# Maple integration test file: "1 Algebraic functions\1.1 Binomial products\1.1.2 Quadratic\1.1.2.3 (a+b x^2)^p (c+d x^2)^q.txt"

lst:=[

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

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

# p>0
[(a+b*x^2)*(c+d*x^2)^4,x,2,a*c^4*x+1/3*c^3*(b*c+4*a*d)*x^3+2/5*c^2*d*(2*b*c+3*a*d)*x^5+2/7*c*d^2*(3*b*c+2*a*d)*x^7+1/9*d^3*(4*b*c+a*d)*x^9+1/11*b*d^4*x^11],
[(a+b*x^2)*(c+d*x^2)^3,x,2,a*c^3*x+1/3*c^2*(b*c+3*a*d)*x^3+3/5*c*d*(b*c+a*d)*x^5+1/7*d^2*(3*b*c+a*d)*x^7+1/9*b*d^3*x^9],
[(a+b*x^2)*(c+d*x^2)^2,x,2,a*c^2*x+1/3*c*(b*c+2*a*d)*x^3+1/5*d*(2*b*c+a*d)*x^5+1/7*b*d^2*x^7],
[(a+b*x^2)*(c+d*x^2),x,2,a*c*x+1/3*(b*c+a*d)*x^3+1/5*b*d*x^5],
[(a+b*x^2)/(c+d*x^2),x,2,b*x/d-(b*c-a*d)*arctan(x*sqrt(d)/sqrt(c))/(d^(3/2)*sqrt(c))],
[(a+b*x^2)/(c+d*x^2)^2,x,2,-1/2*(b*c-a*d)*x/(c*d*(c+d*x^2))+1/2*(b*c+a*d)*arctan(x*sqrt(d)/sqrt(c))/(c^(3/2)*d^(3/2))],
[(a+b*x^2)/(c+d*x^2)^3,x,3,-1/4*(b*c-a*d)*x/(c*d*(c+d*x^2)^2)+1/8*(b*c+3*a*d)*x/(c^2*d*(c+d*x^2))+1/8*(b*c+3*a*d)*arctan(x*sqrt(d)/sqrt(c))/(c^(5/2)*d^(3/2))],
[(a+b*x^2)^2*(c+d*x^2)^3,x,2,a^2*c^3*x+1/3*a*c^2*(2*b*c+3*a*d)*x^3+1/5*c*(b^2*c^2+6*a*b*c*d+3*a^2*d^2)*x^5+1/7*d*(3*b^2*c^2+6*a*b*c*d+a^2*d^2)*x^7+1/9*b*d^2*(3*b*c+2*a*d)*x^9+1/11*b^2*d^3*x^11],
[(a+b*x^2)^2*(c+d*x^2)^2,x,2,a^2*c^2*x+2/3*a*c*(b*c+a*d)*x^3+1/5*(b^2*c^2+4*a*b*c*d+a^2*d^2)*x^5+2/7*b*d*(b*c+a*d)*x^7+1/9*b^2*d^2*x^9],
[(a+b*x^2)^2*(c+d*x^2),x,2,a^2*c*x+1/3*a*(2*b*c+a*d)*x^3+1/5*b*(b*c+2*a*d)*x^5+1/7*b^2*d*x^7],
[(a+b*x^2)^2/(c+d*x^2),x,3,-b*(b*c-2*a*d)*x/d^2+1/3*b^2*x^3/d+(b*c-a*d)^2*arctan(x*sqrt(d)/sqrt(c))/(d^(5/2)*sqrt(c))],
[(a+b*x^2)^2/(c+d*x^2)^2,x,4,b^2*x/d^2+1/2*(b*c-a*d)^2*x/(c*d^2*(c+d*x^2))-1/2*(b*c-a*d)*(3*b*c+a*d)*arctan(x*sqrt(d)/sqrt(c))/(c^(3/2)*d^(5/2))],
[(a+b*x^2)^2/(c+d*x^2)^3,x,3,-1/4*(b*c-a*d)*x*(a+b*x^2)/(c*d*(c+d*x^2)^2)+3/8*(a^2/c^2-b^2/d^2)*x/(c+d*x^2)+1/8*(3*b^2*c^2+2*a*b*c*d+3*a^2*d^2)*arctan(x*sqrt(d)/sqrt(c))/(c^(5/2)*d^(5/2))],
[(a+b*x^2)^3*(c+d*x^2)^3,x,2,a^3*c^3*x+a^2*c^2*(b*c+a*d)*x^3+3/5*a*c*(b^2*c^2+3*a*b*c*d+a^2*d^2)*x^5+1/7*(b*c+a*d)*(b^2*c^2+8*a*b*c*d+a^2*d^2)*x^7+1/3*b*d*(b^2*c^2+3*a*b*c*d+a^2*d^2)*x^9+3/11*b^2*d^2*(b*c+a*d)*x^11+1/13*b^3*d^3*x^13],
