x+sqrt(a+x^2))^(1+n)/(1+n)+1/8*(x+sqrt(a+x^2))^(3+n)/(3+n)],
[(x+sqrt(a+x^2))^n,x,3,-1/2*a*(x+sqrt(a+x^2))^(-1+n)/(1-n)+1/2*(x+sqrt(a+x^2))^(1+n)/(1+n)],
[(x+sqrt(a+x^2))^n/(a+x^2),x,2,2*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],-(x+sqrt(a+x^2))^2/a)*(x+sqrt(a+x^2))^(1+n)/(a*(1+n))],
[(x+sqrt(a+x^2))^n/(a+x^2)^2,x,2,8*hypergeom([3,1/2*(3+n)],[1/2*(5+n)],-(x+sqrt(a+x^2))^2/a)*(x+sqrt(a+x^2))^(3+n)/(a^3*(3+n))],
[(a+x^2)^2*(x-sqrt(a+x^2))^n,x,3,-1/32*a^5*(x-sqrt(a+x^2))^(-5+n)/(5-n)-5/32*a^4*(x-sqrt(a+x^2))^(-3+n)/(3-n)-5/16*a^3*(x-sqrt(a+x^2))^(-1+n)/(1-n)+5/16*a^2*(x-sqrt(a+x^2))^(1+n)/(1+n)+5/32*a*(x-sqrt(a+x^2))^(3+n)/(3+n)+1/32*(x-sqrt(a+x^2))^(5+n)/(5+n)],
[(a+x^2)*(x-sqrt(a+x^2))^n,x,3,-1/8*a^3*(x-sqrt(a+x^2))^(-3+n)/(3-n)-3/8*a^2*(x-sqrt(a+x^2))^(-1+n)/(1-n)+3/8*a*(x-sqrt(a+x^2))^(1+n)/(1+n)+1/8*(x-sqrt(a+x^2))^(3+n)/(3+n)],
[(x-sqrt(a+x^2))^n,x,3,-1/2*a*(x-sqrt(a+x^2))^(-1+n)/(1-n)+1/2*(x-sqrt(a+x^2))^(1+n)/(1+n)],
[(x-sqrt(a+x^2))^n/(a+x^2),x,2,2*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],-(x-sqrt(a+x^2))^2/a)*(x-sqrt(a+x^2))^(1+n)/(a*(1+n))],
[(x-sqrt(a+x^2))^n/(a+x^2)^2,x,2,8*hypergeom([3,1/2*(3+n)],[1/2*(5+n)],-(x-sqrt(a+x^2))^2/a)*(x-sqrt(a+x^2))^(3+n)/(a^3*(3+n))],

# Integrands of the form (a+c x^2)^(m/2) (d x+e Sqrt[a+c x^2])^n
[(a+x^2)^(5/2)*(x+sqrt(a+x^2))^n,x,3,-1/64*a^6*(x+sqrt(a+x^2))^(-6+n)/(6-n)-3/32*a^5*(x+sqrt(a+x^2))^(-4+n)/(4-n)-15/64*a^4*(x+sqrt(a+x^2))^(-2+n)/(2-n)+5/16*a^3*(x+sqrt(a+x^2))^n/n+15/64*a^2*(x+sqrt(a+x^2))^(2+n)/(2+n)+3/32*a*(x+sqrt(a+x^2))^(4+n)/(4+n)+1/64*(x+sqrt(a+x^2))^(6+n)/(6+n)],
[(a+x^2)^(3/2)*(x+sqrt(a+x^2))^n,x,3,-1/16*a^4*(x+sqrt(a+x^2))^(-4+n)/(4-n)-1/4*a^3*(x+sqrt(a+x^2))^(-2+n)/(2-n)+3/8*a^2*(x+sqrt(a+x^2))^n/n+1/4*a*(x+sqrt(a+x^2))^(2+n)/(2+n)+1/16*(x+sqrt(a+x^2))^(4+n)/(4+n)],
[(a+x^2)^(1/2)*(x+sqrt(a+x^2))^n,x,3,-1/4*a^2*(x+sqrt(a+x^2))^(-2+n)/(2-n)+1/2*a*(x+sqrt(a+x^2))^n/n+1/4*(x+sqrt(a+x^2))^(2+n)/(2+n)],
[(x+sqrt(a+x^2))^n/(a+x^2)^(1/2),x,2,(x+sqrt(a+x^2))^n/n],
[(x+sqrt(a+x^2))^n/(a+x^2)^(3/2),x,2,4*hypergeom([2,1/2*(2+n)],[1/2*(4+n)],-(x+sqrt(a+x^2))^2/a)*(x+sqrt(a+x^2))^(2+n)/(a^2*(2+n))],
[(x+sqrt(a+x^2))^n/(a+x^2)^(5/2),x,2,16*hypergeom([4,1/2*(4+n)],[1/2*(6+n)],-(x+sqrt(a+x^2))^2/a)*(x+sqrt(a+x^2))^(4+n)/(a^4*(4+n))],
[(a+x^2)^(5/2)*(x-sqrt(a+x^2))^n,x,3,1/64*a^6*(x-sqrt(a+x^2))^(-6+n)/(6-n)+3/32*a^5*(x-sqrt(a+x^2))^(-4+n)/(4-n)+15/64*a^4*(x-sqrt(a+x^2))^(-2+n)/(2-n)-5/16*a^3*(x-sqrt(a+x^2))^n/n-15/64*a^2*(x-sqrt(a+x^2))^(2+n)/(2+n)-3/32*a*(x-sqrt(a+x^2))^(4+n)/(4+n)-1/64*(x-sqrt(a+x^2))^(6+n)/(6+n)],
[(a+x^2)^(3/2)*(x-sqrt(a+x^2))^n,x,3,1/16*a^4*(x-sqrt(a+x^2))^(-4+n)/(4-n)+1/4*a^3*(x-sqrt(a+x^2))^(-2+n)/(2-n)-3/8*a^2*(x-sqrt(a+x^2))^n/n-1/4*a*(x-sqrt(a+x^2))^(2+n)/(2+n)-1/16*(x-sqrt(a+x^2))^(4+n)/(4+n)],
[(a+x^2)^(1/2)*(x-sqrt(a+x^2))^n,x,3,1/4*a^2*(x-sqrt(a+x^2))^(-2+n)/(2-n)-1/2*a*(x-sqrt(a+x^2))^n/n-1/4*(x-sqrt(a+x^2))^(2+n)/(2+n)],
[(x-sqrt(a+x^2))^n/(a+x^2)^(1/2),x,2,-(x-sqrt(a+x^2))^n/n],
[(x-sqrt(a+x^2))^n/(a+x^2)^(3/2),x,2,-4*hypergeom([2,1/2*(2+n)],[1/2*(4+n)],-(x-sqrt(a+x^2))^2/a)*(x-sqrt(a+x^2))^(2+n)/(a^2*(2+n))],
[(x-sqrt(a+x^2))^n/(a+x^2)^(5/2),x,2,-16*hypergeom([4,1/2*(4+n)],[1/2*(6+n)],-(x-sqrt(a+x^2))^2/a)*(x-sqrt(a+x^2))^(4+n)/(a^4*(4+n))],

# Integrands of the form (g+h x+i x^2)^m (d+e x+f Sqrt[a+b x+c x^2])^n
[(a+2*d*e*x/f^2+e^2*x^2/f^2)^2*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^n,x,4,1/32*(d^2-a*f^2)^5*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(-5+n)/(e*f^4*(5-n))-5/32*(d^2-a*f^2)^4*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(-3+n)/(e*f^4*(3-n))+5/16*(d^2-a*f^2)^3*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(-1+n)/(e*f^4*(1-n))+5/16*(d^2-a*f^2)^2*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(1+n)/(e*f^4*(1+n))-5/32*(d^2-a*f^2)*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(3+n)/(e*f^4*(3+n))+1/32*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(5+n)/(e*f^4*(5+n))],
[(a+2*d*e*x/f^2+e^2*x^2/f^2)*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^n,x,4,1/8*(d^2-a*f^2)^3*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(-3+n)/(e*f^2*(3-n))-3/8*(d^2-a*f^2)^2*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(-1+n)/(e*f^2*(1-n))-3/8*(d^2-a*f^2)*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(1+n)/(e*f^2*(1+n))+1/8*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(3+n)/(e*f^2*(3+n))],
[(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^n,x,4,1/2*(d^2-a*f^2)*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(-1+n)/(e*(1-n))+1/2*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(1+n)/(e*(1+n))],
[(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^n/(a+2*d*e*x/f^2+e^2*x^2/f^2),x,2,-2*f^2*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^2/(d^2-a*f^2))*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(1+n)/(e*(d^2-a*f^2)*(1+n))],
[(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^n/(a+2*d*e*x/f^2+e^2*x^2/f^2)^2,x,3,-8*f^4*hypergeom([3,1/2*(3+n)],[1/2*(5+n)],(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^2/(d^2-a*f^2))*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(3+n)/(e*(d^2-a*f^2)^3*(3+n))],
[(d+e*x+f*sqrt((a*f^2+e*x*(2*d+e*x))/f^2))^n,x,5,1/2*(d^2-a*f^2)*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(-1+n)/(e*(1-n))+1/2*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(1+n)/(e*(1+n))],
[(d+e*x+f*sqrt((a*f^2+e*x*(2*d+e*x))/f^2))^n/(a+2*d*e*x/f^2+e^2*x^2/f^2),x,3,-2*f^2*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^2/(d^2-a*f^2))*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(1+n)/(e*(d^2-a*f^2)*(1+n))],
[(a+2*d*e*x/f^2+e^2*x^2/f^2)^(3/2)*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^n,x,4,-1/16*(d^2-a*f^2)^4*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(-4+n)/(e*f^3*(4-n))+1/4*(d^2-a*f^2)^3*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(-2+n)/(e*f^3*(2-n))+3/8*(d^2-a*f^2)^2*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^n/(e*f^3*n)-1/4*(d^2-a*f^2)*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(2+n)/(e*f^3*(2+n))+1/16*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(4+n)/(e*f^3*(4+n))],
[(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2)*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^n,x,4,-1/4*(d^2-a*f^2)^2*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(-2+n)/(e*f*(2-n))-1/2*(d^2-a*f^2)*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^n/(e*f*n)+1/4*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(2+n)/(e*f*(2+n))],
[(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^n/(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2),x,3,f*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^n/(e*n)],
[(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^n/(a+2*d*e*x/f^2+e^2*x^2/f^2)^(3/2),x,3,4*f^3*hypergeom([2,1/2*(2+n)],[1/2*(4+n)],(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^2/(d^2-a*f^2))*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(2+n)/(e*(d^2-a*f^2)^2*(2+n))],
[(d+e*x+f*sqrt((a*f^2+e*x*(2*d+e*x))/f^2))^n/((a*f^2+e*x*(2*d+e*x))/f^2)^(1/2),x,4,f*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^n/(e*n)],
[(a*g+2*d*e*g*x/f^2+e^2*g*x^2/f^2)^(1/2)*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^n,x,5,-1/4*(d^2-a*f^2)^2*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(-2+n)*sqrt(a*g+2*d*e*g*x/f^2+e^2*g*x^2/f^2)/(e*f*(2-n)*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))-1/2*(d^2-a*f^2)*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^n*sqrt(a*g+2*d*e*g*x/f^2+e^2*g*x^2/f^2)/(e*f*n*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))+1/4*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(2+n)*sqrt(a*g+2*d*e*g*x/f^2+e^2*g*x^2/f^2)/(e*f*(2+n)*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))],
[(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^n/(a*g+2*d*e*g*x/f^2+e^2*g*x^2/f^2)^(1/2),x,4,f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2)*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^n/(e*n*sqrt(a*g+2*d*e*g*x/f^2+e^2*g*x^2/f^2))],
[(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^n/(a*g+2*d*e*g*x/f^2+e^2*g*x^2/f^2)^(3/2),x,4,4*f^3*hypergeom([2,1/2*(2+n)],[1/2*(4+n)],(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^2/(d^2-a*f^2))*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2)*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^(2+n)/(e*(d^2-a*f^2)^2*g*(2+n)*sqrt(a*g+2*d*e*g*x/f^2+e^2*g*x^2/f^2))],
[(d+e*x+f*sqrt((a*f^2+e*x*(2*d+e*x))/f^2))^n/((a*f^2*g+e*g*x*(2*d+e*x))/f^2)^(1/2),x,5,f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2)*(d+e*x+f*sqrt(a+2*d*e*x/f^2+e^2*x^2/f^2))^n/(e*n*sqrt(a*g+2*d*e*g*x/f^2+e^2*g*x^2/f^2))],

# Integrands of the form (a+b x)^m (c+d x^2)^p (e+f x^2)^q
[1/((a+b*x)*sqrt(c+d*x^2)*sqrt(e+f*x^2)),x,7,-b*arctanh(sqrt(b^2*e+a^2*f)*sqrt(c+d*x^2)/(sqrt(b^2*c+a^2*d)*sqrt(e+f*x^2)))/(sqrt(b^2*c+a^2*d)*sqrt(b^2*e+a^2*f))+EllipticPi(x*sqrt(d)/sqrt(-c),-b^2*c/(a^2*d),sqrt(c*f/(d*e)))*sqrt(-c)*sqrt(1+d*x^2/c)*sqrt(1+f*x^2/e)/(a*sqrt(d)*sqrt(c+d*x^2)*sqrt(e+f*x^2))],

# Integrands of the form x^m (A+B x^n) / (a+b x^(2 (m+1))+c x^n+d x^(2 n))

# Integrands of the form (A+B x^n) / (a+b x^2+c x^n+d x^(2 n))
[(e-2*f*x^2)/(e^2+4*d*f*x^2+4*e*f*x^2+4*f^2*x^4),x,4,-1/4*log(e+2*f*x^2-2*x*sqrt(-d)*sqrt(f))/(sqrt(-d)*sqrt(f))+1/4*log(e+2*f*x^2+2*x*sqrt(-d)*sqrt(f))/(sqrt(-d)*sqrt(f))],
[(e-2*f*x^2)/(e^2-4*d*f*x^2+4*e*f*x^2+4*f^2*x^4),x,4,-1/4*log(e+2*f*x^2-2*x*sqrt(d)*sqrt(f))/(sqrt(d)*sqrt(f))+1/4*log(e+2*f*x^2+2*x*sqrt(d)*sqrt(f))/(sqrt(d)*sqrt(f))],
[(e-4*f*x^3)/(e^2+4*d*f*x^2+4*e*f*x^3+4*f^2*x^6),x,2,1/2*arctan(2*x*sqrt(d)*sqrt(f)/(e+2*f*x^3))/(sqrt(d)*sqrt(f))],
[(e-4*f*x^3)/(e^2-4*d*f*x^2+4*e*f*x^3+4*f^2*x^6),x,2,1/2*arctanh(2*x*sqrt(d)*sqrt(f)/(e+2*f*x^3))/(sqrt(d)*sqrt(f))],
[(e-2*f*(-1+n)*x^n)/(e^2+4*d*f*x^2+4*e*f*x^n+4*f^2*x^(2*n)),x,2,1/2*arctan(2*x*sqrt(d)*sqrt(f)/(e+2*f*x^n))/(sqrt(d)*sqrt(f))],
[(e-2*f*(-1+n)*x^n)/(e^2-4*d*f*x^2+4*e*f*x^n+4*f^2*x^(2*n)),x,2,1/2*arctanh(2*x*sqrt(d)*sqrt(f)/(e+2*f*x^n))/(sqrt(d)*sqrt(f))],

# Integrands of the form x^m (A+B x^n) / (a+b x^(2 (m+1))+c x^n+d x^(2 n))
[x/(e^2+4*e*f*x^2+4*d*f*x^4+4*f^2*x^4),x,4,1/4*arctan((e+2*(d+f)*x^2)*sqrt(f)/(e*sqrt(d)))/(e*sqrt(d)*sqrt(f))],
[x/(e^2+4*e*f*x^2-4*d*f*x^4+4*f^2*x^4),x,4,-1/4*arctanh((e-2*(d-f)*x^2)*sqrt(f)/(e*sqrt(d)))/(e*sqrt(d)*sqrt(f))],
[x^2*(3*e+2*f*x^2)/(e^2+4*e*f*x^2+4*f^2*x^4+4*d*f*x^6),x,2,1/2*arctan(2*x^3*sqrt(d)*sqrt(f)/(e+2*f*x^2))/(sqrt(d)*sqrt(f))],
[x^2*(3*e+2*f*x^2)/(e^2+4*e*f*x^2+4*f^2*x^4-4*d*f*x^6),x,2,1/2*arctanh(2*x^3*sqrt(d)*sqrt(f)/(e+2*f*x^2))/(sqrt(d)*sqrt(f))],
[x^m*(e*(1+m)+2*f*(-1+m)*x^2)/(e^2+4*e*f*x^2+4*f^2*x^4+4*d*f*x^(2+2*m)),x,2,1/2*arctan(2*x^(1+m)*sqrt(d)*sqrt(f)/(e+2*f*x^2))/(sqrt(d)*sqrt(f)),1/2*arctan(2*(1-m^2)*x^(1+m)*sqrt(d)*sqrt(f)/((1-m)*(1+m)*(e+2*f*x^2)))/(sqrt(d)*sqrt(f))],
[x^m*(e*(1+m)+2*f*(-1+m)*x^2)/(e^2+4*e*f*x^2+4*f^2*x^4-4*d*f*x^(2+2*m)),x,2,1/2*arctanh(2*x^(1+m)*sqrt(d)*sqrt(f)/(e+2*f*x^2))/(sqrt(d)*sqrt(f)),1/2*arctanh(2*(1-m^2)*x^(1+m)*sqrt(d)*sqrt(f)/((1-m)*(1+m)*(e+2*f*x^2)))/(sqrt(d)*sqrt(f))],
[x*(2*e-2*f*x^3)/(e^2+4*e*f*x^3+4*d*f*x^4+4*f^2*x^6),x,2,1/2*arctan(2*x^2*sqrt(d)*sqrt(f)/(e+2*f*x^3))/(sqrt(d)*sqrt(f))],
[x*(2*e-2*f*x^3)/(e^2+4*e*f*x^3-4*d*f*x^4+4*f^2*x^6),x,2,1/2*arctanh(2*x^2*sqrt(d)*sqrt(f)/(e+2*f*x^3))/(sqrt(d)*sqrt(f))],
[x^2/(e^2+4*e*f*x^3+4*d*f*x^6+4*f^2*x^6),x,4,1/6*arctan((e+2*(d+f)*x^3)*sqrt(f)/(e*sqrt(d)))/(e*sqrt(d)*sqrt(f))],
[x^2/(e^2+4*e*f*x^3-4*d*f*x^6+4*f^2*x^6),x,4,-1/6*arctanh((e-2*(d-f)*x^3)*sqrt(f)/(e*sqrt(d)))/(e*sqrt(d)*sqrt(f))],
[x^m*(e*(1+m)+2*f*(-2+m)*x^3)/(e^2+4*e*f*x^3+4*f^2*x^6+4*d*f*x^(2+2*m)),x,2,1/2*arctan(2*x^(1+m)*sqrt(d)*sqrt(f)/(e+2*f*x^3))/(sqrt(d)*sqrt(f))],
[x^m*(e*(1+m)+2*f*(-2+m)*x^3)/(e^2+4*e*f*x^3+4*f^2*x^6-4*d*f*x^(2+2*m)),x,2,1/2*arctanh(2*x^(1+m)*sqrt(d)*sqrt(f)/(e+2*f*x^3))/(sqrt(d)*sqrt(f))],
[x^m*(e*(1+m)+2*f*(1+m-n)*x^n)/(e^2+4*d*f*x^(2+2*m)+4*e*f*x^n+4*f^2*x^(2*n)),x,2,1/2*arctan(2*x^(1+m)*sqrt(d)*sqrt(f)/(e+2*f*x^n))/(sqrt(d)*sqrt(f))],
[x^m*(e*(1+m)+2*f*(1+m-n)*x^n)/(e^2-4*d*f*x^(2+2*m)+4*e*f*x^n+4*f^2*x^(2*n)),x,2,1/2*arctanh(2*x^(1+m)*sqrt(d)*sqrt(f)/(e+2*f*x^n))/(sqrt(d)*sqrt(f))],

# Integrands of the form u / (c+d x^n+e Sqrt[a+b x^n])
[x^5/(a*c+b*c*x^2+d*sqrt(a+b*x^2)),x,4,-1/2*(2*a*c^2-d^2)*x^2/(b^2*c^3)-1/3*d*(a+b*x^2)^(3/2)/(b^3*c^2)+1/4*(a+b*x^2)^2/(b^3*c)+(a*c^2-d^2)^2*log(d+c*sqrt(a+b*x^2))/(b^3*c^5)+d*(2*a*c^2-d^2)*sqrt(a+b*x^2)/(b^3*c^4)],
[x^3/(a*c+b*c*x^2+d*sqrt(a+b*x^2)),x,4,1/2*x^2/(b*c)-(a*c^2-d^2)*log(d+c*sqrt(a+b*x^2))/(b^2*c^3)-d*sqrt(a+b*x^2)/(b^2*c^2)],
[x/(a*c+b*c*x^2+d*sqrt(a+b*x^2)),x,3,log(d+c*sqrt(a+b*x^2))/(b*c)],
[1/(x*(a*c+b*c*x^2+d*sqrt(a+b*x^2))),x,7,c*log(x)/(a*c^2-d^2)-c*log(d+c*sqrt(a+b*x^2))/(a*c^2-d^2)+d*arctanh(sqrt(a+b*x^2)/sqrt(a))/((a*c^2-d^2)*sqrt(a))],
[1/(x^3*(a*c+b*c*x^2+d*sqrt(a+b*x^2))),x,8,-1/2*b*d*(3*a*c^2-d^2)*arctanh(sqrt(a+b*x^2)/sqrt(a))/(a^(3/2)*(a*c^2-d^2)^2)-b*c^3*log(x)/(a*c^2-d^2)^2+b*c^3*log(d+c*sqrt(a+b*x^2))/(a*c^2-d^2)^2+1/2*(-a*c+d*sqrt(a+b*x^2))/(a*(a*c^2-d^2)*x^2)],
[x^2/(a*c+b*c*x^2+d*sqrt(a+b*x^2)),x,8,x/(b*c)-d*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/(b^(3/2)*c^2)-arctan(c*x*sqrt(b)/sqrt(a*c^2-d^2))*sqrt(a*c^2-d^2)/(b^(3/2)*c^2)+arctan(d*x*sqrt(b)/(sqrt(a*c^2-d^2)*sqrt(a+b*x^2)))*sqrt(a*c^2-d^2)/(b^(3/2)*c^2)],
[1/(a*c+b*c*x^2+d*sqrt(a+b*x^2)),x,4,arctan(c*x*sqrt(b)/sqrt(a*c^2-d^2))/(sqrt(b)*sqrt(a*c^2-d^2))-arctan(d*x*sqrt(b)/(sqrt(a*c^2-d^2)*sqrt(a+b*x^2)))/(sqrt(b)*sqrt(a*c^2-d^2))],
[1/(x^2*(a*c+b*c*x^2+d*sqrt(a+b*x^2))),x,7,-c/((a*c^2-d^2)*x)-c^2*arctan(c*x*sqrt(b)/sqrt(a*c^2-d^2))*sqrt(b)/(a*c^2-d^2)^(3/2)+c^2*arctan(d*x*sqrt(b)/(sqrt(a*c^2-d^2)*sqrt(a+b*x^2)))*sqrt(b)/(a*c^2-d^2)^(3/2)+d*sqrt(a+b*x^2)/(a*(a*c^2-d^2)*x)],
[x^8/(a*c+b*c*x^3+d*sqrt(a+b*x^3)),x,4,-1/3*(2*a*c^2-d^2)*x^3/(b^2*c^3)-2/9*d*(a+b*x^3)^(3/2)/(b^3*c^2)+1/6*(a+b*x^3)^2/(b^3*c)+2/3*(a*c^2-d^2)^2*log(d+c*sqrt(a+b*x^3))/(b^3*c^5)+2/3*d*(2*a*c^2-d^2)*sqrt(a+b*x^3)/(b^3*c^4)],
[x^5/(a*c+b*c*x^3+d*sqrt(a+b*x^3)),x,4,1/3*x^3/(b*c)-2/3*(a*c^2-d^2)*log(d+c*sqrt(a+b*x^3))/(b^2*c^3)-2/3*d*sqrt(a+b*x^3)/(b^2*c^2)],
[x^2/(a*c+b*c*x^3+d*sqrt(a+b*x^3)),x,3,2/3*log(d+c*sqrt(a+b*x^3))/(b*c)],
[1/(x*(a*c+b*c*x^3+d*sqrt(a+b*x^3))),x,7,c*log(x)/(a*c^2-d^2)-2/3*c*log(d+c*sqrt(a+b*x^3))/(a*c^2-d^2)+2/3*d*arctanh(sqrt(a+b*x^3)/sqrt(a))/((a*c^2-d^2)*sqrt(a))],
[1/(x^4*(a*c+b*c*x^3+d*sqrt(a+b*x^3))),x,8,-1/3*b*d*(3*a*c^2-d^2)*arctanh(sqrt(a+b*x^3)/sqrt(a))/(a^(3/2)*(a*c^2-d^2)^2)-b*c^3*log(x)/(a*c^2-d^2)^2+2/3*b*c^3*log(d+c*sqrt(a+b*x^3))/(a*c^2-d^2)^2+1/3*(-a*c+d*sqrt(a+b*x^3))/(a*(a*c^2-d^2)*x^3)],
[x^3/(a*c+b*c*x^3+d*sqrt(a+b*x^3)),x,10,x/(b*c)-1/3*(a*c^2-d^2)^(1/3)*log((a*c^2-d^2)^(1/3)+b^(1/3)*c^(2/3)*x)/(b^(4/3)*c^(5/3))+1/6*(a*c^2-d^2)^(1/3)*log((a*c^2-d^2)^(2/3)-b^(1/3)*c^(2/3)*(a*c^2-d^2)^(1/3)*x+b^(2/3)*c^(4/3)*x^2)/(b^(4/3)*c^(5/3))+(a*c^2-d^2)^(1/3)*arctan((1-2*b^(1/3)*c^(2/3)*x/(a*c^2-d^2)^(1/3))/sqrt(3))/(b^(4/3)*c^(5/3)*sqrt(3))-1/4*d*x^4*AppellF1(4/3,1/2,1,7/3,-b*x^3/a,-b*c^2*x^3/(a*c^2-d^2))*sqrt(1+b*x^3/a)/((a*c^2-d^2)*sqrt(a+b*x^3))],
[x/(a*c+b*c*x^3+d*sqrt(a+b*x^3)),x,9,-1/3*log((a*c^2-d^2)^(1/3)+b^(1/3)*c^(2/3)*x)/(b^(2/3)*c^(1/3)*(a*c^2-d^2)^(1/3))+1/6*log((a*c^2-d^2)^(2/3)-b^(1/3)*c^(2/3)*(a*c^2-d^2)^(1/3)*x+b^(2/3)*c^(4/3)*x^2)/(b^(2/3)*c^(1/3)*(a*c^2-d^2)^(1/3))-arctan((1-2*b^(1/3)*c^(2/3)*x/(a*c^2-d^2)^(1/3))/sqrt(3))/(b^(2/3)*c^(1/3)*(a*c^2-d^2)^(1/3)*sqrt(3))-1/2*d*x^2*AppellF1(2/3,1/2,1,5/3,-b*x^3/a,-b*c^2*x^3/(a*c^2-d^2))*sqrt(1+b*x^3/a)/((a*c^2-d^2)*sqrt(a+b*x^3))],
[1/(a*c+b*c*x^3+d*sqrt(a+b*x^3)),x,9,1/3*c^(1/3)*log((a*c^2-d^2)^(1/3)+b^(1/3)*c^(2/3)*x)/(b^(1/3)*(a*c^2-d^2)^(2/3))-1/6*c^(1/3)*log((a*c^2-d^2)^(2/3)-b^(1/3)*c^(2/3)*(a*c^2-d^2)^(1/3)*x+b^(2/3)*c^(4/3)*x^2)/(b^(1/3)*(a*c^2-d^2)^(2/3))-c^(1/3)*arctan((1-2*b^(1/3)*c^(2/3)*x/(a*c^2-d^2)^(1/3))/sqrt(3))/(b^(1/3)*(a*c^2-d^2)^(2/3)*sqrt(3))-d*x*AppellF1(1/3,1/2,1,4/3,-b*x^3/a,-b*c^2*x^3/(a*c^2-d^2))*sqrt(1+b*x^3/a)/((a*c^2-d^2)*sqrt(a+b*x^3))],
[1/(x^2*(a*c+b*c*x^3+d*sqrt(a+b*x^3))),x,10,-c/((a*c^2-d^2)*x)+1/3*b^(1/3)*c^(5/3)*log((a*c^2-d^2)^(1/3)+b^(1/3)*c^(2/3)*x)/(a*c^2-d^2)^(4/3)-1/6*b^(1/3)*c^(5/3)*log((a*c^2-d^2)^(2/3)-b^(1/3)*c^(2/3)*(a*c^2-d^2)^(1/3)*x+b^(2/3)*c^(4/3)*x^2)/(a*c^2-d^2)^(4/3)+b^(1/3)*c^(5/3)*arctan((1-2*b^(1/3)*c^(2/3)*x/(a*c^2-d^2)^(1/3))/sqrt(3))/((a*c^2-d^2)^(4/3)*sqrt(3))+d*AppellF1(-1/3,1/2,1,2/3,-b*x^3/a,-b*c^2*x^3/(a*c^2-d^2))*sqrt(1+b*x^3/a)/((a*c^2-d^2)*x*sqrt(a+b*x^3))],
[1/(x^3*(a*c+b*c*x^3+d*sqrt(a+b*x^3))),x,10,-1/2*c/((a*c^2-d^2)*x^2)-1/3*b^(2/3)*c^(7/3)*log((a*c^2-d^2)^(1/3)+b^(1/3)*c^(2/3)*x)/(a*c^2-d^2)^(5/3)+1/6*b^(2/3)*c^(7/3)*log((a*c^2-d^2)^(2/3)-b^(1/3)*c^(2/3)*(a*c^2-d^2)^(1/3)*x+b^(2/3)*c^(4/3)*x^2)/(a*c^2-d^2)^(5/3)+b^(2/3)*c^(7/3)*arctan((1-2*b^(1/3)*c^(2/3)*x/(a*c^2-d^2)^(1/3))/sqrt(3))/((a*c^2-d^2)^(5/3)*sqrt(3))+1/2*d*AppellF1(-2/3,1/2,1,1/3,-b*x^3/a,-b*c^2*x^3/(a*c^2-d^2))*sqrt(1+b*x^3/a)/((a*c^2-d^2)*x^2*sqrt(a+b*x^3))],
[1/(a*c+b*c*x^n+d*sqrt(a+b*x^n)),x,4,c*x*hypergeom([1,1/n],[1+1/n],-b*c^2*x^n/(a*c^2-d^2))/(a*c^2-d^2)-d*x*AppellF1(1/n,1/2,1,1+1/n,-b*x^n/a,-b*c^2*x^n/(a*c^2-d^2))*sqrt(1+b*x^n/a)/((a*c^2-d^2)*sqrt(a+b*x^n))],
[x^m/(a*c+b*c*x^n+d*sqrt(a+b*x^n)),x,4,c*x^(1+m)*hypergeom([1,(1+m)/n],[(1+m+n)/n],-b*c^2*x^n/(a*c^2-d^2))/((a*c^2-d^2)*(1+m))-d*x^(1+m)*AppellF1((1+m)/n,1/2,1,(1+m+n)/n,-b*x^n/a,-b*c^2*x^n/(a*c^2-d^2))*sqrt(1+b*x^n/a)/((a*c^2-d^2)*(1+m)*sqrt(a+b*x^n))],
[x^(-1+n)/(a*c+b*c*x^n+d*sqrt(a+b*x^n)),x,3,2*log(d+c*sqrt(a+b*x^n))/(b*c*n)],

# Integrands of the form u (a x^m+b x^n+...)^p
[1/(4*x^(3/2)+sqrt(x)),x,3,arctan(2*sqrt(x))],
[1/(-x^(5/2)+sqrt(x)),x,5,arctan(sqrt(x))+arctanh(sqrt(x))],
[1/(-x^(1/4)+sqrt(x)),x,4,4*x^(1/4)+4*log(1-x^(1/4))+2*sqrt(x)],
[1/(x^(1/3)+sqrt(x)),x,4,6*x^(1/6)-3*x^(1/3)-6*log(1+x^(1/6))+2*sqrt(x)],
[1/(x^(1/4)+sqrt(x)),x,4,-4*x^(1/4)+4*log(1+x^(1/4))+2*sqrt(x)],
[1/(-x^(1/3)+x^(2/3)),x,4,3*x^(1/3)+3*log(1-x^(1/3))],
[1/(1/x^(1/4)+sqrt(x)),x,9,4/3*log(1+x^(1/4))-2/3*log(1-x^(1/4)+sqrt(x))+4*arctan((1-2*x^(1/4))/sqrt(3))/sqrt(3)+2*sqrt(x)],
[1/(x^(1/4)+x^(1/3)),x,4,-12*x^(1/12)+6*x^(1/6)-4*x^(1/4)+3*x^(1/3)-12/5*x^(5/12)-12/7*x^(7/12)+3/2*x^(2/3)+12*log(1+x^(1/12))+2*sqrt(x)],
[1/(1/x^(1/3)+1/x^(1/4)),x,4,12*x^(1/12)-6*x^(1/6)+4*x^(1/4)-3*x^(1/3)+12/5*x^(5/12)+12/7*x^(7/12)-3/2*x^(2/3)+4/3*x^(3/4)-6/5*x^(5/6)+12/11*x^(11/12)-x+12/13*x^(13/12)-6/7*x^(7/6)+4/5*x^(5/4)-12*log(1+x^(1/12))-2*sqrt(x)],
[1/((-1)/x^(1/3)+sqrt(x)),x,9,6/5*log(1-x^(1/6))-3/10*log(2+x^(1/6)+2*x^(1/3)+x^(1/6)*sqrt(5))*(1-sqrt(5))-3/10*log(2+x^(1/6)+2*x^(1/3)-x^(1/6)*sqrt(5))*(1+sqrt(5))+2*sqrt(x)+3/5*arctan((1+4*x^(1/6)-sqrt(5))/sqrt(2*(5+sqrt(5))))*sqrt(2*(5-sqrt(5)))-3/5*arctan(1/2*(1+4*x^(1/6)+sqrt(5))*sqrt(1/10*(5+sqrt(5))))*sqrt(2*(5+sqrt(5)))],
[sqrt(x)/(x+x^2),x,3,2*arctan(sqrt(x))],
[x/(x+4*sqrt(x)),x,4,x+32*log(4+sqrt(x))-8*sqrt(x)],
[sqrt(x)/(x^(1/3)+x),x,13,3*arctan(1-x^(1/6)*sqrt(2))/sqrt(2)-3*arctan(1+x^(1/6)*sqrt(2))/sqrt(2)-3/2*log(1+x^(1/3)-x^(1/6)*sqrt(2))/sqrt(2)+3/2*log(1+x^(1/3)+x^(1/6)*sqrt(2))/sqrt(2)+2*sqrt(x)],
[x^(1/3)/(x^(1/4)+sqrt(x)),x,10,-12*x^(1/12)+3*x^(1/3)-12/7*x^(7/12)+6/5*x^(5/6)+6*log(1+x^(1/12))-2*log(1+x^(1/4))-4*arctan((1-2*x^(1/12))/sqrt(3))*sqrt(3)],
[sqrt(x)/(x^(1/4)+x^(1/3)),x,4,-12*x^(1/12)+6*x^(1/6)-4*x^(1/4)+3*x^(1/3)-12/5*x^(5/12)-12/7*x^(7/12)+3/2*x^(2/3)-4/3*x^(3/4)+6/5*x^(5/6)-12/11*x^(11/12)+x-12/13*x^(13/12)+6/7*x^(7/6)+12*log(1+x^(1/12))+2*sqrt(x)],
[sqrt(x)/((-1)/x^(1/3)+sqrt(x)),x,10,6*x^(1/6)+x+6/5*log(1-x^(1/6))-3/10*log(2+x^(1/6)+2*x^(1/3)-x^(1/6)*sqrt(5))*(1-sqrt(5))-3/10*log(2+x^(1/6)+2*x^(1/3)+x^(1/6)*sqrt(5))*(1+sqrt(5))-3/5*arctan(1/2*(1+4*x^(1/6)+sqrt(5))*sqrt(1/10*(5+sqrt(5))))*sqrt(2*(5-sqrt(5)))-3/5*arctan((1+4*x^(1/6)-sqrt(5))/sqrt(2*(5+sqrt(5))))*sqrt(2*(5+sqrt(5)))],

# Integrands of the form u (a + b x^n)^p when n<0

# Integrands of the form u (a + b/x)^p
[x^m*sqrt(b-a/x)/sqrt(a-b*x),x,3,2*x^(1+m)*sqrt(b-a/x)/((1+2*m)*sqrt(a-b*x))],
[x^2*sqrt(b-a/x)/sqrt(a-b*x),x,3,2/5*x^3*sqrt(b-a/x)/sqrt(a-b*x)],
[x*sqrt(b-a/x)/sqrt(a-b*x),x,3,2/3*x^2*sqrt(b-a/x)/sqrt(a-b*x)],
[sqrt(b-a/x)/sqrt(a-b*x),x,3,2*x*sqrt(b-a/x)/sqrt(a-b*x)],
[sqrt(b-a/x)/(x*sqrt(a-b*x)),x,3,-2*sqrt(b-a/x)/sqrt(a-b*x)],
[sqrt(b-a/x)/(x^2*sqrt(a-b*x)),x,3,-2/3*sqrt(b-a/x)/(x*sqrt(a-b*x))],
[(a+b/x)^m*(c+d*x)^n,x,4,(a+b/x)^m*x*(c+d*x)^n*AppellF1(1-m,-m,-n,2-m,-a*x/b,-d*x/c)/((1-m)*(1+a*x/b)^m*(1+d*x/c)^n)],
[(a+b/x)^m*(c+d*x)^2,x,5,1/6*d*(6*a*c-b*d*(2-m))*(a+b/x)^(1+m)*x^2/a^2+1/3*d^2*(a+b/x)^(1+m)*x^3/a-1/6*b*(6*a^2*c^2-6*a*b*c*d*(1-m)+b^2*d^2*(2-3*m+m^2))*(a+b/x)^(1+m)*hypergeom([2,1+m],[2+m],1+b/(a*x))/(a^4*(1+m))],
[(a+b/x)^m*(c+d*x),x,4,1/2*d*(a+b/x)^(1+m)*x^2/a-1/2*b*(2*a*c-b*d*(1-m))*(a+b/x)^(1+m)*hypergeom([2,1+m],[2+m],1+b/(a*x))/(a^3*(1+m))],
[(a+b/x)^m,x,2,-b*(a+b/x)^(1+m)*hypergeom([2,1+m],[2+m],1+b/(a*x))/(a^2*(1+m))],
[(a+b/x)^m/(c+d*x),x,5,-c*(a+b/x)^(1+m)*hypergeom([1,1+m],[2+m],c*(a+b/x)/(a*c-b*d))/(d*(a*c-b*d)*(1+m))+(a+b/x)^(1+m)*hypergeom([1,1+m],[2+m],1+b/(a*x))/(a*d*(1+m))],
[(a+b/x)^m/(c+d*x)^2,x,3,-b*(a+b/x)^(1+m)*hypergeom([2,1+m],[2+m],c*(a+b/x)/(a*c-b*d))/((a*c-b*d)^2*(1+m))],
[(a+b/x)^m/(c+d*x)^3,x,4,-1/2*d*(a+b/x)^(1+m)/(c*(a*c-b*d)*(d+c/x)^2)-1/2*b*(2*a*c-b*d*(1+m))*(a+b/x)^(1+m)*hypergeom([2,1+m],[2+m],c*(a+b/x)/(a*c-b*d))/(c*(a*c-b*d)^3*(1+m))],
[(a+b/x)^m/(c+d*x)^4,x,5,1/3*d^2*(a+b/x)^(1+m)/(c^2*(a*c-b*d)*(d+c/x)^3)-1/6*d*(6*a*c-b*d*(4+m))*(a+b/x)^(1+m)/(c^2*(a*c-b*d)^2*(d+c/x)^2)-1/6*b*(6*a^2*c^2-6*a*b*c*d*(1+m)+b^2*d^2*(2+3*m+m^2))*(a+b/x)^(1+m)*hypergeom([2,1+m],[2+m],c*(a+b/x)/(a*c-b*d))/(c^2*(a*c-b*d)^4*(1+m))],

# Integrands of the form u (a + b/x^2)^p
[x^m*sqrt(b-a/x^2)/sqrt(a-b*x^2),x,3,x^(1+m)*sqrt(b-a/x^2)/(m*sqrt(a-b*x^2))],
[x^2*sqrt(b-a/x^2)/sqrt(a-b*x^2),x,3,1/2*x^3*sqrt(b-a/x^2)/sqrt(a-b*x^2)],
[x*sqrt(b-a/x^2)/sqrt(a-b*x^2),x,3,x^2*sqrt(b-a/x^2)/sqrt(a-b*x^2)],
[sqrt(b-a/x^2)/sqrt(a-b*x^2),x,3,x*log(x)*sqrt(b-a/x^2)/sqrt(a-b*x^2)],
[sqrt(b-a/x^2)/(x*sqrt(a-b*x^2)),x,3,-sqrt(b-a/x^2)/sqrt(a-b*x^2)],
[sqrt(b-a/x^2)/(x^2*sqrt(a-b*x^2)),x,3,-1/2*sqrt(b-a/x^2)/(x*sqrt(a-b*x^2))],
[(c+d*x)^(3/2)/sqrt(a+b/x^2),x,8,2/5*(c+d*x)^(3/2)*(b+a*x^2)/(a*x*sqrt(a+b/x^2))+2/5*c*(b+a*x^2)*sqrt(c+d*x)/(a*x*sqrt(a+b/x^2))+2/5*(a*c^2-3*b*d^2)*EllipticE(sqrt(1-x*sqrt(-a)/sqrt(b))/sqrt(2),sqrt(-2*d*sqrt(-a)*sqrt(b)/(a*c-d*sqrt(-a)*sqrt(b))))*sqrt(b)*sqrt(c+d*x)*sqrt(1+a*x^2/b)/((-a)^(3/2)*d*x*sqrt(a+b/x^2)*sqrt(a*(c+d*x)/(a*c-d*sqrt(-a)*sqrt(b))))-2/5*c*(a*c^2+b*d^2)*EllipticF(sqrt(1-x*sqrt(-a)/sqrt(b))/sqrt(2),sqrt(-2*d*sqrt(-a)*sqrt(b)/(a*c-d*sqrt(-a)*sqrt(b))))*sqrt(b)*sqrt(1+a*x^2/b)*sqrt(a*(c+d*x)/(a*c-d*sqrt(-a)*sqrt(b)))/((-a)^(3/2)*d*x*sqrt(a+b/x^2)*sqrt(c+d*x))],

# Integrands of the form y'[x] F[y[x]]
[(-1+x^3)/(-4*x+x^4)^(2/3),x,1,3/4*(-4*x+x^4)^(1/3)],
[(2-x^2)*(6*x-x^3)^(1/4),x,1,4/15*(6*x-x^3)^(5/4)],
[(1+x^4)*sqrt(5*x+x^5),x,1,2/15*(5*x+x^5)^(3/2)],
[(2+5*x^4)*sqrt(2*x+x^5),x,1,2/3*(2*x+x^5)^(3/2)],
[(x+3*x^2)/sqrt(x^2+2*x^3),x,1,sqrt(x^2+2*x^3)],
[(2+(1-5*x)^(1/3))/(3+(1-5*x)^(1/3)),x,4,-9/5*(1-5*x)^(1/3)+3/10*(1-5*x)^(2/3)+x+27/5*log(3+(1-5*x)^(1/3))],
[(1+sqrt(x))/(-1+sqrt(x)),x,3,x+4*log(1-sqrt(x))+4*sqrt(x)],
[(1-sqrt(2+3*x))/(1+sqrt(2+3*x)),x,4,-x-4/3*log(1+sqrt(2+3*x))+4/3*sqrt(2+3*x)],
[(-1+sqrt(a+b*x))/(1+sqrt(a+b*x)),x,4,x+4*log(1+sqrt(a+b*x))/b-4*sqrt(a+b*x)/b],
[(a+b*n*x^(-1+n))/(a*x+b*x^n),x,5,log(a*x+b*x^n),n*log(x)+log(b+a*x^(1-n))],
[(a+b*n*x^(-1+n))/(x^n*(b+a*x^(1-n))),x,4,n*log(x)+log(b+a*x^(1-n))],
[x*(a+b*x+c*x^2)^m*(d+e*x+f*x^2+g*x^3)^n*(2*a*d+(3*b*d+3*a*e+b*d*m+a*e*n)*x+(4*c*d+4*b*e+4*a*f+2*c*d*m+b*e*m+b*e*n+2*a*f*n)*x^2+(5*c*e+5*b*f+5*a*g+2*c*e*m+b*f*m+c*e*n+2*b*f*n+3*a*g*n)*x^3+(6*c*f+6*b*g+2*c*f*m+b*g*m+2*c*f*n+3*b*g*n)*x^4+c*g*(7+2*m+3*n)*x^5),x,1,x^2*(a+b*x+c*x^2)^(1+m)*(d+e*x+f*x^2+g*x^3)^(1+n)],
[(a+b*x+c*x^2)^m*(d+e*x+f*x^2+g*x^3)^n*(a*d+(2*b*d+2*a*e+b*d*m+a*e*n)*x+(3*c*d+3*b*e+3*a*f+2*c*d*m+b*e*m+b*e*n+2*a*f*n)*x^2+(4*c*e+4*b*f+4*a*g+2*c*e*m+b*f*m+c*e*n+2*b*f*n+3*a*g*n)*x^3+(5*c*f+5*b*g+2*c*f*m+b*g*m+2*c*f*n+3*b*g*n)*x^4+c*g*(6+2*m+3*n)*x^5),x,1,x*(a+b*x+c*x^2)^(1+m)*(d+e*x+f*x^2+g*x^3)^(1+n)],
[(a+b*x+c*x^2)^m*(d+e*x+f*x^2+g*x^3)^n*(b*d+a*e+b*d*m+a*e*n+(2*c*d+2*b*e+2*a*f+2*c*d*m+b*e*m+b*e*n+2*a*f*n)*x+(3*c*e+3*b*f+3*a*g+2*c*e*m+b*f*m+c*e*n+2*b*f*n+3*a*g*n)*x^2+(4*c*f+4*b*g+2*c*f*m+b*g*m+2*c*f*n+3*b*g*n)*x^3+c*g*(5+2*m+3*n)*x^4),x,1,(a+b*x+c*x^2)^(1+m)*(d+e*x+f*x^2+g*x^3)^(1+n)],
[(a+b*x+c*x^2)^m*(d+e*x+f*x^2+g*x^3)^n*(-a*d+(b*d*m+a*e*n)*x+(c*d+b*e+a*f+2*c*d*m+b*e*m+b*e*n+2*a*f*n)*x^2+(2*c*e+2*b*f+2*a*g+2*c*e*m+b*f*m+c*e*n+2*b*f*n+3*a*g*n)*x^3+(3*c*f+3*b*g+2*c*f*m+b*g*m+2*c*f*n+3*b*g*n)*x^4+c*g*(4+2*m+3*n)*x^5)/x^2,x,-2,(a+b*x+c*x^2)^(1+m)*(d+e*x+f*x^2+g*x^3)^(1+n)/x],
[(a+b*x+c*x^2)^m*(d+e*x+f*x^2+g*x^3)^n*(-2*a*d+(-b*d-a*e+b*d*m+a*e*n)*x+(2*c*d*m+b*e*m+b*e*n+2*a*f*n)*x^2+(c*e+b*f+a*g+2*c*e*m+b*f*m+c*e*n+2*b*f*n+3*a*g*n)*x^3+(2*c*f+2*b*g+2*c*f*m+b*g*m+2*c*f*n+3*b*g*n)*x^4+c*g*(3+2*m+3*n)*x^5)/x^3,x,-2,(a+b*x+c*x^2)^(1+m)*(d+e*x+f*x^2+g*x^3)^(1+n)/x^2],