[(a+b*x^2)^3*(c+d*x^2)^2,x,2,a^3*c^2*x+1/3*a^2*c*(3*b*c+2*a*d)*x^3+1/5*a*(3*b^2*c^2+6*a*b*c*d+a^2*d^2)*x^5+1/7*b*(b^2*c^2+6*a*b*c*d+3*a^2*d^2)*x^7+1/9*b^2*d*(2*b*c+3*a*d)*x^9+1/11*b^3*d^2*x^11],
[(a+b*x^2)^3*(c+d*x^2),x,2,a^3*c*x+1/3*a^2*(3*b*c+a*d)*x^3+3/5*a*b*(b*c+a*d)*x^5+1/7*b^2*(b*c+3*a*d)*x^7+1/9*b^3*d*x^9],
[(a+b*x^2)^3/(c+d*x^2),x,3,b*(b^2*c^2-3*a*b*c*d+3*a^2*d^2)*x/d^3-1/3*b^2*(b*c-3*a*d)*x^3/d^2+1/5*b^3*x^5/d-(b*c-a*d)^3*arctan(x*sqrt(d)/sqrt(c))/(d^(7/2)*sqrt(c))],
[(a+b*x^2)^3/(c+d*x^2)^2,x,4,-b^2*(2*b*c-3*a*d)*x/d^3+1/3*b^3*x^3/d^2-1/2*(b*c-a*d)^3*x/(c*d^3*(c+d*x^2))+1/2*(b*c-a*d)^2*(5*b*c+a*d)*arctan(x*sqrt(d)/sqrt(c))/(c^(3/2)*d^(7/2))],
[(a+b*x^2)^3/(c+d*x^2)^3,x,5,b^3*x/d^3-1/4*(b*c-a*d)^3*x/(c*d^3*(c+d*x^2)^2)+3/8*(b*c-a*d)^2*(3*b*c+a*d)*x/(c^2*d^3*(c+d*x^2))-3/8*(b*c-a*d)*(4*b^2*c^2+(b*c+a*d)^2)*arctan(x*sqrt(d)/sqrt(c))/(c^(5/2)*d^(7/2))],

# p<0
[(c+d*x^2)^4/(a+b*x^2),x,3,d*(2*b*c-a*d)*(2*b^2*c^2-2*a*b*c*d+a^2*d^2)*x/b^4+1/3*d^2*(6*b^2*c^2-4*a*b*c*d+a^2*d^2)*x^3/b^3+1/5*d^3*(4*b*c-a*d)*x^5/b^2+1/7*d^4*x^7/b+(b*c-a*d)^4*arctan(x*sqrt(b)/sqrt(a))/(b^(9/2)*sqrt(a))],
[(c+d*x^2)^3/(a+b*x^2),x,3,d*(3*b^2*c^2-3*a*b*c*d+a^2*d^2)*x/b^3+1/3*d^2*(3*b*c-a*d)*x^3/b^2+1/5*d^3*x^5/b+(b*c-a*d)^3*arctan(x*sqrt(b)/sqrt(a))/(b^(7/2)*sqrt(a))],
[(c+d*x^2)^2/(a+b*x^2),x,3,d*(2*b*c-a*d)*x/b^2+1/3*d^2*x^3/b+(b*c-a*d)^2*arctan(x*sqrt(b)/sqrt(a))/(b^(5/2)*sqrt(a))],
[(c+d*x^2)/(a+b*x^2),x,2,d*x/b+(b*c-a*d)*arctan(x*sqrt(b)/sqrt(a))/(b^(3/2)*sqrt(a))],
[1/((a+b*x^2)*(c+d*x^2)),x,3,arctan(x*sqrt(b)/sqrt(a))*sqrt(b)/((b*c-a*d)*sqrt(a))-arctan(x*sqrt(d)/sqrt(c))*sqrt(d)/((b*c-a*d)*sqrt(c))],
[1/((a+b*x^2)*(c+d*x^2)^2),x,4,-1/2*d*x/(c*(b*c-a*d)*(c+d*x^2))+b^(3/2)*arctan(x*sqrt(b)/sqrt(a))/((b*c-a*d)^2*sqrt(a))-1/2*(3*b*c-a*d)*arctan(x*sqrt(d)/sqrt(c))*sqrt(d)/(c^(3/2)*(b*c-a*d)^2)],
[1/((a+b*x^2)*(c+d*x^2)^3),x,5,-1/4*d*x/(c*(b*c-a*d)*(c+d*x^2)^2)-1/8*d*(7*b*c-3*a*d)*x/(c^2*(b*c-a*d)^2*(c+d*x^2))+b^(5/2)*arctan(x*sqrt(b)/sqrt(a))/((b*c-a*d)^3*sqrt(a))-1/8*(15*b^2*c^2-10*a*b*c*d+3*a^2*d^2)*arctan(x*sqrt(d)/sqrt(c))*sqrt(d)/(c^(5/2)*(b*c-a*d)^3)],
[(c+d*x^2)^5/(a+b*x^2)^2,x,4,d^2*(10*b^3*c^3-20*a*b^2*c^2*d+15*a^2*b*c*d^2-4*a^3*d^3)*x/b^5+1/3*d^3*(10*b^2*c^2-10*a*b*c*d+3*a^2*d^2)*x^3/b^4+1/5*d^4*(5*b*c-2*a*d)*x^5/b^3+1/7*d^5*x^7/b^2+1/2*(b*c-a*d)^5*x/(a*b^5*(a+b*x^2))+1/2*(b*c-a*d)^4*(b*c+9*a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*b^(11/2))],