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

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

# p>0
[x^3*(a+b*sqrt(c+d*x))^2,x,4,-a^2*c^3*x/d^3-4/3*a*b*c^3*(c+d*x)^(3/2)/d^4+1/2*c^2*(3*a^2-b^2*c)*(c+d*x)^2/d^4+12/5*a*b*c^2*(c+d*x)^(5/2)/d^4-c*(a^2-b^2*c)*(c+d*x)^3/d^4-12/7*a*b*c*(c+d*x)^(7/2)/d^4+1/4*(a^2-3*b^2*c)*(c+d*x)^4/d^4+4/9*a*b*(c+d*x)^(9/2)/d^4+1/5*b^2*(c+d*x)^5/d^4],
[x^2*(a+b*sqrt(c+d*x))^2,x,4,a^2*c^2*x/d^2+4/3*a*b*c^2*(c+d*x)^(3/2)/d^3-1/2*c*(2*a^2-b^2*c)*(c+d*x)^2/d^3-8/5*a*b*c*(c+d*x)^(5/2)/d^3+1/3*(a^2-2*b^2*c)*(c+d*x)^3/d^3+4/7*a*b*(c+d*x)^(7/2)/d^3+1/4*b^2*(c+d*x)^4/d^3],
[x*(a+b*sqrt(c+d*x))^2,x,4,-a^2*c*x/d-4/3*a*b*c*(c+d*x)^(3/2)/d^2+1/2*(a^2-b^2*c)*(c+d*x)^2/d^2+4/5*a*b*(c+d*x)^(5/2)/d^2+1/3*b^2*(c+d*x)^3/d^2],
[(a+b*sqrt(c+d*x))^2,x,4,a^2*x+4/3*a*b*(c+d*x)^(3/2)/d+1/2*b^2*(c+d*x)^2/d],
[(a+b*sqrt(c+d*x))^2/x,x,7,b^2*d*x+(a^2+b^2*c)*log(x)-4*a*b*arctanh(sqrt(c+d*x)/sqrt(c))*sqrt(c)+4*a*b*sqrt(c+d*x)],
[(a+b*sqrt(c+d*x))^2/x^2,x,6,b^2*d*log(x)-2*a*b*d*arctanh(sqrt(c+d*x)/sqrt(c))/sqrt(c)-(a+b*sqrt(c+d*x))^2/x],
[(a+b*sqrt(c+d*x))^2/x^3,x,6,1/2*a*b*d^2*arctanh(sqrt(c+d*x)/sqrt(c))/c^(3/2)-1/2*b*d*(b*c+a*sqrt(c+d*x))/(c*x)-1/2*(a+b*sqrt(c+d*x))^2/x^2],
[x^3*sqrt(a+b*sqrt(c+d*x)),x,4,-4/3*a*(a^2-b^2*c)^3*(a+b*sqrt(c+d*x))^(3/2)/(b^8*d^4)+4/5*(a^2-b^2*c)^2*(7*a^2-b^2*c)*(a+b*sqrt(c+d*x))^(5/2)/(b^8*d^4)-12/7*a*(7*a^2-3*b^2*c)*(a^2-b^2*c)*(a+b*sqrt(c+d*x))^(7/2)/(b^8*d^4)+4/9*(35*a^4-30*a^2*b^2*c+3*b^4*c^2)*(a+b*sqrt(c+d*x))^(9/2)/(b^8*d^4)-20/11*a*(7*a^2-3*b^2*c)*(a+b*sqrt(c+d*x))^(11/2)/(b^8*d^4)+12/13*(7*a^2-b^2*c)*(a+b*sqrt(c+d*x))^(13/2)/(b^8*d^4)-28/15*a*(a+b*sqrt(c+d*x))^(15/2)/(b^8*d^4)+4/17*(a+b*sqrt(c+d*x))^(17/2)/(b^8*d^4)],
[x^2*sqrt(a+b*sqrt(c+d*x)),x,4,-4/3*a*(a^2-b^2*c)^2*(a+b*sqrt(c+d*x))^(3/2)/(b^6*d^3)+4/5*(5*a^4-6*a^2*b^2*c+b^4*c^2)*(a+b*sqrt(c+d*x))^(5/2)/(b^6*d^3)-8/7*a*(5*a^2-3*b^2*c)*(a+b*sqrt(c+d*x))^(7/2)/(b^6*d^3)+8/9*(5*a^2-b^2*c)*(a+b*sqrt(c+d*x))^(9/2)/(b^6*d^3)-20/11*a*(a+b*sqrt(c+d*x))^(11/2)/(b^6*d^3)+4/13*(a+b*sqrt(c+d*x))^(13/2)/(b^6*d^3)],
[x*sqrt(a+b*sqrt(c+d*x)),x,4,-4/3*a*(a^2-b^2*c)*(a+b*sqrt(c+d*x))^(3/2)/(b^4*d^2)+4/5*(3*a^2-b^2*c)*(a+b*sqrt(c+d*x))^(5/2)/(b^4*d^2)-12/7*a*(a+b*sqrt(c+d*x))^(7/2)/(b^4*d^2)+4/9*(a+b*sqrt(c+d*x))^(9/2)/(b^4*d^2)],
[sqrt(a+b*sqrt(c+d*x)),x,4,-4/3*a*(a+b*sqrt(c+d*x))^(3/2)/(b^2*d)+4/5*(a+b*sqrt(c+d*x))^(5/2)/(b^2*d)],
[sqrt(a+b*sqrt(c+d*x))/x,x,7,-2*arctanh(sqrt(a+b*sqrt(c+d*x))/sqrt(a-b*sqrt(c)))*sqrt(a-b*sqrt(c))-2*arctanh(sqrt(a+b*sqrt(c+d*x))/sqrt(a+b*sqrt(c)))*sqrt(a+b*sqrt(c))+4*sqrt(a+b*sqrt(c+d*x))],
[sqrt(a+b*sqrt(c+d*x))/x^2,x,8,1/2*b*d*arctanh(sqrt(a+b*sqrt(c+d*x))/sqrt(a-b*sqrt(c)))/(sqrt(c)*sqrt(a-b*sqrt(c)))-1/2*b*d*arctanh(sqrt(a+b*sqrt(c+d*x))/sqrt(a+b*sqrt(c)))/(sqrt(c)*sqrt(a+b*sqrt(c)))-sqrt(a+b*sqrt(c+d*x))/x],
[sqrt(a+b*sqrt(c+d*x))/x^3,x,9,-1/16*b*d^2*arctanh(sqrt(a+b*sqrt(c+d*x))/sqrt(a-b*sqrt(c)))*(2*a-3*b*sqrt(c))/(c^(3/2)*(a-b*sqrt(c))^(3/2))+1/16*b*d^2*arctanh(sqrt(a+b*sqrt(c+d*x))/sqrt(a+b*sqrt(c)))*(2*a+3*b*sqrt(c))/(c^(3/2)*(a+b*sqrt(c))^(3/2))-1/2*sqrt(a+b*sqrt(c+d*x))/x^2+1/8*b*d*(b*c-a*sqrt(c+d*x))*sqrt(a+b*sqrt(c+d*x))/(c*(a^2-b^2*c)*x)],

# p<0
[x^3/(a+b*sqrt(c+d*x)),x,4,-a*(a^4-3*a^2*b^2*c+3*b^4*c^2)*x/(b^6*d^3)+2/3*(a^4-3*a^2*b^2*c+3*b^4*c^2)*(c+d*x)^(3/2)/(b^5*d^4)-1/2*a*(a^2-3*b^2*c)*(c+d*x)^2/(b^4*d^4)+2/5*(a^2-3*b^2*c)*(c+d*x)^(5/2)/(b^3*d^4)-1/3*a*(c+d*x)^3/(b^2*d^4)+2/7*(c+d*x)^(7/2)/(b*d^4)-2*a*(a^2-b^2*c)^3*log(a+b*sqrt(c+d*x))/(b^8*d^4)+2*(a^2-b^2*c)^3*sqrt(c+d*x)/(b^7*d^4)],
[x^2/(a+b*sqrt(c+d*x)),x,4,-a*(a^2-2*b^2*c)*x/(b^4*d^2)+2/3*(a^2-2*b^2*c)*(c+d*x)^(3/2)/(b^3*d^3)-1/2*a*(c+d*x)^2/(b^2*d^3)+2/5*(c+d*x)^(5/2)/(b*d^3)-2*a*(a^2-b^2*c)^2*log(a+b*sqrt(c+d*x))/(b^6*d^3)+2*(a^2-b^2*c)^2*sqrt(c+d*x)/(b^5*d^3)],
[x/(a+b*sqrt(c+d*x)),x,4,-a*x/(b^2*d)+2/3*(c+d*x)^(3/2)/(b*d^2)-2*a*(a^2-b^2*c)*log(a+b*sqrt(c+d*x))/(b^4*d^2)+2*(a^2-b^2*c)*sqrt(c+d*x)/(b^3*d^2)],
[1/(a+b*sqrt(c+d*x)),x,4,-2*a*log(a+b*sqrt(c+d*x))/(b^2*d)+2*sqrt(c+d*x)/(b*d)],
[1/(x*(a+b*sqrt(c+d*x))),x,7,a*log(x)/(a^2-b^2*c)-2*a*log(a+b*sqrt(c+d*x))/(a^2-b^2*c)+2*b*arctanh(sqrt(c+d*x)/sqrt(c))*sqrt(c)/(a^2-b^2*c)],
[1/(x^2*(a+b*sqrt(c+d*x))),x,8,a*b^2*d*log(x)/(a^2-b^2*c)^2-2*a*b^2*d*log(a+b*sqrt(c+d*x))/(a^2-b^2*c)^2+b*(a^2+b^2*c)*d*arctanh(sqrt(c+d*x)/sqrt(c))/((a^2-b^2*c)^2*sqrt(c))+(-a+b*sqrt(c+d*x))/((a^2-b^2*c)*x)],
[1/(x^3*(a+b*sqrt(c+d*x))),x,9,-1/4*b*(a^4-6*a^2*b^2*c-3*b^4*c^2)*d^2*arctanh(sqrt(c+d*x)/sqrt(c))/(c^(3/2)*(a^2-b^2*c)^3)+a*b^4*d^2*log(x)/(a^2-b^2*c)^3-2*a*b^4*d^2*log(a+b*sqrt(c+d*x))/(a^2-b^2*c)^3+1/2*(-a+b*sqrt(c+d*x))/((a^2-b^2*c)*x^2)-1/4*b*d*(4*a*b*c-(a^2+3*b^2*c)*sqrt(c+d*x))/(c*(a^2-b^2*c)^2*x)],
[x^3/(a+b*sqrt(c+d*x))^2,x,4,(5*a^4-9*a^2*b^2*c+3*b^4*c^2)*x/(b^6*d^3)-4/3*a*(2*a^2-3*b^2*c)*(c+d*x)^(3/2)/(b^5*d^4)+3/2*(a^2-b^2*c)*(c+d*x)^2/(b^4*d^4)-4/5*a*(c+d*x)^(5/2)/(b^3*d^4)+1/3*(c+d*x)^3/(b^2*d^4)+2*(a^2-b^2*c)^2*(7*a^2-b^2*c)*log(a+b*sqrt(c+d*x))/(b^8*d^4)-12*a*(a^2-b^2*c)^2*sqrt(c+d*x)/(b^7*d^4)+2*a*(a^2-b^2*c)^3/(b^8*d^4*(a+b*sqrt(c+d*x)))],
[x^2/(a+b*sqrt(c+d*x))^2,x,4,(3*a^2-2*b^2*c)*x/(b^4*d^2)-4/3*a*(c+d*x)^(3/2)/(b^3*d^3)+1/2*(c+d*x)^2/(b^2*d^3)+2*(5*a^4-6*a^2*b^2*c+b^4*c^2)*log(a+b*sqrt(c+d*x))/(b^6*d^3)-8*a*(a^2-b^2*c)*sqrt(c+d*x)/(b^5*d^3)+2*a*(a^2-b^2*c)^2/(b^6*d^3*(a+b*sqrt(c+d*x)))],
[x/(a+b*sqrt(c+d*x))^2,x,4,x/(b^2*d)+2*(3*a^2-b^2*c)*log(a+b*sqrt(c+d*x))/(b^4*d^2)-4*a*sqrt(c+d*x)/(b^3*d^2)+2*a*(a^2-b^2*c)/(b^4*d^2*(a+b*sqrt(c+d*x)))],
[1/(a+b*sqrt(c+d*x))^2,x,4,2*log(a+b*sqrt(c+d*x))/(b^2*d)+2*a/(b^2*d*(a+b*sqrt(c+d*x)))],
[1/(x*(a+b*sqrt(c+d*x))^2),x,7,(a^2+b^2*c)*log(x)/(a^2-b^2*c)^2-2*(a^2+b^2*c)*log(a+b*sqrt(c+d*x))/(a^2-b^2*c)^2+4*a*b*arctanh(sqrt(c+d*x)/sqrt(c))*sqrt(c)/(a^2-b^2*c)^2+2*a/((a^2-b^2*c)*(a+b*sqrt(c+d*x)))],
[1/(x^2*(a+b*sqrt(c+d*x))^2),x,8,b^2*(3*a^2+b^2*c)*d*log(x)/(a^2-b^2*c)^3-2*b^2*(3*a^2+b^2*c)*d*log(a+b*sqrt(c+d*x))/(a^2-b^2*c)^3+2*a*b*(a^2+3*b^2*c)*d*arctanh(sqrt(c+d*x)/sqrt(c))/((a^2-b^2*c)^3*sqrt(c))+4*a*b^2*d/((a^2-b^2*c)^2*(a+b*sqrt(c+d*x)))+(-a+b*sqrt(c+d*x))/((a^2-b^2*c)*x*(a+b*sqrt(c+d*x)))],
[1/(x^3*(a+b*sqrt(c+d*x))^2),x,9,-1/2*a*b*(a^4-10*a^2*b^2*c-15*b^4*c^2)*d^2*arctanh(sqrt(c+d*x)/sqrt(c))/(c^(3/2)*(a^2-b^2*c)^4)+b^4*(5*a^2+b^2*c)*d^2*log(x)/(a^2-b^2*c)^4-2*b^4*(5*a^2+b^2*c)*d^2*log(a+b*sqrt(c+d*x))/(a^2-b^2*c)^4+1/2*a*b^2*(a^2+11*b^2*c)*d^2/(c*(a^2-b^2*c)^3*(a+b*sqrt(c+d*x)))+1/2*(-a+b*sqrt(c+d*x))/((a^2-b^2*c)*x^2*(a+b*sqrt(c+d*x)))-1/2*b*d*(3*a*b*c-(a^2+2*b^2*c)*sqrt(c+d*x))/(c*(a^2-b^2*c)^2*x*(a+b*sqrt(c+d*x)))],
[x^3/sqrt(a+b*sqrt(c+d*x)),x,4,4/3*(a^2-b^2*c)^2*(7*a^2-b^2*c)*(a+b*sqrt(c+d*x))^(3/2)/(b^8*d^4)-12/5*a*(7*a^2-3*b^2*c)*(a^2-b^2*c)*(a+b*sqrt(c+d*x))^(5/2)/(b^8*d^4)+4/7*(35*a^4-30*a^2*b^2*c+3*b^4*c^2)*(a+b*sqrt(c+d*x))^(7/2)/(b^8*d^4)-20/9*a*(7*a^2-3*b^2*c)*(a+b*sqrt(c+d*x))^(9/2)/(b^8*d^4)+12/11*(7*a^2-b^2*c)*(a+b*sqrt(c+d*x))^(11/2)/(b^8*d^4)-28/13*a*(a+b*sqrt(c+d*x))^(13/2)/(b^8*d^4)+4/15*(a+b*sqrt(c+d*x))^(15/2)/(b^8*d^4)-4*a*(a^2-b^2*c)^3*sqrt(a+b*sqrt(c+d*x))/(b^8*d^4)],
[x^2/sqrt(a+b*sqrt(c+d*x)),x,4,4/3*(5*a^4-6*a^2*b^2*c+b^4*c^2)*(a+b*sqrt(c+d*x))^(3/2)/(b^6*d^3)-8/5*a*(5*a^2-3*b^2*c)*(a+b*sqrt(c+d*x))^(5/2)/(b^6*d^3)+8/7*(5*a^2-b^2*c)*(a+b*sqrt(c+d*x))^(7/2)/(b^6*d^3)-20/9*a*(a+b*sqrt(c+d*x))^(9/2)/(b^6*d^3)+4/11*(a+b*sqrt(c+d*x))^(11/2)/(b^6*d^3)-4*a*(a^2-b^2*c)^2*sqrt(a+b*sqrt(c+d*x))/(b^6*d^3)],
[x/sqrt(a+b*sqrt(c+d*x)),x,4,4/3*(3*a^2-b^2*c)*(a+b*sqrt(c+d*x))^(3/2)/(b^4*d^2)-12/5*a*(a+b*sqrt(c+d*x))^(5/2)/(b^4*d^2)+4/7*(a+b*sqrt(c+d*x))^(7/2)/(b^4*d^2)-4*a*(a^2-b^2*c)*sqrt(a+b*sqrt(c+d*x))/(b^4*d^2)],
[1/sqrt(a+b*sqrt(c+d*x)),x,4,4/3*(a+b*sqrt(c+d*x))^(3/2)/(b^2*d)-4*a*sqrt(a+b*sqrt(c+d*x))/(b^2*d)],
[1/(x*sqrt(a+b*sqrt(c+d*x))),x,6,-2*arctanh(sqrt(a+b*sqrt(c+d*x))/sqrt(a-b*sqrt(c)))/sqrt(a-b*sqrt(c))-2*arctanh(sqrt(a+b*sqrt(c+d*x))/sqrt(a+b*sqrt(c)))/sqrt(a+b*sqrt(c))],
[1/(x^2*sqrt(a+b*sqrt(c+d*x))),x,7,-1/2*b*d*arctanh(sqrt(a+b*sqrt(c+d*x))/sqrt(a-b*sqrt(c)))/(sqrt(c)*(a-b*sqrt(c))^(3/2))+1/2*b*d*arctanh(sqrt(a+b*sqrt(c+d*x))/sqrt(a+b*sqrt(c)))/(sqrt(c)*(a+b*sqrt(c))^(3/2))-(a-b*sqrt(c+d*x))*sqrt(a+b*sqrt(c+d*x))/((a^2-b^2*c)*x)],
[1/(x^3*sqrt(a+b*sqrt(c+d*x))),x,8,1/16*b*d^2*arctanh(sqrt(a+b*sqrt(c+d*x))/sqrt(a-b*sqrt(c)))*(2*a-5*b*sqrt(c))/(c^(3/2)*(a-b*sqrt(c))^(5/2))-1/16*b*d^2*arctanh(sqrt(a+b*sqrt(c+d*x))/sqrt(a+b*sqrt(c)))*(2*a+5*b*sqrt(c))/(c^(3/2)*(a+b*sqrt(c))^(5/2))-1/2*(a-b*sqrt(c+d*x))*sqrt(a+b*sqrt(c+d*x))/((a^2-b^2*c)*x^2)-1/8*b*d*(6*a*b*c-(a^2+5*b^2*c)*sqrt(c+d*x))*sqrt(a+b*sqrt(c+d*x))/(c*(a^2-b^2*c)^2*x)],

# p symbolic
[x^3*(a+b*sqrt(c+d*x))^p,x,4,-2*a*(a^2-b^2*c)^3*(a+b*sqrt(c+d*x))^(1+p)/(b^8*d^4*(1+p))+2*(a^2-b^2*c)^2*(7*a^2-b^2*c)*(a+b*sqrt(c+d*x))^(2+p)/(b^8*d^4*(2+p))-6*a*(7*a^2-3*b^2*c)*(a^2-b^2*c)*(a+b*sqrt(c+d*x))^(3+p)/(b^8*d^4*(3+p))+2*(35*a^4-30*a^2*b^2*c+3*b^4*c^2)*(a+b*sqrt(c+d*x))^(4+p)/(b^8*d^4*(4+p))-10*a*(7*a^2-3*b^2*c)*(a+b*sqrt(c+d*x))^(5+p)/(b^8*d^4*(5+p))+6*(7*a^2-b^2*c)*(a+b*sqrt(c+d*x))^(6+p)/(b^8*d^4*(6+p))-14*a*(a+b*sqrt(c+d*x))^(7+p)/(b^8*d^4*(7+p))+2*(a+b*sqrt(c+d*x))^(8+p)/(b^8*d^4*(8+p))],
[x^2*(a+b*sqrt(c+d*x))^p,x,4,-2*a*(a^2-b^2*c)^2*(a+b*sqrt(c+d*x))^(1+p)/(b^6*d^3*(1+p))+2*(5*a^4-6*a^2*b^2*c+b^4*c^2)*(a+b*sqrt(c+d*x))^(2+p)/(b^6*d^3*(2+p))-4*a*(5*a^2-3*b^2*c)*(a+b*sqrt(c+d*x))^(3+p)/(b^6*d^3*(3+p))+4*(5*a^2-b^2*c)*(a+b*sqrt(c+d*x))^(4+p)/(b^6*d^3*(4+p))-10*a*(a+b*sqrt(c+d*x))^(5+p)/(b^6*d^3*(5+p))+2*(a+b*sqrt(c+d*x))^(6+p)/(b^6*d^3*(6+p))],
[x*(a+b*sqrt(c+d*x))^p,x,4,-2*a*(a^2-b^2*c)*(a+b*sqrt(c+d*x))^(1+p)/(b^4*d^2*(1+p))+2*(3*a^2-b^2*c)*(a+b*sqrt(c+d*x))^(2+p)/(b^4*d^2*(2+p))-6*a*(a+b*sqrt(c+d*x))^(3+p)/(b^4*d^2*(3+p))+2*(a+b*sqrt(c+d*x))^(4+p)/(b^4*d^2*(4+p))],
[(a+b*sqrt(c+d*x))^p,x,4,-2*a*(a+b*sqrt(c+d*x))^(1+p)/(b^2*d*(1+p))+2*(a+b*sqrt(c+d*x))^(2+p)/(b^2*d*(2+p))],
[(a+b*sqrt(c+d*x))^p/x,x,6,-hypergeom([1,1+p],[2+p],(a+b*sqrt(c+d*x))/(a-b*sqrt(c)))*(a+b*sqrt(c+d*x))^(1+p)/((1+p)*(a-b*sqrt(c)))-hypergeom([1,1+p],[2+p],(a+b*sqrt(c+d*x))/(a+b*sqrt(c)))*(a+b*sqrt(c+d*x))^(1+p)/((1+p)*(a+b*sqrt(c)))],