[(c+d*x^2)^4/(a+b*x^2)^2,x,4,d^2*(6*b^2*c^2-8*a*b*c*d+3*a^2*d^2)*x/b^4+2/3*d^3*(2*b*c-a*d)*x^3/b^3+1/5*d^4*x^5/b^2+1/2*(b*c-a*d)^4*x/(a*b^4*(a+b*x^2))+1/2*(b*c-a*d)^3*(b*c+7*a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*b^(9/2))],
[(c+d*x^2)^3/(a+b*x^2)^2,x,4,d^2*(3*b*c-2*a*d)*x/b^3+1/3*d^3*x^3/b^2+1/2*(b*c-a*d)^3*x/(a*b^3*(a+b*x^2))+1/2*(b*c-a*d)^2*(b*c+5*a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*b^(7/2))],
[(c+d*x^2)^2/(a+b*x^2)^2,x,4,d^2*x/b^2+1/2*(b*c-a*d)^2*x/(a*b^2*(a+b*x^2))+1/2*(b*c-a*d)*(b*c+3*a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*b^(5/2))],
[(c+d*x^2)/(a+b*x^2)^2,x,2,1/2*(b*c-a*d)*x/(a*b*(a+b*x^2))+1/2*(b*c+a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*b^(3/2))],
[1/((a+b*x^2)^2*(c+d*x^2)),x,4,1/2*b*x/(a*(b*c-a*d)*(a+b*x^2))+1/2*(b*c-3*a*d)*arctan(x*sqrt(b)/sqrt(a))*sqrt(b)/(a^(3/2)*(b*c-a*d)^2)+d^(3/2)*arctan(x*sqrt(d)/sqrt(c))/((b*c-a*d)^2*sqrt(c))],
[1/((a+b*x^2)^2*(c+d*x^2)^2),x,5,1/2*d*(b*c+a*d)*x/(a*c*(b*c-a*d)^2*(c+d*x^2))+1/2*b*x/(a*(b*c-a*d)*(a+b*x^2)*(c+d*x^2))+1/2*b^(3/2)*(b*c-5*a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*(b*c-a*d)^3)+1/2*d^(3/2)*(5*b*c-a*d)*arctan(x*sqrt(d)/sqrt(c))/(c^(3/2)*(b*c-a*d)^3)],
[1/((a+b*x^2)^2*(c+d*x^2)^3),x,6,1/4*d*(2*b*c+a*d)*x/(a*c*(b*c-a*d)^2*(c+d*x^2)^2)+1/2*b*x/(a*(b*c-a*d)*(a+b*x^2)*(c+d*x^2)^2)+1/8*d*(4*b*c-a*d)*(b*c+3*a*d)*x/(a*c^2*(b*c-a*d)^3*(c+d*x^2))+1/2*b^(5/2)*(b*c-7*a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(3/2)*(b*c-a*d)^4)+1/8*d^(3/2)*(35*b^2*c^2-14*a*b*c*d+3*a^2*d^2)*arctan(x*sqrt(d)/sqrt(c))/(c^(5/2)*(b*c-a*d)^4)],
[(c+d*x^2)^5/(a+b*x^2)^3,x,5,d^3*(10*b^2*c^2-15*a*b*c*d+6*a^2*d^2)*x/b^5+1/3*d^4*(5*b*c-3*a*d)*x^3/b^4+1/5*d^5*x^5/b^3+1/4*(b*c-a*d)^5*x/(a*b^5*(a+b*x^2)^2)+1/8*(b*c-a*d)^4*(3*b*c+17*a*d)*x/(a^2*b^5*(a+b*x^2))+1/8*(b*c-a*d)^3*(3*b^2*c^2+14*a*b*c*d+63*a^2*d^2)*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*b^(11/2))],
[(c+d*x^2)^4/(a+b*x^2)^3,x,5,d^3*(4*b*c-3*a*d)*x/b^4+1/3*d^4*x^3/b^3+1/4*(b*c-a*d)^4*x/(a*b^4*(a+b*x^2)^2)+1/8*(b*c-a*d)^3*(3*b*c+13*a*d)*x/(a^2*b^4*(a+b*x^2))+1/8*(b*c-a*d)^2*(3*b^2*c^2+10*a*b*c*d+35*a^2*d^2)*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*b^(9/2))],