# Integrands of the form x^m F[(c x)^n]

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

# F[x^n] / x
[1/(x*sqrt(a+b*x)),x,2,-2*arctanh(sqrt(a+b*x)/sqrt(a))/sqrt(a)],
[1/(x*sqrt(a+b*(c*x)^m)),x,5,-2*arctanh(sqrt(a+b*(c*x)^m)/sqrt(a))/(m*sqrt(a))],
[1/(x*sqrt(a+b*(c*(d*x)^m)^n)),x,6,-2*arctanh(sqrt(a+b*(c*(d*x)^m)^n)/sqrt(a))/(m*n*sqrt(a))],
[1/(x*sqrt(a+b*(c*(d*(e*x)^m)^n)^p)),x,7,-2*arctanh(sqrt(a+b*(c*(d*(e*x)^m)^n)^p)/sqrt(a))/(m*n*p*sqrt(a))],
[1/(x*sqrt(a+b*(c*(d*(e*(f*x)^m)^n)^p)^q)),x,8,-2*arctanh(sqrt(a+b*(c*(d*(e*(f*x)^m)^n)^p)^q)/sqrt(a))/(m*n*p*q*sqrt(a))],
[(-1+x^2)^3*sqrt(-1+1/x^2)/x,x,8,-35/48*(-1+1/x^2)^(3/2)*x^2-7/24*(-1+1/x^2)^(5/2)*x^4-1/6*(-1+1/x^2)^(7/2)*x^6-35/16*arctan(sqrt(-1+1/x^2))+35/16*sqrt(-1+1/x^2)],
[(-1+x^2)^2*sqrt(-1+1/x^2)/x,x,7,5/8*(-1+1/x^2)^(3/2)*x^2+1/4*(-1+1/x^2)^(5/2)*x^4+15/8*arctan(sqrt(-1+1/x^2))-15/8*sqrt(-1+1/x^2)],
[(-1+x^2)*sqrt(-1+1/x^2)/x,x,6,-1/2*(-1+1/x^2)^(3/2)*x^2-3/2*arctan(sqrt(-1+1/x^2))+3/2*sqrt(-1+1/x^2)],
[sqrt(-1+1/x^2)/(x*(-1+x^2)),x,2,sqrt(-1+1/x^2)],
[sqrt(-1+1/x^2)/(x*(-1+x^2)^2),x,4,1/sqrt(-1+1/x^2)-sqrt(-1+1/x^2)],
[sqrt(-1+1/x^2)/(x*(-1+x^2)^3),x,4,(-1/3)/(-1+1/x^2)^(3/2)+(-2)/sqrt(-1+1/x^2)+sqrt(-1+1/x^2)],

# Integrands of the form x^m F[x^n]
[x*sqrt(1+1/x^2)/(1+x^2)^2,x,2,1/sqrt(1+1/x^2)],
[1/(x*(1+x^2)*sqrt(1+1/x^2)),x,2,1/sqrt(1+1/x^2)],
[x/(a+b*x^2+sqrt(a+b*x^2)),x,3,log(1+sqrt(a+b*x^2))/b],
[x/(x^2-(x^2)^(1/3)),x,3,3/4*log(1-(x^2)^(2/3))],
[x*(1+x^2)^3*sqrt(2+2*x^2+x^4),x,3,-1/15*(2+2*x^2+x^4)^(3/2)+1/10*(1+x^2)^2*(2+2*x^2+x^4)^(3/2)],
[x^5*(1+x^9)^2*sqrt(1-x^3),x,3,-8/9*(1-x^3)^(3/2)+32/15*(1-x^3)^(5/2)-22/7*(1-x^3)^(7/2)+86/27*(1-x^3)^(9/2)-74/33*(1-x^3)^(11/2)+14/13*(1-x^3)^(13/2)-14/45*(1-x^3)^(15/2)+2/51*(1-x^3)^(17/2)],

#  Note: Use the substitution u=x^2 instead of algebraic expansion. 
[x/(a+b*x^2)^(3/2)+x/((1+x^2)*sqrt(a+b*x^2)),x,5,-arctanh(sqrt(a+b*x^2)/sqrt(a-b))/sqrt(a-b)+(-1)/(b*sqrt(a+b*x^2))],
[x*(1+a+x^2+b*x^2)/((1+x^2)*(a+b*x^2)^(3/2)),x,5,-arctanh(sqrt(a+b*x^2)/sqrt(a-b))/sqrt(a-b)+(-1)/(b*sqrt(a+b*x^2))],
[x/(a+b*x^2)^(5/2)+x/(a+b*x^2)^(3/2)+x/((1+x^2)*sqrt(a+b*x^2)),x,6,(-1/3)/(b*(a+b*x^2)^(3/2))-arctanh(sqrt(a+b*x^2)/sqrt(a-b))/sqrt(a-b)+(-1)/(b*sqrt(a+b*x^2))],
[x*(1+a+a^2+x^2+a*x^2+b*x^2+2*a*b*x^2+b*x^4+b^2*x^4)/((1+x^2)*(a+b*x^2)^(5/2)),x,9,(-1/3)/(b*(a+b*x^2)^(3/2))-arctanh(sqrt(a+b*x^2)/sqrt(a-b))/sqrt(a-b)+(-1)/(b*sqrt(a+b*x^2))],

# Integrands of the form F[(a + b x)^(1/n), x]
[1/sqrt(x+sqrt(x)),x,4,-2*arctanh(sqrt(x)/sqrt(x+sqrt(x)))+2*sqrt(x+sqrt(x))],
[sqrt(x+sqrt(x)),x,6,1/4*arctanh(sqrt(x)/sqrt(x+sqrt(x)))-1/4*sqrt(x+sqrt(x))+2/3*x*sqrt(x+sqrt(x))+1/6*sqrt(x)*sqrt(x+sqrt(x))],
[sqrt(-x)*(x+sqrt(-x)),x,2,2/5*(-x)^(5/2)-1/2*x^2],
[(5+x^(1/4))/(-6+x),x,8,4*x^(1/4)-2*6^(1/4)*arctan(x^(1/4)/6^(1/4))-2*6^(1/4)*arctanh(x^(1/4)/6^(1/4))+5*log(6-x)],
[1/(4-x+sqrt(4-x)),x,2,-2*log(1+sqrt(4-x))],
[1/(1+x-sqrt(2+x)),x,4,1/5*log(1-sqrt(5)-2*sqrt(2+x))*(5-sqrt(5))+1/5*log(1+sqrt(5)-2*sqrt(2+x))*(5+sqrt(5))],
[1/(4+x+sqrt(1+x)),x,5,log(4+x+sqrt(1+x))-2*arctan((1+2*sqrt(1+x))/sqrt(11))/sqrt(11)],
[1/(x-sqrt(1+x)),x,4,1/5*log(1-sqrt(5)-2*sqrt(1+x))*(5-sqrt(5))+1/5*log(1+sqrt(5)-2*sqrt(1+x))*(5+sqrt(5))],
[1/(x-sqrt(2+x)),x,4,4/3*log(2-sqrt(2+x))+2/3*log(1+sqrt(2+x))],
[1/(x-sqrt(1-x)),x,4,1/5*log(1-sqrt(5)+2*sqrt(1-x))*(5-sqrt(5))+1/5*log(1+sqrt(5)+2*sqrt(1-x))*(5+sqrt(5))],
[sqrt(1+x+sqrt(x)),x,5,-3/8*arcsinh((1+2*sqrt(x))/sqrt(3))+2/3*(1+x+sqrt(x))^(3/2)-1/4*(1+2*sqrt(x))*sqrt(1+x+sqrt(x))],
[sqrt(1+x+sqrt(1+x)),x,6,1/4*arctanh(sqrt(1+x)/sqrt(1+x+sqrt(1+x)))+2/3*(1+x+sqrt(1+x))^(3/2)-1/4*(1+2*sqrt(1+x))*sqrt(1+x+sqrt(1+x))],
[sqrt(x+sqrt(-1+x)),x,5,-3/8*arcsinh((1+2*sqrt(-1+x))/sqrt(3))+2/3*(x+sqrt(-1+x))^(3/2)-1/4*(1+2*sqrt(-1+x))*sqrt(x+sqrt(-1+x))],
[sqrt(2*x+sqrt(-1+2*x)),x,5,-3/16*arcsinh((1+2*sqrt(-1+2*x))/sqrt(3))+1/3*(2*x+sqrt(-1+2*x))^(3/2)-1/8*(1+2*sqrt(-1+2*x))*sqrt(2*x+sqrt(-1+2*x))],
[sqrt(3*x+sqrt(-7+8*x)),x,5,-47/36*arcsinh((4+3*sqrt(-7+8*x))/sqrt(47))/sqrt(6)+1/72*(21-3*(7-8*x)+8*sqrt(-7+8*x))^(3/2)/sqrt(2)-1/36*(4+3*sqrt(-7+8*x))*sqrt(21-3*(7-8*x)+8*sqrt(-7+8*x))/sqrt(2)],
[1/sqrt(x+sqrt(1+x)),x,4,-arctanh(1/2*(1+2*sqrt(1+x))/sqrt(x+sqrt(1+x)))+2*sqrt(x+sqrt(1+x))],
[(1+x)/(4+x+sqrt(-9+6*x)),x,7,x+3*log(4+x+sqrt(3)*sqrt(-3+2*x))+4*arctan(1/2*(3+sqrt(-9+6*x))/sqrt(6))*sqrt(6)-2*sqrt(3)*sqrt(-3+2*x)],
[(12-x)/(4+x+sqrt(-9+6*x)),x,7,-x+10*log(4+x+sqrt(3)*sqrt(-3+2*x))-21*arctan(1/2*(3+sqrt(-9+6*x))/sqrt(6))*sqrt(3/2)+2*sqrt(3)*sqrt(-3+2*x)],
[(-1+x^3)/((1+x^2)*sqrt(x)),x,8,2/3*x^(3/2)+arctan(1-sqrt(2)*sqrt(x))*sqrt(2)-arctan(1+sqrt(2)*sqrt(x))*sqrt(2)],
[1/2/(sqrt(-1+x)*sqrt(x-sqrt(-1+x))),x,4,-arcsinh((1-2*sqrt(-1+x))/sqrt(3))],
[(1+x^(7/2))/(1-x^2),x,10,-2/5*x^(5/2)+arctan(sqrt(x))+1/2*log(1+x)-log(1-sqrt(x))-2*sqrt(x),-2/5*x^(5/2)+arctan(sqrt(x))+arctanh(x)+arctanh(sqrt(x))-2*sqrt(x)],
[(4+2*x)/((-1+2*x)^(1/3)+sqrt(-1+2*x)),x,3,-x+18*(-1+2*x)^(1/6)-9*(-1+2*x)^(1/3)-3/4*(-1+2*x)^(2/3)+3/5*(-1+2*x)^(5/6)+3/7*(-1+2*x)^(7/6)-3/8*(-1+2*x)^(4/3)+1/3*(-1+2*x)^(3/2)-18*log(1+(-1+2*x)^(1/6))+6*sqrt(-1+2*x)],

#  Integrands of the form Sqrt[a+b*Sqrt[c+d*Sqrt[e+f*x]]] 
[1/sqrt(2+sqrt(1+sqrt(x))),x,5,88/3*(2+sqrt(1+sqrt(x)))^(3/2)-48/5*(2+sqrt(1+sqrt(x)))^(5/2)+8/7*(2+sqrt(1+sqrt(x)))^(7/2)-48*sqrt(2+sqrt(1+sqrt(x)))],
[sqrt(2+sqrt(4+sqrt(x))),x,5,64/5*(2+sqrt(4+sqrt(x)))^(5/2)-48/7*(2+sqrt(4+sqrt(x)))^(7/2)+8/9*(2+sqrt(4+sqrt(x)))^(9/2)],
[sqrt(2-sqrt(4+sqrt(-9+5*x))),x,5,64/25*(2-sqrt(4+sqrt(-9+5*x)))^(5/2)-48/35*(2-sqrt(4+sqrt(-9+5*x)))^(7/2)+8/45*(2-sqrt(4+sqrt(-9+5*x)))^(9/2)],
[1/sqrt(2+sqrt(1+sqrt(x))),x,5,88/3*(2+sqrt(1+sqrt(x)))^(3/2)-48/5*(2+sqrt(1+sqrt(x)))^(5/2)+8/7*(2+sqrt(1+sqrt(x)))^(7/2)-48*sqrt(2+sqrt(1+sqrt(x)))],

#  Integrands of the form Sqrt[a+b*Sqrt[c+d*Sqrt[e+f*Sqrt[g+h*x]]]] 
[sqrt(1+sqrt(1+sqrt(1+sqrt(x)))),x,6,-32/5*(1+sqrt(1+sqrt(1+sqrt(x))))^(5/2)+48/7*(1+sqrt(1+sqrt(1+sqrt(x))))^(7/2)+112/9*(1+sqrt(1+sqrt(1+sqrt(x))))^(9/2)-320/11*(1+sqrt(1+sqrt(1+sqrt(x))))^(11/2)+288/13*(1+sqrt(1+sqrt(1+sqrt(x))))^(13/2)-112/15*(1+sqrt(1+sqrt(1+sqrt(x))))^(15/2)+16/17*(1+sqrt(1+sqrt(1+sqrt(x))))^(17/2)],
[sqrt(2+sqrt(3+sqrt(-1+2*sqrt(x)))),x,5,-16/3*(2+sqrt(3+sqrt(-1+2*sqrt(x))))^(3/2)+136/5*(2+sqrt(3+sqrt(-1+2*sqrt(x))))^(5/2)-480/7*(2+sqrt(3+sqrt(-1+2*sqrt(x))))^(7/2)+304/3*(2+sqrt(3+sqrt(-1+2*sqrt(x))))^(9/2)-760/11*(2+sqrt(3+sqrt(-1+2*sqrt(x))))^(11/2)+300/13*(2+sqrt(3+sqrt(-1+2*sqrt(x))))^(13/2)-56/15*(2+sqrt(3+sqrt(-1+2*sqrt(x))))^(15/2)+4/17*(2+sqrt(3+sqrt(-1+2*sqrt(x))))^(17/2)],
[x*sqrt(1+sqrt(1+sqrt(-1+x))),x,5,16/5*(1+sqrt(1+sqrt(-1+x)))^(5/2)-24/7*(1+sqrt(1+sqrt(-1+x)))^(7/2)+8*(1+sqrt(1+sqrt(-1+x)))^(9/2)-160/11*(1+sqrt(1+sqrt(-1+x)))^(11/2)+144/13*(1+sqrt(1+sqrt(-1+x)))^(13/2)-56/15*(1+sqrt(1+sqrt(-1+x)))^(15/2)+8/17*(1+sqrt(1+sqrt(-1+x)))^(17/2)],
[1/(sqrt(-1+x)*sqrt(x-sqrt(-1+x))),x,3,-2*arcsinh((1-2*sqrt(-1+x))/sqrt(3))],
[1/sqrt(1+x+sqrt(-1+2*x)),x,4,-arcsinh((1+sqrt(-1+2*x))/sqrt(2))*sqrt(2)+2*sqrt(1+x+sqrt(-1+2*x)),-arcsinh((1+sqrt(-1+2*x))/sqrt(2))*sqrt(2)+sqrt(2)*sqrt(2+2*x+2*sqrt(-1+2*x))],
[(q+p*x)/(sqrt(b+a*x)*(f+sqrt(b+a*x))),x,3,p*x/a-2*(b*p-f^2*p-a*q)*log(f+sqrt(b+a*x))/a^2-2*f*p*sqrt(b+a*x)/a^2],
[sqrt(1-x-sqrt(x)),x,5,-5/8*arcsin((1+2*sqrt(x))/sqrt(5))-2/3*(1-x-sqrt(x))^(3/2)-1/4*(1+2*sqrt(x))*sqrt(1-x-sqrt(x))],
[(9+x+6*sqrt(x))/(x+4*sqrt(x)),x,4,x+2*log(4+sqrt(x))+4*sqrt(x)],
[(6-8*x^(7/2))/(5-9*sqrt(x)),x,8,125000/4782969*x+50000/1594323*x^(3/2)+2500/59049*x^2+400/6561*x^(5/2)+200/2187*x^3+80/567*x^(7/2)+2/9*x^4-280728140/387420489*log(5-9*sqrt(x))-56145628/43046721*sqrt(x)],

#  Although the following optimal antiderivative contains the imaginary unit, it is continuous for x along the real line. 
[(1+x^3)*sqrt(1+x)/(1+x^2),x,16,-2/3*(1+x)^(3/2)+2/5*(1+x)^(5/2)+(1-I)^(3/2)*arctanh(sqrt(1+x)/sqrt(1-I))+(1+I)^(3/2)*arctanh(sqrt(1+x)/sqrt(1+I))-2*sqrt(1+x),-2/3*(1+x)^(3/2)+2/5*(1+x)^(5/2)-2*sqrt(1+x)-1/2*log(1+x+sqrt(2)-sqrt(1+x)*sqrt(2*(1+sqrt(2))))/sqrt(1+sqrt(2))+1/2*log(1+x+sqrt(2)+sqrt(1+x)*sqrt(2*(1+sqrt(2))))/sqrt(1+sqrt(2))-arctan((-2*sqrt(1+x)+sqrt(2*(1+sqrt(2))))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))+arctan((2*sqrt(1+x)+sqrt(2*(1+sqrt(2))))/sqrt(2*(-1+sqrt(2))))*sqrt(1+sqrt(2))],
[sqrt(-1+x-sqrt(x))/((-1+x)*sqrt(x)),x,9,arctan(1/2*(3-sqrt(x))/sqrt(-1+x-sqrt(x)))-2*arctanh(1/2*(1-2*sqrt(x))/sqrt(-1+x-sqrt(x)))-arctanh(1/2*(1+3*sqrt(x))/sqrt(-1+x-sqrt(x)))],
[(1+2*sqrt(1+x))/(x*sqrt(1+x)*sqrt(x+sqrt(1+x))),x,6,-arctan(1/2*(3+sqrt(1+x))/sqrt(x+sqrt(1+x)))+3*arctanh(1/2*(1-3*sqrt(1+x))/sqrt(x+sqrt(1+x)))],

# Integrands of the form F[((a + b x)/(c + d x))^(1/n), x]

#  Following pairs of integrands are equal: 
[1/(sqrt(x)*sqrt(1+x)),x,2,2*arcsinh(sqrt(x))],
[sqrt(x/(1+x))/x,x,3,2*arcsinh(sqrt(x))],
[sqrt(x)/sqrt(1+x),x,3,-arcsinh(sqrt(x))+sqrt(x)*sqrt(1+x)],
[sqrt(x/(1+x)),x,4,-arcsinh(sqrt(x))+sqrt(x)*sqrt(1+x)],
[sqrt(-1+x)/(x^2*sqrt(1+x)),x,3,arctan(sqrt(-1+x)*sqrt(1+x))-sqrt(-1+x)*sqrt(1+x)/x],
[sqrt((-1+x)/(1+x))/x^2,x,4,arctan(sqrt(-1+x)*sqrt(1+x))-sqrt(-1+x)*sqrt(1+x)/x],
[x^3*sqrt(-1+x)/sqrt(1+x),x,4,3/8*arccosh(x)+1/24*(7-2*x)*(-1+x)^(3/2)*sqrt(1+x)+1/4*(-1+x)^(3/2)*x^2*sqrt(1+x)-3/8*sqrt(-1+x)*sqrt(1+x)],
[x^3*sqrt((-1+x)/(1+x)),x,5,3/8*arccosh(x)+1/24*(7-2*x)*(-1+x)^(3/2)*sqrt(1+x)+1/4*(-1+x)^(3/2)*x^2*sqrt(1+x)-3/8*sqrt(-1+x)*sqrt(1+x)],
[sqrt(-x/(1+x))/x,x,2,2*arctan(sqrt(-x/(1+x)))],
[sqrt((1-x)/(1+x))/(-1+x),x,2,2*arctan(sqrt((1-x)/(1+x)))],
[sqrt((a+b*x)/(c-b*x))/(a+b*x),x,3,2*arctan(sqrt((a+b*x)/(c-b*x)))/b],
[sqrt((a+b*x)/(c+d*x))/(a+b*x),x,3,2*arctanh(sqrt(d)*sqrt((a+b*x)/(c+d*x))/sqrt(b))/(sqrt(b)*sqrt(d))],
[sqrt(-x/(1+x)),x,3,-arctan(sqrt(-x/(1+x)))+(1+x)*sqrt(-x/(1+x))],
[sqrt((1-x)/(1+x)),x,3,-2*arctan(sqrt((1-x)/(1+x)))+(1+x)*sqrt((1-x)/(1+x))],
[sqrt((a+x)/(a-x)),x,3,2*a*arctan(sqrt((a+x)/(a-x)))-(a-x)*sqrt((a+x)/(a-x))],
[sqrt((-a+x)/(a+x)),x,3,-2*a*arctanh(sqrt((-a+x)/(a+x)))+(a+x)*sqrt((-a+x)/(a+x))],
[sqrt((a+b*x)/(c+d*x)),x,3,-(b*c-a*d)*arctanh(sqrt(d)*sqrt((a+b*x)/(c+d*x))/sqrt(b))/(d^(3/2)*sqrt(b))+(c+d*x)*sqrt((a+b*x)/(c+d*x))/d],
[sqrt((-1+x)/(5+3*x)),x,4,-8/3*arcsinh(1/2*sqrt(3/2)*sqrt(-1+x))/sqrt(3)+1/3*sqrt(-1+x)*sqrt(5+3*x)],
[sqrt((-1+5*x)/(1+7*x))/x^2,x,4,-12*arctan(sqrt(1+7*x)/sqrt(-1+5*x))-sqrt(-1+5*x)*sqrt(1+7*x)/x],
[x/((1+x)*sqrt((1-x)/(1+x))),x,3,-(1+x)*sqrt((1-x)/(1+x))],
[x/((1+x)*sqrt(-1+2/(1+x))),x,4,-(1+x)*sqrt(-1+2/(1+x))],
[x/((1+x)*sqrt((2+x)/(3+x))),x,7,-arcsinh(sqrt(2+x))+2*arctanh(sqrt(2)*sqrt(2+x)/sqrt(3+x))*sqrt(2)+sqrt(2+x)*sqrt(3+x)],
[sqrt(1+1/x)/(1+x)^2,x,2,2/sqrt(1+1/x)],
[sqrt(1+1/x)/sqrt(1-x^2),x,5,-arcsin(1-2*x)*sqrt(1+1/x)*sqrt(x)/sqrt(1+x)],

# Integrands of the form F[(a + b x + c x^2)^(n/2), x]

# Euler substitution #1 for Sqrt[a+b x+c x^2] when a>0
[1/(x+sqrt(3-2*x-x^2)),x,8,arctan((sqrt(3)-sqrt(3-2*x-x^2))/x)-1/2*log((-3+x+sqrt(3)*sqrt(3-2*x-x^2))/x^2)+1/14*log(1+sqrt(3)+sqrt(7)-sqrt(3)*(sqrt(3)-sqrt(3-2*x-x^2))/x)*(7-sqrt(7))+1/14*log(1+sqrt(3)-sqrt(7)-sqrt(3)*(sqrt(3)-sqrt(3-2*x-x^2))/x)*(7+sqrt(7))],
[1/(x+sqrt(3-2*x-x^2))^2,x,5,8/7*arctanh((3-x-x*sqrt(3)-sqrt(3)*sqrt(3-2*x-x^2))/(x*sqrt(7)))/sqrt(7)+2/7*(4-sqrt(3)+3*(sqrt(3)-sqrt(3-2*x-x^2))/x)/(2-sqrt(3)-2*(1+sqrt(3))*(sqrt(3)-sqrt(3-2*x-x^2))/x+sqrt(3)*(sqrt(3)-sqrt(3-2*x-x^2))^2/x^2)],
[1/(x+sqrt(3-2*x-x^2))^3,x,6,12/49*arctanh((3-x-x*sqrt(3)-sqrt(3)*sqrt(3-2*x-x^2))/(x*sqrt(7)))/sqrt(7)-4/21*(9-5*sqrt(3)+(21+5*sqrt(3))*(sqrt(3)-sqrt(3-2*x-x^2))/x)/(2-sqrt(3)-2*(1+sqrt(3))*(sqrt(3)-sqrt(3-2*x-x^2))/x+sqrt(3)*(sqrt(3)-sqrt(3-2*x-x^2))^2/x^2)^2+2/147*(18-43*sqrt(3)-(18+49*sqrt(3))*(sqrt(3)-sqrt(3-2*x-x^2))/x)/(2-sqrt(3)-2*(1+sqrt(3))*(sqrt(3)-sqrt(3-2*x-x^2))/x+sqrt(3)*(sqrt(3)-sqrt(3-2*x-x^2))^2/x^2)],

# Euler substitution #2 for Sqrt[a+b x+c x^2] when c>0
[1/(x+sqrt(-3-2*x+x^2)),x,3,2*log(1-x-sqrt(-3-2*x+x^2))-3/2*log(x+sqrt(-3-2*x+x^2))+(-2)/(1-x-sqrt(-3-2*x+x^2))],
[1/(x+sqrt(-3-2*x+x^2))^2,x,3,4*log(1-x-sqrt(-3-2*x+x^2))-4*log(x+sqrt(-3-2*x+x^2))+(-2)/(1-x-sqrt(-3-2*x+x^2))+3/2/(x+sqrt(-3-2*x+x^2))],
[1/(x+sqrt(-3-2*x+x^2))^3,x,3,6*log(1-x-sqrt(-3-2*x+x^2))-6*log(x+sqrt(-3-2*x+x^2))+(-2)/(1-x-sqrt(-3-2*x+x^2))+3/4/(x+sqrt(-3-2*x+x^2))^2+4/(x+sqrt(-3-2*x+x^2))],

# Euler substitution #3 for Sqrt[a+b x+c x^2] when a<0 and c<0
[1/(x+sqrt(-3-4*x-x^2)),x,10,-arctan(sqrt(-1-x)/sqrt(3+x))+1/2*log(3+x)+1/2*log((3*sqrt(-1-x)+x*sqrt(-1-x)+x*sqrt(3+x))/(3+x)^(3/2))-arctan((1-3*sqrt(-1-x)/sqrt(3+x))/sqrt(2))*sqrt(2)],
[1/(x+sqrt(-3-4*x-x^2))^2,x,5,arctan((1-3*sqrt(-1-x)/sqrt(3+x))/sqrt(2))/sqrt(2)+(1-sqrt(-1-x)/sqrt(3+x))/(1-3*(1+x)/(3+x)-2*sqrt(-1-x)/sqrt(3+x))],
[1/(x+sqrt(-3-4*x-x^2))^3,x,6,-3/2*arctan((1-3*sqrt(-1-x)/sqrt(3+x))/sqrt(2))/sqrt(2)+1/18*(-13+27*sqrt(-1-x)/sqrt(3+x))/(1-3*(1+x)/(3+x)-2*sqrt(-1-x)/sqrt(3+x))-2/9*(2-sqrt(-1-x)/sqrt(3+x))/(1-3*(1+x)/(3+x)-2*sqrt(-1-x)/sqrt(3+x))^2],

# Integrands of the form F[a + b x + c x^2 + d x^3 + e x^4, x] when d^3 - 4 c d e + 8 b e^2=0

#  It would be better to make the substitution u=x+x^2 than u=x+1/2, but that is tough to know... 
[x^3*(1+x)^3*(1+2*x)*sqrt(1-x^2-2*x^3-x^4),x,5,-1/15*(1-x^2-2*x^3-x^4)^(3/2)*(2+3*x^2+6*x^3+3*x^4),-2/15*(1-x^2-2*x^3-x^4)^(3/2)-1/5*x^2*(1+x)^2*(1-x^2-2*x^3-x^4)^(3/2)],
[(1+2*x)*(x+x^2)^3*sqrt(1-(x+x^2)^2),x,6,-1/15*(1-x^2-2*x^3-x^4)^(3/2)*(2+3*x^2+6*x^3+3*x^4),-2/15*(1-x^2-2*x^3-x^4)^(3/2)-1/5*x^2*(1+x)^2*(1-x^2-2*x^3-x^4)^(3/2)],

# Integrands of the form (0 + b x + c x^2 + d x^3 + e x^4)^p when d^3 - 4 c d e + 8 b e^2=0
[(8*x-8*x^2+4*x^3-x^4)^(3/2),x,7,1/7*(3-2*(-1+x)^2-(-1+x)^4)^(3/2)*(-1+x)+16/5*EllipticE(1-x,sqrt(-1/3))*sqrt(3)-176/35*EllipticF(1-x,sqrt(-1/3))*sqrt(3)+2/35*(13-3*(-1+x)^2)*(-1+x)*sqrt(3-2*(-1+x)^2-(-1+x)^4)],
[(8*x-8*x^2+4*x^3-x^4)^(1/2),x,6,2*EllipticE(1-x,sqrt(-1/3))/sqrt(3)-4*EllipticF(1-x,sqrt(-1/3))/sqrt(3)+1/3*(-1+x)*sqrt(3-2*(-1+x)^2-(-1+x)^4)],
[1/(8*x-8*x^2+4*x^3-x^4)^(1/2),x,3,-EllipticF(1-x,sqrt(-1/3))/sqrt(3)],
[1/(8*x-8*x^2+4*x^3-x^4)^(3/2),x,6,1/8*EllipticE(1-x,sqrt(-1/3))/sqrt(3)-1/4*EllipticF(1-x,sqrt(-1/3))/sqrt(3)+1/24*(5+(-1+x)^2)*(-1+x)/sqrt(3-2*(-1+x)^2-(-1+x)^4)],
[1/(8*x-8*x^2+4*x^3-x^4)^(5/2),x,7,1/72*(5+(-1+x)^2)*(-1+x)/(3-2*(-1+x)^2-(-1+x)^4)^(3/2)+7/144*EllipticE(1-x,sqrt(-1/3))/sqrt(3)-11/144*EllipticF(1-x,sqrt(-1/3))/sqrt(3)+1/432*(26+7*(-1+x)^2)*(-1+x)/sqrt(3-2*(-1+x)^2-(-1+x)^4)],
[((2-x)*x*(4-2*x+x^2))^(3/2),x,7,1/7*(3-2*(-1+x)^2-(-1+x)^4)^(3/2)*(-1+x)+16/5*EllipticE(1-x,sqrt(-1/3))*sqrt(3)-176/35*EllipticF(1-x,sqrt(-1/3))*sqrt(3)+2/35*(13-3*(-1+x)^2)*(-1+x)*sqrt(3-2*(-1+x)^2-(-1+x)^4)],
[((2-x)*x*(4-2*x+x^2))^(1/2),x,6,2*EllipticE(1-x,sqrt(-1/3))/sqrt(3)-4*EllipticF(1-x,sqrt(-1/3))/sqrt(3)+1/3*(-1+x)*sqrt(3-2*(-1+x)^2-(-1+x)^4)],
[1/((2-x)*x*(4-2*x+x^2))^(1/2),x,3,-EllipticF(1-x,sqrt(-1/3))/sqrt(3)],
[1/((2-x)*x*(4-2*x+x^2))^(3/2),x,6,1/8*EllipticE(1-x,sqrt(-1/3))/sqrt(3)-1/4*EllipticF(1-x,sqrt(-1/3))/sqrt(3)+1/24*(5+(-1+x)^2)*(-1+x)/sqrt(3-2*(-1+x)^2-(-1+x)^4)],
[1/((2-x)*x*(4-2*x+x^2))^(5/2),x,7,1/72*(5+(-1+x)^2)*(-1+x)/(3-2*(-1+x)^2-(-1+x)^4)^(3/2)+7/144*EllipticE(1-x,sqrt(-1/3))/sqrt(3)-11/144*EllipticF(1-x,sqrt(-1/3))/sqrt(3)+1/432*(26+7*(-1+x)^2)*(-1+x)/sqrt(3-2*(-1+x)^2-(-1+x)^4)],

# 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)^(3/2),x,6,1/7*(c/d+x)*(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)^(3/2)+2/35*c*(c/d+x)*(7*c^3+20*a*d^2-3*c*d^2*(c/d+x)^2)*sqrt(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)/d^2-16/35*c^3*(c^3+8*a*d^2)*(c/d+x)*sqrt(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)/(d^2*(sqrt(c)+d^2*(c/d+x)^2/sqrt(c^3+4*a*d^2))*sqrt(c^3+4*a*d^2))+16/35*c^(13/4)*(c^3+4*a*d^2)^(3/4)*(c^3+8*a*d^2)*sqrt(cos(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4))))^2)/cos(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4))))*EllipticE(sin(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4)))),sqrt(1/2*(1+c^(3/2)/sqrt(c^3+4*a*d^2))))*(sqrt(c)+d^2*(c/d+x)^2/sqrt(c^3+4*a*d^2))*sqrt(d^2*(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)/((c^3+4*a*d^2)*(sqrt(c)+d^2*(c/d+x)^2/sqrt(c^3+4*a*d^2))^2))/(d^5*sqrt(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4))+8/35*c^(7/4)*(c^3+4*a*d^2)^(3/4)*sqrt(cos(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4))))^2)/cos(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4))))*EllipticF(sin(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4)))),sqrt(1/2*(1+c^(3/2)/sqrt(c^3+4*a*d^2))))*(sqrt(c)+d^2*(c/d+x)^2/sqrt(c^3+4*a*d^2))*(-c^(3/2)*(c^3+8*a*d^2)+(c^3+5*a*d^2)*sqrt(c^3+4*a*d^2))*sqrt(d^2*(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)/((c^3+4*a*d^2)*(sqrt(c)+d^2*(c/d+x)^2/sqrt(c^3+4*a*d^2))^2))/(d^5*sqrt(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4))],
[(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)^(1/2),x,5,1/3*(c/d+x)*sqrt(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)-2/3*c^2*(c/d+x)*sqrt(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)/((sqrt(c)+d^2*(c/d+x)^2/sqrt(c^3+4*a*d^2))*sqrt(c^3+4*a*d^2))+2/3*c^(9/4)*(c^3+4*a*d^2)^(3/4)*sqrt(cos(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4))))^2)/cos(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4))))*EllipticE(sin(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4)))),sqrt(1/2*(1+c^(3/2)/sqrt(c^3+4*a*d^2))))*(sqrt(c)+d^2*(c/d+x)^2/sqrt(c^3+4*a*d^2))*sqrt(d^2*(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)/((c^3+4*a*d^2)*(sqrt(c)+d^2*(c/d+x)^2/sqrt(c^3+4*a*d^2))^2))/(d^3*sqrt(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4))+1/3*c^(3/4)*(c^3+4*a*d^2)^(1/4)*sqrt(cos(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4))))^2)/cos(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4))))*EllipticF(sin(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4)))),sqrt(1/2*(1+c^(3/2)/sqrt(c^3+4*a*d^2))))*(sqrt(c)+d^2*(c/d+x)^2/sqrt(c^3+4*a*d^2))*(c^3+4*a*d^2-c^(3/2)*sqrt(c^3+4*a*d^2))*sqrt(d^2*(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)/((c^3+4*a*d^2)*(sqrt(c)+d^2*(c/d+x)^2/sqrt(c^3+4*a*d^2))^2))/(d^3*sqrt(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4))],
[1/(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)^(1/2),x,2,1/2*(c^3+4*a*d^2)^(1/4)*sqrt(cos(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4))))^2)/cos(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4))))*EllipticF(sin(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4)))),sqrt(1/2*(1+c^(3/2)/sqrt(c^3+4*a*d^2))))*(sqrt(c)+d^2*(c/d+x)^2/sqrt(c^3+4*a*d^2))*sqrt(d^2*(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)/((c^3+4*a*d^2)*(sqrt(c)+d^2*(c/d+x)^2/sqrt(c^3+4*a*d^2))^2))/(c^(1/4)*d*sqrt(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4))],
[1/(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)^(3/2),x,5,-1/8*(c/d+x)*(c^3-4*a*d^2-c*d^2*(c/d+x)^2)/(a*c*(c^3+4*a*d^2)*sqrt(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4))-1/8*d^2*(c/d+x)*sqrt(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)/(a*(c^3+4*a*d^2)^(3/2)*(sqrt(c)+d^2*(c/d+x)^2/sqrt(c^3+4*a*d^2)))+1/8*c^(1/4)*sqrt(cos(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4))))^2)/cos(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4))))*EllipticE(sin(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4)))),sqrt(1/2*(1+c^(3/2)/sqrt(c^3+4*a*d^2))))*(sqrt(c)+d^2*(c/d+x)^2/sqrt(c^3+4*a*d^2))*sqrt(d^2*(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)/((c^3+4*a*d^2)*(sqrt(c)+d^2*(c/d+x)^2/sqrt(c^3+4*a*d^2))^2))/(a*d*(c^3+4*a*d^2)^(1/4)*sqrt(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4))+1/16*sqrt(cos(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4))))^2)/cos(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4))))*EllipticF(sin(2*arctan((c+d*x)/(c^(1/4)*(c^3+4*a*d^2)^(1/4)))),sqrt(1/2*(1+c^(3/2)/sqrt(c^3+4*a*d^2))))*(sqrt(c)+d^2*(c/d+x)^2/sqrt(c^3+4*a*d^2))*(c^3+4*a*d^2-c^(3/2)*sqrt(c^3+4*a*d^2))*sqrt(d^2*(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4)/((c^3+4*a*d^2)*(sqrt(c)+d^2*(c/d+x)^2/sqrt(c^3+4*a*d^2))^2))/(a*c^(5/4)*d*(c^3+4*a*d^2)^(3/4)*sqrt(4*a*c+4*c^2*x^2+4*c*d*x^3+d^2*x^4))],

# 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)^(3/2), x, 7, ((d/(4*e) + x)*(43*d^4 + 1280*a*e^3 - 144*d^2*e^2*(d/(4*e) + x)^2)*Sqrt[(5*d^4)/e + 256*a*e^2 - 96*d^2*e*(d/(4*e) + x)^2 + 256*e^3*(d/(4*e) + x)^4])/(2240*Sqrt[2]*e) + ((d/(4*e) + x)*((5*d^4)/e + 256*a*e^2 - 96*d^2*e*(d/(4*e) + x)^2 + 256*e^3*(d/(4*e) + x)^4)^(3/2))/(896*Sqrt[2]) + (3*d^2*(d^4 + 512*a*e^3)*(3*d^2 - 2*Sqrt[d^4 - 64*a*e^3])*Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]]*Sqrt[1 - (16*e^2*(d/(4*e) + x)^2)/(3*d^2 - 2*Sqrt[d^4 - 64*a*e^3])]*Sqrt[1 - (16*e^2*(d/(4*e) + x)^2)/(3*d^2 + 2*Sqrt[d^4 - 64*a*e^3])]*EllipticE[ArcSin[(4*e*(d/(4*e) + x))/Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]]], (3*d^2 + 2*Sqrt[d^4 - 64*a*e^3])/(3*d^2 - 2*Sqrt[d^4 - 64*a*e^3])])/(1120*Sqrt[2]*e^3*Sqrt[(5*d^4)/e + 256*a*e^2 - 96*d^2*e*(d/(4*e) + x)^2 + 256*e^3*(d/(4*e) + x)^4]) + (Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]]*(11*d^8 - 1984*a*d^4*e^3 + 81920*a^2*e^6 + 6*d^2*Sqrt[d^4 - 64*a*e^3]*(d^4 + 512*a*e^3))*Sqrt[1 - (16*e^2*(d/(4*e) + x)^2)/(3*d^2 - 2*Sqrt[d^4 - 64*a*e^3])]*Sqrt[1 - (16*e^2*(d/(4*e) + x)^2)/(3*d^2 + 2*Sqrt[d^4 - 64*a*e^3])]*EllipticF[ArcSin[(4*e*(d/(4*e) + x))/Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]]], (3*d^2 + 2*Sqrt[d^4 - 64*a*e^3])/(3*d^2 - 2*Sqrt[d^4 - 64*a*e^3])])/(1120*Sqrt[2]*e^3*Sqrt[(5*d^4)/e + 256*a*e^2 - 96*d^2*e*(d/(4*e) + x)^2 + 256*e^3*(d/(4*e) + x)^4])} 
[(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4)^(1/2),x,5,1/3*(1/4*d/e+x)*sqrt(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4)-2*d^2*(1/4*d/e+x)*sqrt(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4)/((1+16*e^2*(1/4*d/e+x)^2/sqrt(5*d^4+256*a*e^3))*sqrt(5*d^4+256*a*e^3))+1/8*d^2*(5*d^4+256*a*e^3)^(3/4)*sqrt(cos(2*arctan((d+4*e*x)/(5*d^4+256*a*e^3)^(1/4)))^2)/cos(2*arctan((d+4*e*x)/(5*d^4+256*a*e^3)^(1/4)))*EllipticE(sin(2*arctan((d+4*e*x)/(5*d^4+256*a*e^3)^(1/4))),sqrt(1/2*(1+3*d^2/sqrt(5*d^4+256*a*e^3))))*(1+16*e^2*(1/4*d/e+x)^2/sqrt(5*d^4+256*a*e^3))*sqrt(e*(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4)/((5*d^4+256*a*e^3)*(1+16*e^2*(1/4*d/e+x)^2/sqrt(5*d^4+256*a*e^3))^2))/(e^2*sqrt(2)*sqrt(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4))+1/48*(5*d^4+256*a*e^3)^(1/4)*sqrt(cos(2*arctan((d+4*e*x)/(5*d^4+256*a*e^3)^(1/4)))^2)/cos(2*arctan((d+4*e*x)/(5*d^4+256*a*e^3)^(1/4)))*EllipticF(sin(2*arctan((d+4*e*x)/(5*d^4+256*a*e^3)^(1/4))),sqrt(1/2*(1+3*d^2/sqrt(5*d^4+256*a*e^3))))*(1+16*e^2*(1/4*d/e+x)^2/sqrt(5*d^4+256*a*e^3))*(5*d^4+256*a*e^3-3*d^2*sqrt(5*d^4+256*a*e^3))*sqrt(e*(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4)/((5*d^4+256*a*e^3)*(1+16*e^2*(1/4*d/e+x)^2/sqrt(5*d^4+256*a*e^3))^2))/(e^2*sqrt(2)*sqrt(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4))],
[1/(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4)^(1/2),x,2,(5*d^4+256*a*e^3)^(1/4)*sqrt(cos(2*arctan((d+4*e*x)/(5*d^4+256*a*e^3)^(1/4)))^2)/cos(2*arctan((d+4*e*x)/(5*d^4+256*a*e^3)^(1/4)))*EllipticF(sin(2*arctan((d+4*e*x)/(5*d^4+256*a*e^3)^(1/4))),sqrt(1/2*(1+3*d^2/sqrt(5*d^4+256*a*e^3))))*(1+16*e^2*(1/4*d/e+x)^2/sqrt(5*d^4+256*a*e^3))*sqrt(e*(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4)/((5*d^4+256*a*e^3)*(1+16*e^2*(1/4*d/e+x)^2/sqrt(5*d^4+256*a*e^3))^2))/(e*sqrt(2)*sqrt(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4))],
[1/(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4)^(3/2),x,5,4*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)*sqrt(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4))+384*d^2*e^2*(1/4*d/e+x)*sqrt(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4)/((d^4-64*a*e^3)*(5*d^4+256*a*e^3)^(3/2)*(1+16*e^2*(1/4*d/e+x)^2/sqrt(5*d^4+256*a*e^3)))-12*d^2*sqrt(cos(2*arctan((d+4*e*x)/(5*d^4+256*a*e^3)^(1/4)))^2)/cos(2*arctan((d+4*e*x)/(5*d^4+256*a*e^3)^(1/4)))*EllipticE(sin(2*arctan((d+4*e*x)/(5*d^4+256*a*e^3)^(1/4))),sqrt(1/2*(1+3*d^2/sqrt(5*d^4+256*a*e^3))))*sqrt(2)*(1+16*e^2*(1/4*d/e+x)^2/sqrt(5*d^4+256*a*e^3))*sqrt(e*(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4)/((5*d^4+256*a*e^3)*(1+16*e^2*(1/4*d/e+x)^2/sqrt(5*d^4+256*a*e^3))^2))/((d^4-64*a*e^3)*(5*d^4+256*a*e^3)^(1/4)*sqrt(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4))-2*sqrt(cos(2*arctan((d+4*e*x)/(5*d^4+256*a*e^3)^(1/4)))^2)/cos(2*arctan((d+4*e*x)/(5*d^4+256*a*e^3)^(1/4)))*EllipticF(sin(2*arctan((d+4*e*x)/(5*d^4+256*a*e^3)^(1/4))),sqrt(1/2*(1+3*d^2/sqrt(5*d^4+256*a*e^3))))*sqrt(2)*(1+16*e^2*(1/4*d/e+x)^2/sqrt(5*d^4+256*a*e^3))*(5*d^4+256*a*e^3-3*d^2*sqrt(5*d^4+256*a*e^3))*sqrt(e*(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4)/((5*d^4+256*a*e^3)*(1+16*e^2*(1/4*d/e+x)^2/sqrt(5*d^4+256*a*e^3))^2))/((d^4-64*a*e^3)*(5*d^4+256*a*e^3)^(3/4)*sqrt(8*a*e^2-d^3*x+8*d*e^2*x^3+8*e^3*x^4))],