[(c+d*x^2)^3/(a+b*x^2)^3,x,5,d^3*x/b^3+1/4*(b*c-a*d)^3*x/(a*b^3*(a+b*x^2)^2)+3/8*(b*c-a*d)^2*(b*c+3*a*d)*x/(a^2*b^3*(a+b*x^2))+3/8*(b*c-a*d)*(4*a^2*d^2+(b*c+a*d)^2)*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*b^(7/2))],
[(c+d*x^2)^2/(a+b*x^2)^3,x,3,3/8*(c^2/a^2-d^2/b^2)*x/(a+b*x^2)+1/4*(b*c-a*d)*x*(c+d*x^2)/(a*b*(a+b*x^2)^2)+1/8*(3*b^2*c^2+2*a*b*c*d+3*a^2*d^2)*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*b^(5/2))],
[(c+d*x^2)/(a+b*x^2)^3,x,3,1/4*(b*c-a*d)*x/(a*b*(a+b*x^2)^2)+1/8*(3*b*c+a*d)*x/(a^2*b*(a+b*x^2))+1/8*(3*b*c+a*d)*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*b^(3/2))],
[1/((a+b*x^2)^3*(c+d*x^2)),x,5,1/4*b*x/(a*(b*c-a*d)*(a+b*x^2)^2)+1/8*b*(3*b*c-7*a*d)*x/(a^2*(b*c-a*d)^2*(a+b*x^2))+1/8*(3*b^2*c^2-10*a*b*c*d+15*a^2*d^2)*arctan(x*sqrt(b)/sqrt(a))*sqrt(b)/(a^(5/2)*(b*c-a*d)^3)-d^(5/2)*arctan(x*sqrt(d)/sqrt(c))/((b*c-a*d)^3*sqrt(c))],
[1/((a+b*x^2)^3*(c+d*x^2)^2),x,6,1/8*d*(b*c-4*a*d)*(3*b*c+a*d)*x/(a^2*c*(b*c-a*d)^3*(c+d*x^2))+1/4*b*x/(a*(b*c-a*d)*(a+b*x^2)^2*(c+d*x^2))+3/8*b*(b*c-3*a*d)*x/(a^2*(b*c-a*d)^2*(a+b*x^2)*(c+d*x^2))+1/8*b^(3/2)*(3*b^2*c^2-14*a*b*c*d+35*a^2*d^2)*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*(b*c-a*d)^4)-1/2*d^(5/2)*(7*b*c-a*d)*arctan(x*sqrt(d)/sqrt(c))/(c^(3/2)*(b*c-a*d)^4)],
[1/((a+b*x^2)^3*(c+d*x^2)^3),x,7,1/8*d*(3*b^2*c^2-13*a*b*c*d-2*a^2*d^2)*x/(a^2*c*(b*c-a*d)^3*(c+d*x^2)^2)+1/4*b*x/(a*(b*c-a*d)*(a+b*x^2)^2*(c+d*x^2)^2)+1/8*b*(3*b*c-11*a*d)*x/(a^2*(b*c-a*d)^2*(a+b*x^2)*(c+d*x^2)^2)+3/8*d*(b*c+a*d)*(b^2*c^2-6*a*b*c*d+a^2*d^2)*x/(a^2*c^2*(b*c-a*d)^4*(c+d*x^2))+3/8*b^(5/2)*(b^2*c^2-6*a*b*c*d+21*a^2*d^2)*arctan(x*sqrt(b)/sqrt(a))/(a^(5/2)*(b*c-a*d)^5)-3/8*d^(5/2)*(21*b^2*c^2-6*a*b*c*d+a^2*d^2)*arctan(x*sqrt(d)/sqrt(c))/(c^(5/2)*(b*c-a*d)^5)],
[(-1+x^2)^3/(1+x^2)^4,x,3,-1/3*x*(1-x^2)^2/(1+x^2)^3-2/3*x/(1+x^2)],
[(-1+x^2)^4/(1+x^2)^5,x,5,1/4*x*(1-x^2)^3/(1+x^2)^4+3/8*x*(1-x^2)/(1+x^2)^2+3/8*arctan(x)],

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

# p>0
[(a+b*x^2)^(1/2)*(c+d*x^2)^3,x,6,1/192*d*(72*b^2*c^2-52*a*b*c*d+15*a^2*d^2)*x*(a+b*x^2)^(3/2)/b^3+1/48*d*(12*b*c-5*a*d)*x*(a+b*x^2)^(3/2)*(c+d*x^2)/b^2+1/8*d*x*(a+b*x^2)^(3/2)*(c+d*x^2)^2/b+1/128*a*(64*b^3*c^3-48*a*b^2*c^2*d+24*a^2*b*c*d^2-5*a^3*d^3)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(7/2)+1/128*(64*b^3*c^3-48*a*b^2*c^2*d+24*a^2*b*c*d^2-5*a^3*d^3)*x*sqrt(a+b*x^2)/b^3],