# Integrands of the form x^m (a + b x + c x^2 + d x^3 + e x^4)^(p/2) when d^3 - 4 c d e + 8 b e^2=0
[(a+8*x-8*x^2+4*x^3-x^4)^(3/2),x,8,1/7*(3+a-2*(-1+x)^2-(-1+x)^4)^(3/2)*(-1+x)-16/35*(7+2*a)*(-1+x)*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))/sqrt(3+a-2*(-1+x)^2-(-1+x)^4)+2/35*(13+5*a-3*(-1+x)^2)*(-1+x)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)+4/35*(3+a)*(16+5*a)*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticF(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*sqrt(1+sqrt(4+a))/(sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))+16/35*(7+2*a)*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticE(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))*sqrt(1+sqrt(4+a))/(sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))],
[(a+8*x-8*x^2+4*x^3-x^4)^(1/2),x,7,-2/3*(-1+x)*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))/sqrt(3+a-2*(-1+x)^2-(-1+x)^4)+1/3*(-1+x)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)+2/3*(3+a)*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticF(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*sqrt(1+sqrt(4+a))/(sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))+2/3*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticE(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))*sqrt(1+sqrt(4+a))/(sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))],
[1/(a+8*x-8*x^2+4*x^3-x^4)^(1/2),x,3,sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticF(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*sqrt(1+sqrt(4+a))/(sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))],
[1/(a+8*x-8*x^2+4*x^3-x^4)^(3/2),x,7,1/2*(5+a+(-1+x)^2)*(-1+x)/((12+7*a+a^2)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4))-1/2*(-1+x)*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))/((12+7*a+a^2)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4))+1/2*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticF(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*sqrt(1+sqrt(4+a))/((4+a)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))+1/2*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticE(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))*sqrt(1+sqrt(4+a))/((12+7*a+a^2)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))],
[1/(a+8*x-8*x^2+4*x^3-x^4)^(5/2),x,8,1/6*(5+a+(-1+x)^2)*(-1+x)/((12+7*a+a^2)*(3+a-2*(-1+x)^2-(-1+x)^4)^(3/2))+1/12*(104+47*a+5*a^2+4*(7+2*a)*(-1+x)^2)*(-1+x)/((12+7*a+a^2)^2*sqrt(3+a-2*(-1+x)^2-(-1+x)^4))-1/3*(7+2*a)*(-1+x)*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))/((12+7*a+a^2)^2*sqrt(3+a-2*(-1+x)^2-(-1+x)^4))+1/12*(16+5*a)*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticF(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*sqrt(1+sqrt(4+a))/((3+a)*(4+a)^2*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))+1/3*(7+2*a)*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticE(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))*sqrt(1+sqrt(4+a))/((12+7*a+a^2)^2*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))],
[x*(a+8*x-8*x^2+4*x^3-x^4)^(3/2),x,14,1/8*(1+(-1+x)^2)*(3+a-2*(-1+x)^2-(-1+x)^4)^(3/2)+1/7*(3+a-2*(-1+x)^2-(-1+x)^4)^(3/2)*(-1+x)+3/16*(4+a)^2*arctan((1+(-1+x)^2)/sqrt(3+a-2*(-1+x)^2-(-1+x)^4))-16/35*(7+2*a)*(-1+x)*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))/sqrt(3+a-2*(-1+x)^2-(-1+x)^4)+3/16*(4+a)*(1+(-1+x)^2)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)+2/35*(13+5*a-3*(-1+x)^2)*(-1+x)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)+4/35*(3+a)*(16+5*a)*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticF(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*sqrt(1+sqrt(4+a))/(sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))+16/35*(7+2*a)*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticE(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))*sqrt(1+sqrt(4+a))/(sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))],
[x*(a+8*x-8*x^2+4*x^3-x^4)^(1/2),x,12,1/4*(4+a)*arctan((1+(-1+x)^2)/sqrt(3+a-2*(-1+x)^2-(-1+x)^4))-2/3*(-1+x)*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))/sqrt(3+a-2*(-1+x)^2-(-1+x)^4)+1/4*(1+(-1+x)^2)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)+1/3*(-1+x)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)+2/3*(3+a)*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticF(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*sqrt(1+sqrt(4+a))/(sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))+2/3*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticE(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))*sqrt(1+sqrt(4+a))/(sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))],
[x/(a+8*x-8*x^2+4*x^3-x^4)^(1/2),x,7,1/2*arctan((1+(-1+x)^2)/sqrt(3+a-2*(-1+x)^2-(-1+x)^4))+sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticF(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*sqrt(1+sqrt(4+a))/(sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))],
[x/(a+8*x-8*x^2+4*x^3-x^4)^(3/2),x,10,1/2*(1+(-1+x)^2)/((4+a)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4))+1/2*(5+a+(-1+x)^2)*(-1+x)/((12+7*a+a^2)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4))-1/2*(-1+x)*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))/((12+7*a+a^2)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4))+1/2*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticF(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*sqrt(1+sqrt(4+a))/((4+a)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))+1/2*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticE(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))*sqrt(1+sqrt(4+a))/((12+7*a+a^2)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))],
[x/(a+8*x-8*x^2+4*x^3-x^4)^(5/2),x,12,1/6*(1+(-1+x)^2)/((4+a)*(3+a-2*(-1+x)^2-(-1+x)^4)^(3/2))+1/6*(5+a+(-1+x)^2)*(-1+x)/((12+7*a+a^2)*(3+a-2*(-1+x)^2-(-1+x)^4)^(3/2))+1/3*(1+(-1+x)^2)/((4+a)^2*sqrt(3+a-2*(-1+x)^2-(-1+x)^4))+1/12*(104+47*a+5*a^2+4*(7+2*a)*(-1+x)^2)*(-1+x)/((12+7*a+a^2)^2*sqrt(3+a-2*(-1+x)^2-(-1+x)^4))-1/3*(7+2*a)*(-1+x)*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))/((12+7*a+a^2)^2*sqrt(3+a-2*(-1+x)^2-(-1+x)^4))+1/12*(16+5*a)*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticF(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*sqrt(1+sqrt(4+a))/((3+a)*(4+a)^2*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))+1/3*(7+2*a)*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticE(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))*sqrt(1+sqrt(4+a))/((12+7*a+a^2)^2*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))],
[x^2*(a+8*x-8*x^2+4*x^3-x^4)^(3/2),x,15,1/4*(1+(-1+x)^2)*(3+a-2*(-1+x)^2-(-1+x)^4)^(3/2)+1/63*(15+7*(-1+x)^2)*(3+a-2*(-1+x)^2-(-1+x)^4)^(3/2)*(-1+x)+3/8*(4+a)^2*arctan((1+(-1+x)^2)/sqrt(3+a-2*(-1+x)^2-(-1+x)^4))+4/315*(140+111*a+21*a^2)*(-1+x)*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))/sqrt(3+a-2*(-1+x)^2-(-1+x)^4)+3/8*(4+a)*(1+(-1+x)^2)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)+2/315*(2*(80+27*a)+3*(20+7*a)*(-1+x)^2)*(-1+x)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)+4/315*(3+a)*(100+33*a)*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticF(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*sqrt(1+sqrt(4+a))/(sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))-4/315*(140+111*a+21*a^2)*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticE(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))*sqrt(1+sqrt(4+a))/(sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))],
[x^2*(a+8*x-8*x^2+4*x^3-x^4)^(1/2),x,13,1/2*(4+a)*arctan((1+(-1+x)^2)/sqrt(3+a-2*(-1+x)^2-(-1+x)^4))+2/15*(8+3*a)*(-1+x)*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))/sqrt(3+a-2*(-1+x)^2-(-1+x)^4)+1/2*(1+(-1+x)^2)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)+1/15*(7+3*(-1+x)^2)*(-1+x)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)+8/15*(3+a)*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticF(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*sqrt(1+sqrt(4+a))/(sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))-2/15*(8+3*a)*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticE(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))*sqrt(1+sqrt(4+a))/(sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))],
[x^2/(a+8*x-8*x^2+4*x^3-x^4)^(1/2),x,11,arctan((1+(-1+x)^2)/sqrt(3+a-2*(-1+x)^2-(-1+x)^4))+(-1+x)*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))/sqrt(3+a-2*(-1+x)^2-(-1+x)^4)+sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticF(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*sqrt(1+sqrt(4+a))/(sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))-sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticE(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))*sqrt(1+sqrt(4+a))/(sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))],
[x^2/(a+8*x-8*x^2+4*x^3-x^4)^(3/2),x,10,(1+(-1+x)^2)/((4+a)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4))+1/2*(4+a)*(2+(-1+x)^2)*(-1+x)/((12+7*a+a^2)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4))-1/2*(-1+x)*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))/((3+a)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4))+1/2*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticE(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))*sqrt(1+sqrt(4+a))/((3+a)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))],
[x^2/(a+8*x-8*x^2+4*x^3-x^4)^(5/2),x,13,1/3*(1+(-1+x)^2)/((4+a)*(3+a-2*(-1+x)^2-(-1+x)^4)^(3/2))+1/6*(4+a)*(2+(-1+x)^2)*(-1+x)/((12+7*a+a^2)*(3+a-2*(-1+x)^2-(-1+x)^4)^(3/2))+2/3*(1+(-1+x)^2)/((4+a)^2*sqrt(3+a-2*(-1+x)^2-(-1+x)^4))+1/12*(29+7*a+(13+3*a)*(-1+x)^2)*(-1+x)/((3+a)^2*(4+a)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4))-1/12*(13+3*a)*(-1+x)*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))/((3+a)^2*(4+a)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4))+1/12*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticF(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*sqrt(1+sqrt(4+a))/((12+7*a+a^2)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))+1/12*(13+3*a)*sqrt(cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))^2)/cos(arctan((-1+x)/sqrt(1+sqrt(4+a))))*EllipticE(sin(arctan((-1+x)/sqrt(1+sqrt(4+a)))),sqrt(-2*sqrt(4+a)/(1-sqrt(4+a))))*(1+(-1+x)^2/(1-sqrt(4+a)))*(1-sqrt(4+a))*sqrt(1+sqrt(4+a))/((3+a)^2*(4+a)*sqrt(3+a-2*(-1+x)^2-(-1+x)^4)*sqrt((1+(-1+x)^2/(1-sqrt(4+a)))/(1+(-1+x)^2/(1+sqrt(4+a)))))],

# Integrands of the form F[a + b x + c x^2 + d x^3 + e x^4, x] when b^3 - 4 a b c + 8 a^2 d=0

# 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)^(1/2), x, 6, 0} 
[1/(8+8*x-x^3+8*x^4)^(1/2),x,4,-1/8*x^2*sqrt(cos(2*arctan((4+x)/(29^(1/4)*x*sqrt(3))))^2)/cos(2*arctan((4+x)/(29^(1/4)*x*sqrt(3))))*EllipticF(sin(2*arctan((4+x)/(29^(1/4)*x*sqrt(3)))),sqrt(1/58*(29+sqrt(29))))*(87+(4+x)^2*sqrt(29)/x^2)*sqrt((261-6*(1+4/x)^2+(1+4/x)^4)/(87+(4+x)^2*sqrt(29)/x^2)^2)/(29^(1/4)*sqrt(3)*sqrt(8+8*x-x^3+8*x^4))],
[1/(8+8*x-x^3+8*x^4)^(3/2),x,10,-1/1008*(66-(1+4/x)^2)*x^2/sqrt(8+8*x-x^3+8*x^4)+1/12528*(216-7*(1+4/x)^2)*(1+4/x)*x^2/sqrt(8+8*x-x^3+8*x^4)+7/432*(261-6*(1+4/x)^2+(1+4/x)^4)*(1+4/x)*x^2/(sqrt(29)*(87+(4+x)^2*sqrt(29)/x^2)*sqrt(8+8*x-x^3+8*x^4))-7/144*x^2*sqrt(cos(2*arctan((4+x)/(29^(1/4)*x*sqrt(3))))^2)/cos(2*arctan((4+x)/(29^(1/4)*x*sqrt(3))))*EllipticE(sin(2*arctan((4+x)/(29^(1/4)*x*sqrt(3)))),sqrt(1/58*(29+sqrt(29))))*(87+(4+x)^2*sqrt(29)/x^2)*sqrt((261-6*(1+4/x)^2+(1+4/x)^4)/(87+(4+x)^2*sqrt(29)/x^2)^2)/(29^(3/4)*sqrt(3)*sqrt(8+8*x-x^3+8*x^4))+1/576*x^2*sqrt(cos(2*arctan((4+x)/(29^(1/4)*x*sqrt(3))))^2)/cos(2*arctan((4+x)/(29^(1/4)*x*sqrt(3))))*EllipticF(sin(2*arctan((4+x)/(29^(1/4)*x*sqrt(3)))),sqrt(1/58*(29+sqrt(29))))*(14-5*sqrt(29))*(87+(4+x)^2*sqrt(29)/x^2)*sqrt((261-6*(1+4/x)^2+(1+4/x)^4)/(87+(4+x)^2*sqrt(29)/x^2)^2)/(29^(3/4)*sqrt(3)*sqrt(8+8*x-x^3+8*x^4))],

# 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)^(1/2), x, 0, 0} 
[1/(1+4*x+4*x^2+4*x^4)^(1/2),x,3,-1/2*x^2*sqrt(cos(2*arctan((1+1/x)/5^(1/4)))^2)/cos(2*arctan((1+1/x)/5^(1/4)))*EllipticF(sin(2*arctan((1+1/x)/5^(1/4))),sqrt(1/10*(5+sqrt(5))))*((1+1/x)^2+sqrt(5))*sqrt((5-2*(1+1/x)^2+(1+1/x)^4)/((1+1/x)^2+sqrt(5))^2)/(5^(1/4)*sqrt(1+4*x+4*x^2+4*x^4))],
[1/(1+4*x+4*x^2+4*x^4)^(3/2),x,9,-(3-(1+1/x)^2)*x^2/sqrt(1+4*x+4*x^2+4*x^4)+1/10*(13-9*(1+1/x)^2)*(1+1/x)*x^2/sqrt(1+4*x+4*x^2+4*x^4)+9/10*(5-2*(1+1/x)^2+(1+1/x)^4)*(1+1/x)*x^2/(((1+1/x)^2+sqrt(5))*sqrt(1+4*x+4*x^2+4*x^4))-9/2*x^2*sqrt(cos(2*arctan((1+1/x)/5^(1/4)))^2)/cos(2*arctan((1+1/x)/5^(1/4)))*EllipticE(sin(2*arctan((1+1/x)/5^(1/4))),sqrt(1/10*(5+sqrt(5))))*((1+1/x)^2+sqrt(5))*sqrt((5-2*(1+1/x)^2+(1+1/x)^4)/((1+1/x)^2+sqrt(5))^2)/(5^(3/4)*sqrt(1+4*x+4*x^2+4*x^4))+3/4*x^2*sqrt(cos(2*arctan((1+1/x)/5^(1/4)))^2)/cos(2*arctan((1+1/x)/5^(1/4)))*EllipticF(sin(2*arctan((1+1/x)/5^(1/4))),sqrt(1/10*(5+sqrt(5))))*(3-sqrt(5))*((1+1/x)^2+sqrt(5))*sqrt((5-2*(1+1/x)^2+(1+1/x)^4)/((1+1/x)^2+sqrt(5))^2)/(5^(3/4)*sqrt(1+4*x+4*x^2+4*x^4))],

# 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)^(1/2), x, 0, 0} 
[1/(8+24*x+8*x^2-15*x^3+8*x^4)^(1/2),x,4,-1/8*x^2*sqrt(cos(2*arctan((4+3*x)/(517^(1/4)*x)))^2)/cos(2*arctan((4+3*x)/(517^(1/4)*x)))*EllipticF(sin(2*arctan((4+3*x)/(517^(1/4)*x))),sqrt(1/1034*(517+19*sqrt(517))))*((3+4/x)^2+sqrt(517))*sqrt((517-38*(3+4/x)^2+(3+4/x)^4)/((3+4/x)^2+sqrt(517))^2)/(517^(1/4)*sqrt(8+24*x+8*x^2-15*x^3+8*x^4))],
[1/(8+24*x+8*x^2-15*x^3+8*x^4)^(3/2),x,10,-1/208*(172-7*(3+4/x)^2)*x^2/sqrt(8+24*x+8*x^2-15*x^3+8*x^4)+1/322608*(50896-2455*(3+4/x)^2)*(3+4/x)*x^2/sqrt(8+24*x+8*x^2-15*x^3+8*x^4)+2455/322608*(517-38*(3+4/x)^2+(3+4/x)^4)*(3+4/x)*x^2/(((3+4/x)^2+sqrt(517))*sqrt(8+24*x+8*x^2-15*x^3+8*x^4))-2455/624*x^2*sqrt(cos(2*arctan((4+3*x)/(517^(1/4)*x)))^2)/cos(2*arctan((4+3*x)/(517^(1/4)*x)))*EllipticE(sin(2*arctan((4+3*x)/(517^(1/4)*x))),sqrt(1/1034*(517+19*sqrt(517))))*((3+4/x)^2+sqrt(517))*sqrt((517-38*(3+4/x)^2+(3+4/x)^4)/((3+4/x)^2+sqrt(517))^2)/(517^(3/4)*sqrt(8+24*x+8*x^2-15*x^3+8*x^4))+1/2496*x^2*sqrt(cos(2*arctan((4+3*x)/(517^(1/4)*x)))^2)/cos(2*arctan((4+3*x)/(517^(1/4)*x)))*EllipticF(sin(2*arctan((4+3*x)/(517^(1/4)*x))),sqrt(1/1034*(517+19*sqrt(517))))*(4910-203*sqrt(517))*((3+4/x)^2+sqrt(517))*sqrt((517-38*(3+4/x)^2+(3+4/x)^4)/((3+4/x)^2+sqrt(517))^2)/(517^(3/4)*sqrt(8+24*x+8*x^2-15*x^3+8*x^4))],
[1/(8+24*x+8*x^2-15*x^3+8*x^4)^(5/2),x,12,-1/97344*(124415-6308*(3+4/x)^2)*x^2/sqrt(8+24*x+8*x^2-15*x^3+8*x^4)-1/624*(64489-1399*(3+4/x)^2)*x^2/((517-38*(3+4/x)^2+(3+4/x)^4)*sqrt(8+24*x+8*x^2-15*x^3+8*x^4))+1/78056941248*(18932921731-1086525994*(3+4/x)^2)*(3+4/x)*x^2/sqrt(8+24*x+8*x^2-15*x^3+8*x^4)+1/483912*(11921698-359497*(3+4/x)^2)*(3+4/x)*x^2/((517-38*(3+4/x)^2+(3+4/x)^4)*sqrt(8+24*x+8*x^2-15*x^3+8*x^4))+543262997/39028470624*(517-38*(3+4/x)^2+(3+4/x)^4)*(3+4/x)*x^2/(((3+4/x)^2+sqrt(517))*sqrt(8+24*x+8*x^2-15*x^3+8*x^4))-543262997/75490272*x^2*sqrt(cos(2*arctan((4+3*x)/(517^(1/4)*x)))^2)/cos(2*arctan((4+3*x)/(517^(1/4)*x)))*EllipticE(sin(2*arctan((4+3*x)/(517^(1/4)*x))),sqrt(1/1034*(517+19*sqrt(517))))*((3+4/x)^2+sqrt(517))*sqrt((517-38*(3+4/x)^2+(3+4/x)^4)/((3+4/x)^2+sqrt(517))^2)/(517^(3/4)*sqrt(8+24*x+8*x^2-15*x^3+8*x^4))+1/1207844352*x^2*sqrt(cos(2*arctan((4+3*x)/(517^(1/4)*x)))^2)/cos(2*arctan((4+3*x)/(517^(1/4)*x)))*EllipticF(sin(2*arctan((4+3*x)/(517^(1/4)*x))),sqrt(1/1034*(517+19*sqrt(517))))*(4346103976-175318963*sqrt(517))*((3+4/x)^2+sqrt(517))*sqrt((517-38*(3+4/x)^2+(3+4/x)^4)/((3+4/x)^2+sqrt(517))^2)/(517^(3/4)*sqrt(8+24*x+8*x^2-15*x^3+8*x^4))],

#  {(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4)^(1/2), x, 6, 0} 
[1/(9-6*x-44*x^2+15*x^3+3*x^4)^(1/2),x,4,-1/12*x^2*sqrt(cos(2*arctan((6-x)/(613^(1/4)*x)))^2)/cos(2*arctan((6-x)/(613^(1/4)*x)))*EllipticF(sin(2*arctan((6-x)/(613^(1/4)*x))),sqrt(1/1226*(613+91*sqrt(613))))*((6-x)^2/x^2+sqrt(613))*sqrt((613-182*(1+(-6)/x)^2+(-1+6/x)^4)/((6-x)^2/x^2+sqrt(613))^2)/(613^(1/4)*sqrt(9-6*x-44*x^2+15*x^3+3*x^4))],
[1/(9-6*x-44*x^2+15*x^3+3*x^4)^(3/2),x,10,-1/51759*(176-23*(1+(-6)/x)^2)*x^2/sqrt(9-6*x-44*x^2+15*x^3+3*x^4)+1/31728267*(45401-3722*(1+(-6)/x)^2)*(1+(-6)/x)*x^2/sqrt(9-6*x-44*x^2+15*x^3+3*x^4)+3722/31728267*(613-182*(1+(-6)/x)^2+(-1+6/x)^4)*(1+(-6)/x)*x^2/(((6-x)^2/x^2+sqrt(613))*sqrt(9-6*x-44*x^2+15*x^3+3*x^4))+3722/51759*x^2*sqrt(cos(2*arctan((6-x)/(613^(1/4)*x)))^2)/cos(2*arctan((6-x)/(613^(1/4)*x)))*EllipticE(sin(2*arctan((6-x)/(613^(1/4)*x))),sqrt(1/1226*(613+91*sqrt(613))))*((6-x)^2/x^2+sqrt(613))*sqrt((613-182*(1+(-6)/x)^2+(-1+6/x)^4)/((6-x)^2/x^2+sqrt(613))^2)/(613^(3/4)*sqrt(9-6*x-44*x^2+15*x^3+3*x^4))-1/207036*x^2*sqrt(cos(2*arctan((6-x)/(613^(1/4)*x)))^2)/cos(2*arctan((6-x)/(613^(1/4)*x)))*EllipticF(sin(2*arctan((6-x)/(613^(1/4)*x))),sqrt(1/1226*(613+91*sqrt(613))))*(7444-145*sqrt(613))*((6-x)^2/x^2+sqrt(613))*sqrt((613-182*(1+(-6)/x)^2+(-1+6/x)^4)/((6-x)^2/x^2+sqrt(613))^2)/(613^(3/4)*sqrt(9-6*x-44*x^2+15*x^3+3*x^4))],

# Integrands requiring algebraic expansion
[(2*sqrt(3-x)+3/sqrt(1+x))^2/x,x,12,-4*x+12*arcsin(1/2*(1-x))+21*log(x)-9*log(1+x)-24*arctanh(sqrt(3)*sqrt(1+x)/sqrt(3-x))*sqrt(3)],
[(-1+x+x^2)/(1+sqrt(1+x^2)),x,14,(-1)/x-x-1/2*arcsinh(x)-log(1+sqrt(1+x^2))+sqrt(1+x^2)+sqrt(1+x^2)/x+1/2*x*sqrt(1+x^2)],
[(-1+x+x^2)/(1+x+sqrt(1+x^2)),x,12,1/12*(6*x^2+2*x^3-3*arcsinh(x)-6*log(1+sqrt(1+x^2))+(4-3*x-2*x^2)*sqrt(1+x^2)),1/2*x+1/2*x^2+1/6*x^3-1/6*(1+x^2)^(3/2)-1/4*arcsinh(x)+1/2*log(x+sqrt(1+x^2))-log(1+x+sqrt(1+x^2))-1/4*x*sqrt(1+x^2)+1/2/(x+sqrt(1+x^2))],
[(x+2*sqrt(-1+x))/(x*sqrt(-1+x)),x,2,2*log(x)+2*sqrt(-1+x)],

#  Positive integer powers of monomial sums 
[(a+b*x^(2/3)+c*sqrt(x))^2,x,4,a^2*x+4/3*a*c*x^(3/2)+6/5*a*b*x^(5/3)+1/2*c^2*x^2+12/13*b*c*x^(13/6)+3/7*b^2*x^(7/3)],
[(a+b*x^(2/3)+c*sqrt(x))^3,x,4,a^3*x+2*a^2*c*x^(3/2)+9/5*a^2*b*x^(5/3)+3/2*a*c^2*x^2+36/13*a*b*c*x^(13/6)+9/7*a*b^2*x^(7/3)+2/5*c^3*x^(5/2)+9/8*b*c^2*x^(8/3)+18/17*b^2*c*x^(17/6)+1/3*b^3*x^3],
[(-1+x^2)/(x^3*sqrt(a-b+b/x^2)),x,5,arctanh(sqrt(a-b*(1+(-1)/x^2))/sqrt(a-b))/sqrt(a-b)+sqrt(a-b*(1+(-1)/x^2))/b],
[(-1+x^2)/(x^3*sqrt(a+b*(-1+1/x^2))),x,6,arctanh(sqrt(a-b*(1+(-1)/x^2))/sqrt(a-b))/sqrt(a-b)+sqrt(a-b*(1+(-1)/x^2))/b],
[(1+x)/((4+x^2)*sqrt(9+x^2)),x,6,1/2*arctan(1/2*x*sqrt(5)/sqrt(9+x^2))/sqrt(5)-arctanh(sqrt(9+x^2)/sqrt(5))/sqrt(5)],

#  Checks to ensure that expansion occurs before substitution for fractional powers of linears: 
[x*(1+sqrt(1-x^2)),x,3,1/2*x^2-1/3*(1-x^2)^(3/2)],
[x*(1+sqrt(1-x)*sqrt(1+x)),x,3,1/2*x^2-1/3*(1-x^2)^(3/2)],
[x*(1+1/(sqrt(2+x)*sqrt(3+x))),x,5,1/2*x^2-5*arcsinh(sqrt(2+x))+sqrt(2+x)*sqrt(3+x)],
[(x-sqrt(x^6))/(x*(1-x^4)),x,9,1/2*arctan(x)+1/2*arctanh(x)+1/2*arctan(x)*sqrt(x^6)/x^3-1/2*arctanh(x)*sqrt(x^6)/x^3],
[(1-sqrt(x^6)/x)/(1-x^4),x,9,1/2*arctan(x)+1/2*arctanh(x)+1/2*arctan(x)*sqrt(x^6)/x^3-1/2*arctanh(x)*sqrt(x^6)/x^3],
[(x-sqrt(x^6))/(x-x^5),x,10,1/2*arctan(x)+1/2*arctanh(x)+1/2*arctan(x)*sqrt(x^6)/x^3-1/2*arctanh(x)*sqrt(x^6)/x^3],
[x/(x+sqrt(x^6)),x,11,1/2*arctan(x)+1/2*arctanh(x)+1/2*arctan(x)*sqrt(x^6)/x^3-1/2*arctanh(x)*sqrt(x^6)/x^3],
[(sqrt(x)-sqrt(x^3))/(x-x^3),x,12,arctan(sqrt(x))+arctanh(sqrt(x))+arctan(sqrt(x))*sqrt(x^3)/x^(3/2)-arctanh(sqrt(x))*sqrt(x^3)/x^(3/2)],
[1/(sqrt(x)+sqrt(x^3)),x,13,arctan(sqrt(x))+arctanh(sqrt(x))+arctan(sqrt(x))*sqrt(x^3)/x^(3/2)-arctanh(sqrt(x))*sqrt(x^3)/x^(3/2)],
[1/(sqrt(-1+x)+sqrt((-1+x)^3)),x,14,arctan(sqrt(-1+x))+arctanh(sqrt(-1+x))+arctan(sqrt(-1+x))*sqrt((-1+x)^3)/(-1+x)^(3/2)-arctanh(sqrt(-1+x))*sqrt((-1+x)^3)/(-1+x)^(3/2)],

#  Following integrands are equal. 
[(-3)/(4+5*x)^2+(-5-4*x)/((4+5*x)^2*sqrt(1-x^2)),x,2,3/5/(4+5*x)+sqrt(1-x^2)/(4+5*x)],
[(-5-4*x-3*sqrt(1-x^2))/((4+5*x)^2*sqrt(1-x^2)),x,8,3/5/(4+5*x)+sqrt(1-x^2)/(4+5*x)],
[1/(3*(1-x^2)+(-5-4*x)*sqrt(1-x^2)),x,16,3/5/(4+5*x)+sqrt(1-x^2)/(4+5*x)],
[1/(3-3*x^2-5*sqrt(1-x^2)-4*x*sqrt(1-x^2)),x,16,3/5/(4+5*x)+sqrt(1-x^2)/(4+5*x)],
[(-1+sqrt(1-x^2))/((2+x-2*sqrt(1-x^2))^2*sqrt(1-x^2)),x,31,3/5/(4+5*x)+sqrt(1-x^2)/(4+5*x)],
[(a+b*x^(-1+n))/(c*x+d*x^n),x,5,b*log(x)/d-(b*c-a*d)*log(d+c*x^(1-n))/(c*d*(1-n))],
[sqrt(1+2*x^2)/(1+sqrt(1+2*x^2)),x,6,(-1/2)/x+x-arcsinh(x*sqrt(2))/sqrt(2)+1/2*sqrt(1+2*x^2)/x],
[sqrt(-1+4*x^2)/(x+sqrt(-1+4*x^2)),x,8,4/3*x-1/3*arctanh(x*sqrt(3))/sqrt(3)+1/3*arctanh(sqrt(3)*sqrt(-1+4*x^2))/sqrt(3)-1/3*sqrt(-1+4*x^2)],
[(a+b*x+c*x^2)/((d+e*x)^3*sqrt(-1+x^2)),x,4,-1/2*(3*b*d*e-a*(2*d^2+e^2)-c*(d^2+2*e^2))*arctanh((e+d*x)/(sqrt(d^2-e^2)*sqrt(-1+x^2)))/(d^2-e^2)^(5/2)-1/2*(c*d^2-b*d*e+a*e^2)*sqrt(-1+x^2)/(e*(d^2-e^2)*(d+e*x)^2)+1/2*(c*(d^3-4*d*e^2)-e*(3*a*d*e-b*(d^2+2*e^2)))*sqrt(-1+x^2)/(e*(d^2-e^2)^2*(d+e*x))],

# Integrands requiring algebraic simplification

#  Following pairs or triples of integrands are equal. 
[(1+2*x^8)/(x*(1+x^8)^(3/2)),x,4,-1/4*arctanh(sqrt(1+x^8))+(-1/4)/sqrt(1+x^8)],
[(1+2*x^8)*sqrt(1+x^8)/(x+2*x^9+x^17),x,6,-1/4*arctanh(sqrt(1+x^8))+(-1/4)/sqrt(1+x^8)],
[1-9*x^2+x/sqrt(1-9*x^2),x,2,x-3*x^3-1/9*sqrt(1-9*x^2)],
[(x+(1-9*x^2)^(3/2))/sqrt(1-9*x^2),x,3,x-3*x^3-1/9*sqrt(1-9*x^2)],
[(x-3*sqrt(x))^(2/3)*(-3+2*sqrt(x))/sqrt(x),x,2,6/5*(x-3*sqrt(x))^(5/3)],
[(9+2*x-9*sqrt(x))/(x-3*sqrt(x))^(1/3),x,3,6/5*(x-3*sqrt(x))^(5/3)],
[1/sqrt(4-9*x^2),x,1,1/3*arcsin(3/2*x)],
[1/(sqrt(2-3*x)*sqrt(2+3*x)),x,2,1/3*arcsin(3/2*x)],
[1/sqrt((2-3*x)*(2+3*x)),x,2,1/3*arcsin(3/2*x)],
[1/sqrt(15-2*x-x^2),x,2,-arcsin(1/4*(-1-x))],
[1/(sqrt(3-x)*sqrt(5+x)),x,3,-arcsin(1/4*(-1-x))],
[1/sqrt((3-x)*(5+x)),x,3,-arcsin(1/4*(-1-x))],
[1/sqrt(-15-8*x-x^2),x,2,arcsin(4+x)],
[1/(sqrt(-3-x)*sqrt(5+x)),x,3,arcsin(4+x)],
[1/sqrt((-3-x)*(5+x)),x,3,arcsin(4+x)],
[1-sqrt(x),x,1,x-2/3*x^(3/2)],
[(1-x)/(1+sqrt(x)),x,4,x-2/3*x^(3/2)],
[sqrt(1/(1-x^2)),x,2,arcsin(x)*sqrt(1/(1-x^2))*sqrt(1-x^2)],
[sqrt((1+x^2)/(1-x^4)),x,3,arcsin(x)*sqrt(1/(1-x^2))*sqrt(1-x^2)],
[sqrt(1/(-1+x^2)),x,2,arcsin(x)*sqrt(1-x^2)*sqrt(1/(-1+x^2))],
[sqrt((1+x^2)/(-1+x^4)),x,3,arcsin(x)*sqrt(1-x^2)*sqrt(1/(-1+x^2))],

#  Following pairs of integrands are equal. 
[1/sqrt(1-x),x,1,-2*sqrt(1-x)],
[sqrt(1+x)/sqrt(1-x^2),x,2,-2*sqrt(1-x)],
[1/sqrt(1+x),x,1,2*sqrt(1+x)],
[sqrt(1-x)/sqrt(1-x^2),x,2,2*sqrt(1+x)],
[sqrt(1-x),x,1,-2/3*(1-x)^(3/2)],
[sqrt(1-x^2)/sqrt(1+x),x,2,-2/3*(1-x)^(3/2)],
[sqrt(1+x),x,1,2/3*(1+x)^(3/2)],
[sqrt(1-x^2)/sqrt(1-x),x,2,2/3*(1+x)^(3/2)],
[sqrt(2+3*x)/sqrt(1+x),x,3,-arcsinh(sqrt(2+3*x))/sqrt(3)+sqrt(1+x)*sqrt(2+3*x)],
[sqrt(1-x)*sqrt(2+3*x)/sqrt(1-x^2),x,4,-arcsinh(sqrt(2+3*x))/sqrt(3)+sqrt(1+x)*sqrt(2+3*x)],
[(1+x)^(3/2)/((1-x)^(3/2)*x),x,7,-arcsin(x)-arctanh(sqrt(1-x)*sqrt(1+x))+4*sqrt(1+x)/sqrt(1-x)],
[(1+x)^3/(x*(1-x^2)^(3/2)),x,6,-arcsin(x)-arctanh(sqrt(1-x^2))+4*(1+x)/sqrt(1-x^2)],
[(1+a*x)^(3/2)/(x*(1-a*x)^(3/2)),x,7,-arcsin(a*x)-arctanh(sqrt(1-a*x)*sqrt(1+a*x))+4*sqrt(1+a*x)/sqrt(1-a*x)],
[(1+a*x)^3/(x*(1-a^2*x^2)^(3/2)),x,6,-arcsin(a*x)-arctanh(sqrt(1-a^2*x^2))+4*(1+a*x)/sqrt(1-a^2*x^2)],

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

# Integrands requiring rationalization of denominator
[1/(x-sqrt(1+x^2)),x,4,-1/2*x^2-1/2*arcsinh(x)-1/2*x*sqrt(1+x^2)],
[1/(x-sqrt(1-x^2)),x,7,-1/2*arcsin(x)-1/2*arctanh(x/sqrt(1-x^2))+1/4*log(1-2*x^2)],
[1/(x-sqrt(1+2*x^2)),x,7,arctanh(x/sqrt(1+2*x^2))-1/2*log(1+x^2)-arcsinh(x*sqrt(2))*sqrt(2)],

#  Integrands are equal.  Denominators needs to be rationalized before expansion. 
[(2*x-x^3+x^2*sqrt(2-x^2))/(-2+2*x^2),x,10,-1/4*x^2-1/2*arctanh(x/sqrt(2-x^2))+1/4*log(1-x^2)+1/4*x*sqrt(2-x^2)],
[x*sqrt(2-x^2)/(x-sqrt(2-x^2)),x,12,-1/4*x^2-1/2*arctanh(x/sqrt(2-x^2))+1/4*log(1-x)+1/4*log(1+x)+1/4*x*sqrt(2-x^2)],
[x/(-x+sqrt(2*x-x^2)),x,5,-1/2*x+1/2*arctanh(sqrt(2*x-x^2))-1/2*log(1-x)-1/2*sqrt(2*x-x^2)],
[(x+sqrt(2*x-x^2))/(2-2*x),x,7,-1/2*x+1/2*arctanh(sqrt(2*x-x^2))-1/2*log(1-x)-1/2*sqrt(2*x-x^2)],
[(x+sqrt(2-x)*sqrt(x))/(2-2*x),x,9,-1/2*x+1/2*arctanh(sqrt(2*x-x^2))-1/2*log(1-x)-1/2*sqrt(2*x-x^2)],
[sqrt(x)/(sqrt(2-x)-sqrt(x)),x,7,-1/2*x+1/2*arctanh(sqrt(2-x)*sqrt(x))-1/2*log(1-x)-1/2*sqrt(2-x)*sqrt(x)],

# Integrands requiring piecewise constant extraction
[1/((1+x)*(-1+x^2))^(2/3),x,3,-3/2*(1-x^2)/(-(1+x)*(1-x^2))^(2/3),-3/2*(1-x)*(1+x)/(-1-x+x^2+x^3)^(2/3)],
[(-1+x^2)/((1+x^2)*sqrt(x*(1+x^2))),x,2,-2*x/sqrt(x*(1+x^2))],
[(-1+x^2)/((1+x^2)*sqrt(x+x^3)),x,2,-2*x/sqrt(x+x^3)],
[sqrt((-1+x^2)^2/(x*(1+x^2)))/(1+x^2),x,2,2*x*sqrt((1-x^2)^2/(x*(1+x^2)))/(1-x^2)],
[sqrt((-1+x^2)^2/(x+x^3))/(1+x^2),x,3,2*x*sqrt((1-x^2)^2/(x+x^3))/(1-x^2)],
[1/(sqrt(a+b/x^2)*sqrt(c+d*x^2)),x,5,arctanh(sqrt(d)*sqrt(b+a*x^2)/(sqrt(a)*sqrt(c+d*x^2)))*sqrt(b+a*x^2)/(x*sqrt(a)*sqrt(d)*sqrt(a+b/x^2))],
[sqrt(-2*x^2+x^4)/((-1+x^2)*(2+x^2)),x,7,2/3*arctan(1/2*sqrt(-2+x^2))*sqrt(-2*x^2+x^4)/(x*sqrt(-2+x^2))-1/3*arctan(sqrt(-2+x^2))*sqrt(-2*x^2+x^4)/(x*sqrt(-2+x^2))],
[sqrt(1+(-1)/(-1+x^2)^2)/(2-x^2),x,13,(1-x^2)*arctan(sqrt(-2+x^2))*sqrt(1+(-1)/(1-x^2)^2)/(x*sqrt(-2+x^2)),(1-x^2)*arctan(sqrt(-2+x^2))*sqrt(-2*x^2+x^4)*sqrt(1+(-1)/(1-x^2)^2)/(x*sqrt(-2+x^2)*sqrt(-1+(-1+x^2)^2))],
[sqrt((-2*x^2+x^4)/(-1+x^2)^2)/(2+x^2),x,8,-2/3*(1-x^2)*arctan(1/2*sqrt(-2+x^2))*sqrt((-2*x^2+x^4)/(1-x^2)^2)/(x*sqrt(-2+x^2))+1/3*(1-x^2)*arctan(sqrt(-2+x^2))*sqrt((-2*x^2+x^4)/(1-x^2)^2)/(x*sqrt(-2+x^2))],
[(1+2*x/(1+x^2))^(5/2),x,6,-4/3*(1-2*x)*(1+x)*sqrt(1+2*x/(1+x^2))-1/3*(1-x)*(1+x)^3*sqrt(1+2*x/(1+x^2))/(1+x^2)-(4+3*x)*(1+x^2)*sqrt(1+2*x/(1+x^2))/(1+x)+5*arcsinh(x)*sqrt(1+x^2)*sqrt(1+2*x/(1+x^2))/(1+x)],
[(1+2*x/(1+x^2))^(3/2),x,6,-(1-x)*(1+x)*sqrt(1+2*x/(1+x^2))-x*(1+x^2)*sqrt(1+2*x/(1+x^2))/(1+x)+3*arcsinh(x)*sqrt(1+x^2)*sqrt(1+2*x/(1+x^2))/(1+x)],
[(1+2*x/(1+x^2))^(1/2),x,4,(1+x^2)*sqrt(1+2*x/(1+x^2))/(1+x)+arcsinh(x)*sqrt(1+x^2)*sqrt(1+2*x/(1+x^2))/(1+x)],
[1/(1+2*x/(1+x^2))^(1/2),x,7,(1+x)/sqrt(1+2*x/(1+x^2))-(1+x)*arcsinh(x)/(sqrt(1+x^2)*sqrt(1+2*x/(1+x^2)))-(1+x)*arctanh((1-x)/(sqrt(2)*sqrt(1+x^2)))*sqrt(2)/(sqrt(1+x^2)*sqrt(1+2*x/(1+x^2)))],
[1/(1+2*x/(1+x^2))^(3/2),x,8,3/2*(2+x)/sqrt(1+2*x/(1+x^2))+1/2*(-1-x^2)/((1+x)*sqrt(1+2*x/(1+x^2)))-3*(1+x)*arcsinh(x)/(sqrt(1+x^2)*sqrt(1+2*x/(1+x^2)))-9/2*(1+x)*arctanh((1-x)/(sqrt(2)*sqrt(1+x^2)))/(sqrt(2)*sqrt(1+x^2)*sqrt(1+2*x/(1+x^2)))],
[sqrt(1+2*x/(1+x^2))/(1+x^2),x,3,-(1-x)*sqrt(1+2*x/(1+x^2))/(1+x)],

#  Piecewise constant extraction and simplification caused infinite recursion prior to version 4.89. 
[F(x)*sqrt(x-x^2),x,0,CannotIntegrate(F(x)*sqrt(x-x^2),x)],
[F(x)/sqrt(x-x^2),x,0,CannotIntegrate(F(x)/sqrt(x-x^2),x)],
[F(x)*sqrt(1-x)*sqrt(x),x,1,CannotIntegrate(F(x)*sqrt(x-x^2),x)],
[F(x)/(sqrt(1-x)*sqrt(x)),x,1,CannotIntegrate(F(x)/sqrt(x-x^2),x)],

# Integrands involving roots of improper binomials

#  Integrands of the form F[x^m*(a+b*x^n)^p] where m==-n*p 
[F((a+b*x)/x),x,1,CannotIntegrate(F(b+a/x),x)],
[F((a+b*x^2)/x^2),x,1,CannotIntegrate(F(b+a/x^2),x)],
[F(x/(a+b*x)),x,0,CannotIntegrate(F(x/(a+b*x)),x)],
[F(x^2/(a+b*x^2)),x,0,CannotIntegrate(F(x^2/(a+b*x^2)),x)],
[F(x^2/(a+b*x)^2),x,0,CannotIntegrate(F(x^2/(a+b*x)^2),x)],
[F(x^4/(a+b*x^2)^2),x,0,CannotIntegrate(F(x^4/(a+b*x^2)^2),x)],

# Integrands involving nested radicals
[sqrt(b*x^2+sqrt(a+b^2*x^4))/sqrt(a+b^2*x^4),x,2,arctanh(x*sqrt(2)*sqrt(b)/sqrt(b*x^2+sqrt(a+b^2*x^4)))/(sqrt(2)*sqrt(b))],
[sqrt(-b*x^2+sqrt(a+b^2*x^4))/sqrt(a+b^2*x^4),x,2,arctan(x*sqrt(2)*sqrt(b)/sqrt(-b*x^2+sqrt(a+b^2*x^4)))/(sqrt(2)*sqrt(b))],
[sqrt(2*x^2+sqrt(3+4*x^4))/((c+d*x)*sqrt(3+4*x^4)),x,5,(1/2-1/2*I)*arctan((2*I*c*x+d*sqrt(3))/(sqrt(-2*I*x^2+sqrt(3))*sqrt(2*I*c^2-d^2*sqrt(3))))/sqrt(2*I*c^2-d^2*sqrt(3))+(-1/2-1/2*I)*arctanh((-2*I*c*x+d*sqrt(3))/(sqrt(2*I*x^2+sqrt(3))*sqrt(2*I*c^2+d^2*sqrt(3))))/sqrt(2*I*c^2+d^2*sqrt(3))],
[sqrt(2*x^2+sqrt(3+4*x^4))/((c+d*x)^2*sqrt(3+4*x^4)),x,7,(1+I)*c*arctan((2*I*c*x+d*sqrt(3))/(sqrt(-2*I*x^2+sqrt(3))*sqrt(2*I*c^2-d^2*sqrt(3))))/(2*I*c^2-d^2*sqrt(3))^(3/2)+(1-I)*c*arctanh((-2*I*c*x+d*sqrt(3))/(sqrt(2*I*x^2+sqrt(3))*sqrt(2*I*c^2+d^2*sqrt(3))))/(2*I*c^2+d^2*sqrt(3))^(3/2)+(1/2-1/2*I)*d*sqrt(-2*I*x^2+sqrt(3))/((c+d*x)*(2*I*c^2-d^2*sqrt(3)))+(-1/2-1/2*I)*d*sqrt(2*I*x^2+sqrt(3))/((c+d*x)*(2*I*c^2+d^2*sqrt(3)))],

# Miscellaneous algebraic function integrands
[(-4+x)/((1+x^(1/3))*sqrt(x)),x,6,-30*x^(1/6)-6/5*x^(5/6)+6/7*x^(7/6)+30*arctan(x^(1/6))+2*sqrt(x)],
[(1+sqrt(x))/(x^(5/6)+x^(7/6)),x,7,3*x^(1/3)+6*arctan(x^(1/6))-3*log(1+x^(1/3))],
[(1+sqrt(x))/((1+x^(1/3))*sqrt(x)),x,8,6*x^(1/6)-3*x^(1/3)+3/2*x^(2/3)-6*arctan(x^(1/6))+3*log(1+x^(1/3))],
[sqrt(2+b/x^2)/(b+2*x^2),x,3,-arccsch(x*sqrt(2)/sqrt(b))/sqrt(b)],
[sqrt(2-b/x^2)/(-b+2*x^2),x,3,-arccsc(x*sqrt(2)/sqrt(b))/sqrt(b)],
[sqrt(a+c/x^2)/(d+e*x),x,11,arctanh(sqrt(a+c/x^2)/sqrt(a))*sqrt(a)/e-arctanh(sqrt(c)/(x*sqrt(a+c/x^2)))*sqrt(c)/d-arctanh((a*d-c*e/x)/(sqrt(a*d^2+c*e^2)*sqrt(a+c/x^2)))*sqrt(a*d^2+c*e^2)/(d*e)],
[sqrt(a+c/x^2+b/x)/(d+e*x),x,10,arctanh(1/2*(2*a+b/x)/(sqrt(a)*sqrt(a+c/x^2+b/x)))*sqrt(a)/e-arctanh(1/2*(b+2*c/x)/(sqrt(c)*sqrt(a+c/x^2+b/x)))*sqrt(c)/d-arctanh(1/2*(2*a*d-b*e+(b*d-2*c*e)/x)/(sqrt(a*d^2-e*(b*d-c*e))*sqrt(a+c/x^2+b/x)))*sqrt(a*d^2-e*(b*d-c*e))/(d*e)],
[(x^(1/6)+(x^3)^(1/5))/sqrt(x),x,4,3/2*x^(2/3)+10/11*(x^3)^(1/5)*sqrt(x)],
[(2+x)/sqrt(4*x-x^2),x,3,-4*arcsin(1-1/2*x)-sqrt(4*x-x^2)],
[(3+x)/(6*x+x^2)^(1/3),x,1,3/4*(6*x+x^2)^(2/3)],
[(4+x)/(6*x-x^2)^(3/2),x,1,1/9*(-12+7*x)/sqrt(6*x-x^2)],
[1/((1+x)*sqrt(2*x+x^2)),x,2,arctan(sqrt(2*x+x^2))],
[1/((1+2*x)*sqrt(x+x^2)),x,2,arctan(2*sqrt(x+x^2))],
[(-1+x)/sqrt(2*x-x^2),x,1,-sqrt(2*x-x^2)],
[sqrt(x-x^2)/(1+x),x,6,-3/2*arcsin(1-2*x)+arctan(1/2*(1-3*x)/(sqrt(2)*sqrt(x-x^2)))*sqrt(2)+sqrt(x-x^2)],
[sqrt(x^(1/4)+x),x,5,-1/3*arctanh(sqrt(x)/sqrt(x^(1/4)+x))+1/3*x^(1/4)*sqrt(x^(1/4)+x)+2/3*x*sqrt(x^(1/4)+x)],
[sqrt(x+x^(3/2)),x,3,32/105*(x+x^(3/2))^(3/2)/x^(3/2)-16/35*(x+x^(3/2))^(3/2)/x+4/7*(x+x^(3/2))^(3/2)/sqrt(x)],
[x*sqrt(x+x^(3/2)),x,5,-32/99*(x+x^(3/2))^(3/2)+512/3465*(x+x^(3/2))^(3/2)/x^(3/2)-256/1155*(x+x^(3/2))^(3/2)/x+64/231*(x+x^(3/2))^(3/2)/sqrt(x)+4/11*(x+x^(3/2))^(3/2)*sqrt(x)],
[(1-x^2)*sqrt(1/(2-x^2)),x,2,1/2*x/sqrt(1/(2-x^2))],
[sqrt(x^2+x^3-x^4),x,5,-1/8*(1-2*x)*sqrt(x^2+x^3-x^4)/x-1/3*(1+x-x^2)*sqrt(x^2+x^3-x^4)/x-5/16*arcsin((1-2*x)/sqrt(5))*sqrt(x^2+x^3-x^4)/(x*sqrt(1+x-x^2))],
[1/sqrt((a^2+x^2)^3),x,2,x*(a^2+x^2)/(a^2*sqrt((a^2+x^2)^3))],
[sqrt(x)/(1+x+sqrt(x)),x,6,-log(1+x+sqrt(x))-2*arctan((1+2*sqrt(x))/sqrt(3))/sqrt(3)+2*sqrt(x)],
[x/(1+x+sqrt(x)),x,5,x+4*arctan((1+2*sqrt(x))/sqrt(3))/sqrt(3)-2*sqrt(x)],
[1/(sqrt(x)*(1+x+sqrt(x))^(7/2)),x,4,4/15*(1+2*sqrt(x))/(1+x+sqrt(x))^(5/2)+64/135*(1+2*sqrt(x))/(1+x+sqrt(x))^(3/2)+512/405*(1+2*sqrt(x))/sqrt(1+x+sqrt(x))],