[(a+b*x^2)^(1/2)*(c+d*x^2)^2,x,5,1/24*d*(8*b*c-3*a*d)*x*(a+b*x^2)^(3/2)/b^2+1/6*d*x*(a+b*x^2)^(3/2)*(c+d*x^2)/b+1/16*a*(8*b^2*c^2-4*a*b*c*d+a^2*d^2)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(5/2)+1/16*(8*b^2*c^2-4*a*b*c*d+a^2*d^2)*x*sqrt(a+b*x^2)/b^2],
[(a+b*x^2)^(1/2)*(c+d*x^2),x,4,1/4*d*x*(a+b*x^2)^(3/2)/b+1/8*a*(4*b*c-a*d)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(3/2)+1/8*(4*b*c-a*d)*x*sqrt(a+b*x^2)/b],
[(a+b*x^2)^(1/2),x,3,1/2*a*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/sqrt(b)+1/2*x*sqrt(a+b*x^2)],
[(a+b*x^2)^(1/2)/(c+d*x^2),x,5,arctanh(x*sqrt(b)/sqrt(a+b*x^2))*sqrt(b)/d-arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))*sqrt(b*c-a*d)/(d*sqrt(c))],
[(a+b*x^2)^(1/2)/(c+d*x^2)^2,x,3,1/2*a*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))/(c^(3/2)*sqrt(b*c-a*d))+1/2*x*sqrt(a+b*x^2)/(c*(c+d*x^2))],
[(a+b*x^2)^(1/2)/(c+d*x^2)^3,x,4,-1/4*d*x*(a+b*x^2)^(3/2)/(c*(b*c-a*d)*(c+d*x^2)^2)+1/8*a*(4*b*c-3*a*d)*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))/(c^(5/2)*(b*c-a*d)^(3/2))+1/8*(4*b*c-3*a*d)*x*sqrt(a+b*x^2)/(c^2*(b*c-a*d)*(c+d*x^2))],
[(a+b*x^2)^(1/2)/(c+d*x^2)^4,x,6,1/16*a*(8*b^2*c^2-12*a*b*c*d+5*a^2*d^2)*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))/(c^(7/2)*(b*c-a*d)^(5/2))+1/6*x*sqrt(a+b*x^2)/(c*(c+d*x^2)^3)+1/24*(4*b*c-5*a*d)*x*sqrt(a+b*x^2)/(c^2*(b*c-a*d)*(c+d*x^2)^2)+1/48*(2*b*c-5*a*d)*(4*b*c-3*a*d)*x*sqrt(a+b*x^2)/(c^3*(b*c-a*d)^2*(c+d*x^2))],
[(a+b*x^2)^(3/2)*(c+d*x^2)^3,x,7,1/128*(4*b*c-a*d)*(8*b^2*c^2-2*a*b*c*d+a^2*d^2)*x*(a+b*x^2)^(3/2)/b^3+1/160*d*(36*b^2*c^2-20*a*b*c*d+5*a^2*d^2)*x*(a+b*x^2)^(5/2)/b^3+1/80*d*(14*b*c-5*a*d)*x*(a+b*x^2)^(5/2)*(c+d*x^2)/b^2+1/10*d*x*(a+b*x^2)^(5/2)*(c+d*x^2)^2/b+3/256*a^2*(4*b*c-a*d)*(8*b^2*c^2-2*a*b*c*d+a^2*d^2)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(7/2)+3/256*a*(4*b*c-a*d)*(8*b^2*c^2-2*a*b*c*d+a^2*d^2)*x*sqrt(a+b*x^2)/b^3],
[(a+b*x^2)^(3/2)*(c+d*x^2)^2,x,6,1/192*(48*b^2*c^2-16*a*b*c*d+3*a^2*d^2)*x*(a+b*x^2)^(3/2)/b^2+1/48*d*(10*b*c-3*a*d)*x*(a+b*x^2)^(5/2)/b^2+1/8*d*x*(a+b*x^2)^(5/2)*(c+d*x^2)/b+1/128*a^2*(48*b^2*c^2-16*a*b*c*d+3*a^2*d^2)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(5/2)+1/128*a*(48*b^2*c^2-16*a*b*c*d+3*a^2*d^2)*x*sqrt(a+b*x^2)/b^2],
[(a+b*x^2)^(3/2)*(c+d*x^2),x,5,1/24*(6*b*c-a*d)*x*(a+b*x^2)^(3/2)/b+1/6*d*x*(a+b*x^2)^(5/2)/b+1/16*a^2*(6*b*c-a*d)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(3/2)+1/16*a*(6*b*c-a*d)*x*sqrt(a+b*x^2)/b],
[(a+b*x^2)^(3/2),x,4,1/4*x*(a+b*x^2)^(3/2)+3/8*a^2*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/sqrt(b)+3/8*a*x*sqrt(a+b*x^2)],
[(a+b*x^2)^(3/2)/(c+d*x^2),x,6,-1/2*(2*b*c-3*a*d)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))*sqrt(b)/d^2+(b*c-a*d)^(3/2)*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))/(d^2*sqrt(c))+1/2*b*x*sqrt(a+b*x^2)/d],
[(a+b*x^2)^(3/2)/(c+d*x^2)^2,x,6,b^(3/2)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/d^2-1/2*(2*b*c+a*d)*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))*sqrt(b*c-a*d)/(c^(3/2)*d^2)-1/2*(b*c-a*d)*x*sqrt(a+b*x^2)/(c*d*(c+d*x^2))],
[(a+b*x^2)^(3/2)/(c+d*x^2)^3,x,4,1/4*x*(a+b*x^2)^(3/2)/(c*(c+d*x^2)^2)+3/8*a^2*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))/(c^(5/2)*sqrt(b*c-a*d))+3/8*a*x*sqrt(a+b*x^2)/(c^2*(c+d*x^2))],
[(a+b*x^2)^(3/2)/(c+d*x^2)^4,x,5,-1/6*d*x*(a+b*x^2)^(5/2)/(c*(b*c-a*d)*(c+d*x^2)^3)+1/24*(6*b*c-5*a*d)*x*(a+b*x^2)^(3/2)/(c^2*(b*c-a*d)*(c+d*x^2)^2)+1/16*a^2*(6*b*c-5*a*d)*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))/(c^(7/2)*(b*c-a*d)^(3/2))+1/16*a*(6*b*c-5*a*d)*x*sqrt(a+b*x^2)/(c^3*(b*c-a*d)*(c+d*x^2))],
[(a+b*x^2)^(3/2)/(c+d*x^2)^5,x,7,1/128*a^2*(48*b^2*c^2-80*a*b*c*d+35*a^2*d^2)*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))/(c^(9/2)*(b*c-a*d)^(5/2))-1/8*(b*c-a*d)*x*sqrt(a+b*x^2)/(c*d*(c+d*x^2)^4)+1/48*(2*b*c+7*a*d)*x*sqrt(a+b*x^2)/(c^2*d*(c+d*x^2)^3)+1/192*(8*b^2*c^2+24*a*b*c*d-35*a^2*d^2)*x*sqrt(a+b*x^2)/(c^3*d*(b*c-a*d)*(c+d*x^2)^2)+1/384*(16*b^3*c^3+40*a*b^2*c^2*d-170*a^2*b*c*d^2+105*a^3*d^3)*x*sqrt(a+b*x^2)/(c^4*d*(b*c-a*d)^2*(c+d*x^2))],
[(a+b*x^2)^(5/2)*(c+d*x^2)^3,x,8,1/1536*a*(320*b^3*c^3-120*a*b^2*c^2*d+36*a^2*b*c*d^2-5*a^3*d^3)*x*(a+b*x^2)^(3/2)/b^3+1/1920*(320*b^3*c^3-120*a*b^2*c^2*d+36*a^2*b*c*d^2-5*a^3*d^3)*x*(a+b*x^2)^(5/2)/b^3+1/960*d*(152*b^2*c^2-68*a*b*c*d+15*a^2*d^2)*x*(a+b*x^2)^(7/2)/b^3+1/120*d*(16*b*c-5*a*d)*x*(a+b*x^2)^(7/2)*(c+d*x^2)/b^2+1/12*d*x*(a+b*x^2)^(7/2)*(c+d*x^2)^2/b+1/1024*a^3*(320*b^3*c^3-120*a*b^2*c^2*d+36*a^2*b*c*d^2-5*a^3*d^3)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(7/2)+1/1024*a^2*(320*b^3*c^3-120*a*b^2*c^2*d+36*a^2*b*c*d^2-5*a^3*d^3)*x*sqrt(a+b*x^2)/b^3],
[(a+b*x^2)^(5/2)*(c+d*x^2)^2,x,7,1/384*a*(80*b^2*c^2-20*a*b*c*d+3*a^2*d^2)*x*(a+b*x^2)^(3/2)/b^2+1/480*(80*b^2*c^2-20*a*b*c*d+3*a^2*d^2)*x*(a+b*x^2)^(5/2)/b^2+3/80*d*(4*b*c-a*d)*x*(a+b*x^2)^(7/2)/b^2+1/10*d*x*(a+b*x^2)^(7/2)*(c+d*x^2)/b+1/256*a^3*(80*b^2*c^2-20*a*b*c*d+3*a^2*d^2)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(5/2)+1/256*a^2*(80*b^2*c^2-20*a*b*c*d+3*a^2*d^2)*x*sqrt(a+b*x^2)/b^2],
[(a+b*x^2)^(5/2)*(c+d*x^2),x,6,5/192*a*(8*b*c-a*d)*x*(a+b*x^2)^(3/2)/b+1/48*(8*b*c-a*d)*x*(a+b*x^2)^(5/2)/b+1/8*d*x*(a+b*x^2)^(7/2)/b+5/128*a^3*(8*b*c-a*d)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(3/2)+5/128*a^2*(8*b*c-a*d)*x*sqrt(a+b*x^2)/b],
[(a+b*x^2)^(5/2),x,5,5/24*a*x*(a+b*x^2)^(3/2)+1/6*x*(a+b*x^2)^(5/2)+5/16*a^3*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/sqrt(b)+5/16*a^2*x*sqrt(a+b*x^2)],
[(a+b*x^2)^(5/2)/(c+d*x^2),x,7,1/4*b*x*(a+b*x^2)^(3/2)/d+1/8*(8*b^2*c^2-20*a*b*c*d+15*a^2*d^2)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))*sqrt(b)/d^3-(b*c-a*d)^(5/2)*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))/(d^3*sqrt(c))-1/8*b*(4*b*c-7*a*d)*x*sqrt(a+b*x^2)/d^2],
[(a+b*x^2)^(5/2)/(c+d*x^2)^2,x,7,-1/2*(b*c-a*d)*x*(a+b*x^2)^(3/2)/(c*d*(c+d*x^2))-1/2*b^(3/2)*(4*b*c-5*a*d)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/d^3+1/2*(b*c-a*d)^(3/2)*(4*b*c+a*d)*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))/(c^(3/2)*d^3)+1/2*b*(2*b*c-a*d)*x*sqrt(a+b*x^2)/(c*d^2)],
[(a+b*x^2)^(5/2)/(c+d*x^2)^3,x,7,-1/4*(b*c-a*d)*x*(a+b*x^2)^(3/2)/(c*d*(c+d*x^2)^2)+b^(5/2)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/d^3-1/8*(8*b^2*c^2+4*a*b*c*d+3*a^2*d^2)*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))*sqrt(b*c-a*d)/(c^(5/2)*d^3)-1/8*(b*c-a*d)*(4*b*c+3*a*d)*x*sqrt(a+b*x^2)/(c^2*d^2*(c+d*x^2))],
[(a+b*x^2)^(5/2)/(c+d*x^2)^4,x,5,1/6*x*(a+b*x^2)^(5/2)/(c*(c+d*x^2)^3)+5/24*a*x*(a+b*x^2)^(3/2)/(c^2*(c+d*x^2)^2)+5/16*a^3*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))/(c^(7/2)*sqrt(b*c-a*d))+5/16*a^2*x*sqrt(a+b*x^2)/(c^3*(c+d*x^2))],
[(a+b*x^2)^(5/2)/(c+d*x^2)^5,x,6,-1/8*d*x*(a+b*x^2)^(7/2)/(c*(b*c-a*d)*(c+d*x^2)^4)+1/48*(8*b*c-7*a*d)*x*(a+b*x^2)^(5/2)/(c^2*(b*c-a*d)*(c+d*x^2)^3)+5/192*a*(8*b*c-7*a*d)*x*(a+b*x^2)^(3/2)/(c^3*(b*c-a*d)*(c+d*x^2)^2)+5/128*a^3*(8*b*c-7*a*d)*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))/(c^(9/2)*(b*c-a*d)^(3/2))+5/128*a^2*(8*b*c-7*a*d)*x*sqrt(a+b*x^2)/(c^4*(b*c-a*d)*(c+d*x^2))],
[sqrt(1-x^2)/(1+x^2),x,4,-arcsin(x)+arctan(x*sqrt(2)/sqrt(1-x^2))*sqrt(2)],
[sqrt(1+x^2)/(-1+x^2),x,4,arcsinh(x)-arctanh(x*sqrt(2)/sqrt(1+x^2))*sqrt(2)],
[sqrt(1-x^2)/(-1+2*x^2),x,4,-1/2*arcsin(x)-1/2*arctanh(x/sqrt(1-x^2))],

# p<0
[(c+d*x^2)^3/(a+b*x^2)^(1/2),x,5,1/16*(2*b*c-a*d)*(8*b^2*c^2-8*a*b*c*d+5*a^2*d^2)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(7/2)+1/48*d*(44*b^2*c^2-44*a*b*c*d+15*a^2*d^2)*x*sqrt(a+b*x^2)/b^3+5/24*d*(2*b*c-a*d)*x*(c+d*x^2)*sqrt(a+b*x^2)/b^2+1/6*d*x*(c+d*x^2)^2*sqrt(a+b*x^2)/b],
[(c+d*x^2)^2/(a+b*x^2)^(1/2),x,4,1/8*(8*b^2*c^2-8*a*b*c*d+3*a^2*d^2)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(5/2)+3/8*d*(2*b*c-a*d)*x*sqrt(a+b*x^2)/b^2+1/4*d*x*(c+d*x^2)*sqrt(a+b*x^2)/b],
[(c+d*x^2)/(a+b*x^2)^(1/2),x,3,1/2*(2*b*c-a*d)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(3/2)+1/2*d*x*sqrt(a+b*x^2)/b],
[1/(a+b*x^2)^(1/2),x,2,arctanh(x*sqrt(b)/sqrt(a+b*x^2))/sqrt(b)],
[1/((a+b*x^2)^(1/2)*(c+d*x^2)),x,2,arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))/(sqrt(c)*sqrt(b*c-a*d))],
[1/((a+b*x^2)^(1/2)*(c+d*x^2)^2),x,3,1/2*(2*b*c-a*d)*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))/(c^(3/2)*(b*c-a*d)^(3/2))-1/2*d*x*sqrt(a+b*x^2)/(c*(b*c-a*d)*(c+d*x^2))],
[1/((a+b*x^2)^(1/2)*(c+d*x^2)^3),x,5,1/8*(8*b^2*c^2-8*a*b*c*d+3*a^2*d^2)*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))/(c^(5/2)*(b*c-a*d)^(5/2))-1/4*d*x*sqrt(a+b*x^2)/(c*(b*c-a*d)*(c+d*x^2)^2)-3/8*d*(2*b*c-a*d)*x*sqrt(a+b*x^2)/(c^2*(b*c-a*d)^2*(c+d*x^2))],
[(c+d*x^2)^4/(a+b*x^2)^(3/2),x,6,1/16*d*(64*b^3*c^3-144*a*b^2*c^2*d+120*a^2*b*c*d^2-35*a^3*d^3)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(9/2)+(b*c-a*d)*x*(c+d*x^2)^3/(a*b*sqrt(a+b*x^2))-1/48*d*(48*b^3*c^3-248*a*b^2*c^2*d+290*a^2*b*c*d^2-105*a^3*d^3)*x*sqrt(a+b*x^2)/(a*b^4)-1/24*d*(