#  {Sqrt[1+x^2]/(1-x^3), x, 0} 
[(-1+x)/(1+sqrt(1+x^2)),x,10,(-1)/x-arcsinh(x)-log(1+sqrt(1+x^2))+sqrt(1+x^2)+sqrt(1+x^2)/x],
[1/((1+x)^(2/3)*(-1+x^2)^(2/3)),x,1,3/2*(-1+x^2)^(1/3)/(1+x)^(2/3)],
[(1-x^6)^(2/3)+(1-x^6)^(2/3)/x^6,x,-3,-1/5*(1-x^6)^(2/3)/x^5+1/5*x*(1-x^6)^(2/3)],
[1/2*x^(-1+m)*(2*a*m+b*(2*m-n)*x^n)/(a+b*x^n)^(3/2),x,2,x^m/sqrt(a+b*x^n)],
[(x-2*x^3)/sqrt(2+3*x),x,3,-10/81*(2+3*x)^(3/2)+8/135*(2+3*x)^(5/2)-4/567*(2+3*x)^(7/2)-4/81*sqrt(2+3*x)],
[1/((1+x)^(1/4)+sqrt(1+x)),x,5,-4*(1+x)^(1/4)+4*log(1+(1+x)^(1/4))+2*sqrt(1+x)],
[(1+2*x)/sqrt(x+x^2),x,1,2*sqrt(x+x^2)],
[1/2/((1+x)*sqrt(x)),x,3,arctan(sqrt(x))],
[1/(x*sqrt(6*x-x^2)),x,1,-1/3*sqrt(6*x-x^2)/x],
[sqrt(x)*(1+sqrt(x)),x,2,2/3*x^(3/2)+1/2*x^2],
[(1-sqrt(x))/x^(1/3),x,2,3/2*x^(2/3)-6/7*x^(7/6)],
[sqrt(x)/(1+x^(1/3)),x,7,-6*x^(1/6)-6/5*x^(5/6)+6/7*x^(7/6)+6*arctan(x^(1/6))+2*sqrt(x)],
[(1+sqrt(x))^(1/3)/x,x,6,-1/2*log(x)+3*log(1-(1+sqrt(x))^(1/3))-2*arctan((1+2*(1+sqrt(x))^(1/3))/sqrt(3))*sqrt(3)+6*(1+sqrt(x))^(1/3)],
[1-sqrt(x),x,1,x-2/3*x^(3/2)],
[1-x^(1/4),x,1,x-4/5*x^(5/4)],
[(1-sqrt(x))/(1+x^(1/4)),x,2,x-4/5*x^(5/4)],
[1/sqrt((a+b*x)*(c+d*x)),x,3,arctanh(1/2*(b*c+a*d+2*b*d*x)/(sqrt(b)*sqrt(d)*sqrt(a*c+(b*c+a*d)*x+b*d*x^2)))/(sqrt(b)*sqrt(d))],
[1/sqrt((a+b*x)*(c-d*x)),x,3,-arctan(1/2*(b*c-a*d-2*b*d*x)/(sqrt(b)*sqrt(d)*sqrt(a*c+(b*c-a*d)*x-b*d*x^2)))/(sqrt(b)*sqrt(d))],
[1/((1-x^2)*sqrt(x)),x,4,arctan(sqrt(x))+arctanh(sqrt(x))],
[sqrt(x)/(x-x^3),x,5,arctan(sqrt(x))+arctanh(sqrt(x))],
[x/(2+x^2-sqrt(3)+x*(1+sqrt(3))),x,4,1/2*log(2+x^2-sqrt(3)+x*(1+sqrt(3)))+arctanh((1+2*x+sqrt(3))/sqrt(2*(-2+3*sqrt(3))))*sqrt(1/23*(13+8*sqrt(3)))],
[sqrt(x^2+x^3),x,2,-4/15*(x^2+x^3)^(3/2)/x^3+2/5*(x^2+x^3)^(3/2)/x^2],
[1/((1+x)*sqrt(2*x+x^2)),x,2,arctan(sqrt(2*x+x^2))],
[sqrt(x)*sqrt(1-x-sqrt(x)),x,6,45/64*arcsin((1+2*sqrt(x))/sqrt(5))+5/12*(1-x-sqrt(x))^(3/2)-1/2*(1-x-sqrt(x))^(3/2)*sqrt(x)+9/32*(1+2*sqrt(x))*sqrt(1-x-sqrt(x))],
[(1+sqrt(-3+x))^(1/3),x,4,-3/2*(1+sqrt(-3+x))^(4/3)+6/7*(1+sqrt(-3+x))^(7/3)],
[1/sqrt(3+sqrt(-1+2*x)),x,4,2/3*(3+sqrt(-1+2*x))^(3/2)-6*sqrt(3+sqrt(-1+2*x))],

#  {(Sqrt[x]+x)^(2/3), x, 0} 

#  {(-3*x+x^2)^(-1/3), x, 0} 
[sqrt(1-x)/(1+sqrt(x)),x,4,-arcsin(sqrt(x))-sqrt(1-x)*(2-sqrt(x))],
[sqrt(1-x)/(1-sqrt(x)),x,4,arcsin(sqrt(x))-sqrt(1-x)*(2+sqrt(x))],
[x/(x-sqrt(1+x^2)),x,3,-1/3*x^3-1/3*(1+x^2)^(3/2)],
[x/(x-sqrt(1-x^2)),x,7,1/2*x-1/2*arctanh(x*sqrt(2))/sqrt(2)-1/2*arctanh(sqrt(2)*sqrt(1-x^2))/sqrt(2)+1/2*sqrt(1-x^2)],
[x/(x-sqrt(1+2*x^2)),x,7,-x+arctan(x)+arctan(sqrt(1+2*x^2))-sqrt(1+2*x^2)],
[sqrt(x)*sqrt(x+sqrt(x)),x,6,-5/32*arctanh(sqrt(x)/sqrt(x+sqrt(x)))-5/12*(x+sqrt(x))^(3/2)+1/2*sqrt(x)*(x+sqrt(x))^(3/2)+5/32*(1+2*sqrt(x))*sqrt(x+sqrt(x))],
[(1+x^(1/3))/(1+sqrt(x)),x,10,-3*x^(1/3)+6/5*x^(5/6)-4*log(1+x^(1/6))-log(1-x^(1/6)+x^(1/3))-2*arctan((1-2*x^(1/6))/sqrt(3))*sqrt(3)+2*sqrt(x)],
[(1+x^(1/3))/(1+x^(1/4)),x,11,12*x^(1/12)+4*x^(1/4)-3*x^(1/3)+12/7*x^(7/12)+4/3*x^(3/4)-6/5*x^(5/6)+12/13*x^(13/12)-8*log(1+x^(1/12))-2*log(1-x^(1/12)+x^(1/6))+4*arctan((1-2*x^(1/12))/sqrt(3))*sqrt(3)-2*sqrt(x)],

#  {1/Sqrt[a*x+b*x^3], x, 0} 
[x^2/(-1+x^2+sqrt(1-x^2)),x,3,x+arcsin(x)],
[sqrt((1+x)/x),x,5,arctanh(sqrt(1+1/x))+x*sqrt(1+1/x)],
[sqrt((1-x)/x),x,5,-arctan(sqrt(-1+1/x))+x*sqrt(-1+1/x)],
[sqrt((-1+x)/x),x,5,-arcsinh(sqrt(-1+x))+sqrt(-1+x)*sqrt(x),-arctanh(sqrt((-1+x)/x))+x*sqrt((-1+x)/x)],
[sqrt((1+x)/x)/x,x,5,2*arctanh(sqrt(1+1/x))-2*sqrt(1+1/x)],
[sqrt(x/(1+x)),x,4,-arcsinh(sqrt(x))+sqrt(x)*sqrt(1+x)],
[1/sqrt((-1-x)/x),x,5,arctan(sqrt((-1-x)/x))-x*sqrt((-1-x)/x)],
[sqrt((4-x)*x),x,4,-2*arcsin(1-1/2*x)-1/2*(2-x)*sqrt(4*x-x^2)],
[1/sqrt((1-x)*x),x,3,-arcsin(1-2*x)],
[x/(x*(2+x))^(3/2),x,2,x/sqrt(2*x+x^2)],
[sqrt(1+1/x)/(1-x^2),x,5,arctanh(sqrt(1+1/x)/sqrt(2))*sqrt(2)],
[1/(1-x^2+sqrt(5)+x^2*sqrt(5)),x,2,1/2*arctan(x*sqrt(1/2*(3-sqrt(5))))],

#  Integrands equivalent to expressions of the form 1/Sqrt[a*x + b*x^2] 
[1/sqrt(a*x+b*x^2),x,2,2*arctanh(x*sqrt(b)/sqrt(a*x+b*x^2))/sqrt(b)],
[1/sqrt(x*(a+b*x)),x,3,2*arctanh(x*sqrt(b)/sqrt(a*x+b*x^2))/sqrt(b)],
[1/sqrt((b+a/x)*x^2),x,3,2*arctanh(x*sqrt(b)/sqrt(a*x+b*x^2))/sqrt(b)],
[1/sqrt((a/x^2+b/x)*x^3),x,3,2*arctanh(x*sqrt(b)/sqrt(a*x+b*x^2))/sqrt(b)],
[1/sqrt((a*x^2+b*x^3)/x),x,3,2*arctanh(x*sqrt(b)/sqrt(a*x+b*x^2))/sqrt(b)],
[1/sqrt((a*x^3+b*x^4)/x^2),x,3,2*arctanh(x*sqrt(b)/sqrt(a*x+b*x^2))/sqrt(b)],

#  Integrands equivalent to expressions of the form 1/Sqrt[a*c*x + b*c*x^2] 
[1/sqrt(a*c*x+b*c*x^2),x,2,2*arctanh(x*sqrt(b)*sqrt(c)/sqrt(a*c*x+b*c*x^2))/(sqrt(b)*sqrt(c))],
[1/sqrt(c*(a*x+b*x^2)),x,3,2*arctanh(x*sqrt(b)*sqrt(c)/sqrt(a*c*x+b*c*x^2))/(sqrt(b)*sqrt(c))],
[1/sqrt(c*x*(a+b*x)),x,3,2*arctanh(x*sqrt(b)*sqrt(c)/sqrt(a*c*x+b*c*x^2))/(sqrt(b)*sqrt(c))],
[1/sqrt(c*(b+a/x)*x^2),x,3,2*arctanh(x*sqrt(b)*sqrt(c)/sqrt(a*c*x+b*c*x^2))/(sqrt(b)*sqrt(c))],

#  Subproblems of Charlwood Fifty problems 
[sqrt(1-x^2+x*sqrt(-1+x^2)),x,-1,3/4*arcsin(x-sqrt(-1+x^2))/sqrt(2)+1/4*(3*x+sqrt(-1+x^2))*sqrt(1-x^2+x*sqrt(-1+x^2))],
[sqrt(-x+sqrt(x)*sqrt(1+x))/sqrt(1+x),x,-1,-3/2*arcsin(sqrt(x)-sqrt(1+x))/sqrt(2)+1/2*(sqrt(x)+3*sqrt(1+x))*sqrt(-x+sqrt(x)*sqrt(1+x))],
[(-x-2*sqrt(1+x^2))/(x+x^3+sqrt(1+x^2)),x,-25,arctanh((x+sqrt(1+x^2))*sqrt(2+sqrt(5)))*sqrt(2*(-1+sqrt(5)))-arctan((x+sqrt(1+x^2))*sqrt(-2+sqrt(5)))*sqrt(2*(1+sqrt(5)))],
[(1+2*x)/((1+x^2)*sqrt(2+2*x+x^2)),x,5,-arctanh((x*(5-sqrt(5))+2*sqrt(5))/(sqrt(2+2*x+x^2)*sqrt(10*(-1+sqrt(5)))))*sqrt(1/2*(-1+sqrt(5)))-arctan((2*sqrt(5)-x*(5+sqrt(5)))/(sqrt(2+2*x+x^2)*sqrt(10*(1+sqrt(5)))))*sqrt(1/2*(1+sqrt(5)))],
[1/((1+x^4)*sqrt(-x^2+sqrt(1+x^4))),x,2,arctan(x/sqrt(-x^2+sqrt(1+x^4)))],
[1/((a+b*x^4)*sqrt(c*x^2+d*sqrt(a+b*x^4))),x,2,arctanh(x*sqrt(c)/sqrt(c*x^2+d*sqrt(a+b*x^4)))/(a*sqrt(c))],
[1/((a+b*x^4)*sqrt(-c*x^2+d*sqrt(a+b*x^4))),x,2,arctan(x*sqrt(c)/sqrt(-c*x^2+d*sqrt(a+b*x^4)))/(a*sqrt(c))],
[x/sqrt(a+b*c^4+4*b*c^3*d*x+6*b*c^2*d^2*x^2+4*b*c*d^3*x^3+b*d^4*x^4),x,7,1/2*arctanh(d^2*(c/d+x)^2*sqrt(b)/sqrt(a+b*d^4*(c/d+x)^4))/(d^2*sqrt(b))-1/2*c*sqrt(cos(2*arctan(b^(1/4)*(c+d*x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*(c+d*x)/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*(c+d*x)/a^(1/4))),sqrt(1/2))*(sqrt(a)+d^2*(c/d+x)^2*sqrt(b))*sqrt((a+b*d^4*(c/d+x)^4)/(sqrt(a)+d^2*(c/d+x)^2*sqrt(b))^2)/(a^(1/4)*b^(1/4)*d^2*sqrt(a+b*d^4*(c/d+x)^4))],
[1/sqrt(a+b*c^4+4*b*c^3*d*x+6*b*c^2*d^2*x^2+4*b*c*d^3*x^3+b*d^4*x^4),x,2,1/2*sqrt(cos(2*arctan(b^(1/4)*(c+d*x)/a^(1/4)))^2)/cos(2*arctan(b^(1/4)*(c+d*x)/a^(1/4)))*EllipticF(sin(2*arctan(b^(1/4)*(c+d*x)/a^(1/4))),sqrt(1/2))*(sqrt(a)+d^2*(c/d+x)^2*sqrt(b))*sqrt((a+b*d^4*(c/d+x)^4)/(sqrt(a)+d^2*(c/d+x)^2*sqrt(b))^2)/(a^(1/4)*b^(1/4)*d*sqrt(a+b*d^4*(c/d+x)^4))],
[(a-c*x^4)/((a*d+a*e*x^2+c*d*x^4)*sqrt(a+b*x^2+c*x^4)),x,2,arctanh(x*sqrt(b*d-a*e)/(sqrt(d)*sqrt(a+b*x^2+c*x^4)))/(sqrt(d)*sqrt(b*d-a*e))],
[(a-c*x^4)/((a*d+a*e*x^2+c*d*x^4)*sqrt(a-b*x^2+c*x^4)),x,2,arctan(x*sqrt(b*d+a*e)/(sqrt(d)*sqrt(a-b*x^2+c*x^4)))/(sqrt(d)*sqrt(b*d+a*e))],
[1/((8+x^3)*sqrt(5-2*x+x^2)),x,9,1/12*arctanh(sqrt(5-2*x+x^2))-1/4*arctan((1-x)/(sqrt(3)*sqrt(5-2*x+x^2)))/sqrt(3)-1/12*arctanh((7-3*x)/(sqrt(13)*sqrt(5-2*x+x^2)))/sqrt(13)],
[sqrt(x^2/(1+x^2)),x,3,sqrt(x^2)*sqrt(1+x^2)/x],
[sqrt(x^n/(1+x^n)),x,3,2*x*hypergeom([1/2,1/2*(1+2/n)],[1/2*(3+2/n)],-x^n)*sqrt(x^n)/(2+n)],
[(e*f-e*f*x^2)/((a*d+b*d*x+a*d*x^2)*sqrt(a+b*x+c*x^2+b*x^3+a*x^4)),x,1,e*f*arctan(1/2*(a*b+(4*a^2+b^2-2*a*c)*x+a*b*x^2)/(a*sqrt(2*a-c)*sqrt(a+b*x+c*x^2+b*x^3+a*x^4)))/(a*d*sqrt(2*a-c))],
[(e*f-e*f*x^2)/((-a*d+b*d*x-a*d*x^2)*sqrt(-a+b*x+c*x^2+b*x^3-a*x^4)),x,1,e*f*arctanh(1/2*(a*b-(4*a^2+b^2+2*a*c)*x+a*b*x^2)/(a*sqrt(2*a+c)*sqrt(-a+b*x+c*x^2+b*x^3-a*x^4)))/(a*d*sqrt(2*a+c))],
[sqrt(a*x^2+b*x*sqrt(-a/b^2+a^2*x^2/b^2))/(x*sqrt(-a/b^2+a^2*x^2/b^2)),x,2,b*arcsinh((a*x+b*sqrt(-a/b^2+a^2*x^2/b^2))/sqrt(a))*sqrt(2)/sqrt(a)],
[sqrt(-a*x^2+b*x*sqrt(a/b^2+a^2*x^2/b^2))/(x*sqrt(a/b^2+a^2*x^2/b^2)),x,2,b*arcsin((a*x-b*sqrt(a/b^2+a^2*x^2/b^2))/sqrt(a))*sqrt(2)/sqrt(a)],
[sqrt(x*(a*x+b*sqrt(-a/b^2+a^2*x^2/b^2)))/(x*sqrt(-a/b^2+a^2*x^2/b^2)),x,3,b*arcsinh((a*x+b*sqrt(-a/b^2+a^2*x^2/b^2))/sqrt(a))*sqrt(2)/sqrt(a)],
[sqrt(x*(-a*x+b*sqrt(a/b^2+a^2*x^2/b^2)))/(x*sqrt(a/b^2+a^2*x^2/b^2)),x,3,b*arcsin((a*x-b*sqrt(a/b^2+a^2*x^2/b^2))/sqrt(a))*sqrt(2)/sqrt(a)],
[(-sqrt(-4+x)+x*sqrt(-4+x)-4*sqrt(-1+x)+x*sqrt(-1+x))/((4-5*x+x^2)*(1+sqrt(-4+x)+sqrt(-1+x))),x,3,2*log(1+sqrt(-4+x)+sqrt(-1+x))],
[1/(x*(3+3*x+x^2)*(3+3*x+3*x^2+x^3)^(1/3)),x,3,-arctan((1+2*3^(1/3)*(1+x)/(2+(1+x)^3)^(1/3))/sqrt(3))/3^(5/6)-1/6*log(1-(1+x)^3)/3^(1/3)+1/2*log(3^(1/3)*(1+x)-(2+(1+x)^3)^(1/3))/3^(1/3)],
[(1-x^2)/((1-x+x^2)*(1-x^3)^(2/3)),x,-42,-1/2*log(1+2*(1-x)^3-x^3)/2^(2/3)+3/2*log(2^(1/3)*(1-x)+(1-x^3)^(1/3))/2^(2/3)+arctan((1-2*2^(1/3)*(1-x)/(1-x^3)^(1/3))/sqrt(3))*sqrt(3)/2^(2/3)],
[x^2/((1+x^4)*sqrt(-1+x^4)),x,-9,-1/4*arctan((1+x^2)/(x*sqrt(-1+x^4)))-1/4*arctanh((1-x^2)/(x*sqrt(-1+x^4)))],
[(a-c*x^4)/((a*e+c*d*x^2)*(d+e*x^2)*sqrt(a+b*x^2+c*x^4)),x,2,arctan(x*sqrt(c*d^2-b*d*e+a*e^2)/(sqrt(d)*sqrt(e)*sqrt(a+b*x^2+c*x^4)))/(sqrt(d)*sqrt(e)*sqrt(c*d^2-b*d*e+a*e^2))],
#  {(1 - Sqrt[3] + x)/((1 + Sqrt[3] + x)*Sqrt[-4 + 2*Sqrt[3] + x^2]*Sqrt[2 - (-2 + Sqrt[3])*x^2]), x, -10, (1/3^(3/4))*ArcTanh[(1 - Sqrt[3] + x)^2/(3^(3/4)*Sqrt[-4 + 2*Sqrt[3] + x^2]*Sqrt[2 - (-2 + Sqrt[3])*x^2])]}

# {(1 - Sqrt[3] + x)/((1 + Sqrt[3] + x)*Sqrt[(-4 + 2*Sqrt[3] + x^2)*(2 - (Sqrt[3] - 2)*x^2)]), x, -11, (1/3^(3/4))*ArcTanh[(1 - Sqrt[3] + x)^2/(3^(3/4)*Sqrt[(-4 + 2*Sqrt[3] + x^2)*(2 - (Sqrt[3] - 2)*x^2)])]} 

#  Lack of gcd cancellation used to cause a zero-divide error in IntSum. 
[x+(1-x^2)/(1+x),x,1,x],
[1/(1/x+sqrt(1-x^2)),x,12,arcsin(x)-arctan((1-2*x^2)/sqrt(3))/sqrt(3)-arctan(x/(sqrt(1-x^2)*sqrt((-I+sqrt(3))/(I+sqrt(3)))))/sqrt(3)-arctan(x*sqrt((-I+sqrt(3))/(I+sqrt(3)))/sqrt(1-x^2))/sqrt(3)],
[x*sqrt(1-x^2)/(x-x^3+sqrt(1-x^2)),x,13,arcsin(x)-arctan((1-2*x^2)/sqrt(3))/sqrt(3)-arctan(x/(sqrt(1-x^2)*sqrt((-I+sqrt(3))/(I+sqrt(3)))))/sqrt(3)-arctan(x*sqrt((-I+sqrt(3))/(I+sqrt(3)))/sqrt(1-x^2))/sqrt(3),1/4*(1-x)^2-1/2*x^2+1/4*(1+x)^2+arcsin(x)-arctan((1-2*x^2)/sqrt(3))/sqrt(3)-arctan(x/(sqrt(1-x^2)*sqrt((-I+sqrt(3))/(I+sqrt(3)))))/sqrt(3)-arctan(x*sqrt((-I+sqrt(3))/(I+sqrt(3)))/sqrt(1-x^2))/sqrt(3)],
[(1-x^4)^n/(1+x+x^2+x^3)^n,x,-1,-(1-x)*(1-x^4)^n/((1+n)*(1+x+x^2+x^3)^n)],

#  Manuel Bronstein pseudo-elliptic integrals: 
[x/sqrt(-44375*b^4+576000*b^3*c*x+576000*b^2*c^2*x^2+5308416*c^4*x^4),x,1,1/18432*log(20738073600000000*b^8*c^4+597005697024000000*b^6*c^6*x^2+2583100705996800000*b^5*c^7*x^3+951050714480640000*b^4*c^8*x^4+21641687369515008000*b^3*c^9*x^5+32462531054272512000*b^2*c^10*x^6+149587343098087735296*c^12*x^8+5308416*(12203125*b^6*c^4+79200000*b^5*c^5*x+38880000*b^4*c^6*x^2+1105920000*b^3*c^7*x^3+1990656000*b^2*c^8*x^4+12230590464*c^10*x^6)*sqrt(-44375*b^4+576000*b^3*c*x+576000*b^2*c^2*x^2+5308416*c^4*x^4))/c^2],
[(1+4*x)/sqrt(9+120*x+64*x^2+64*x^3+64*x^4),x,2,1/16*log(921+2864*x+9280*x^2+13440*x^3+17024*x^4+19456*x^5+12288*x^6+8192*x^7+4096*x^8+(179+444*x+744*x^2+1280*x^3+960*x^4+768*x^5+512*x^6)*sqrt(9+120*x+64*x^2+64*x^3+64*x^4)),1/16*log(921+2864*x+9280*x^2+13440*x^3+17024*x^4+19456*x^5+12288*x^6+8192*x^7+4096*x^8+179*sqrt(9+120*x+64*x^2+64*x^3+64*x^4)+444*x*sqrt(9+120*x+64*x^2+64*x^3+64*x^4)+744*x^2*sqrt(9+120*x+64*x^2+64*x^3+64*x^4)+1280*x^3*sqrt(9+120*x+64*x^2+64*x^3+64*x^4)+960*x^4*sqrt(9+120*x+64*x^2+64*x^3+64*x^4)+768*x^5*sqrt(9+120*x+64*x^2+64*x^3+64*x^4)+512*x^6*sqrt(9+120*x+64*x^2+64*x^3+64*x^4))]]:
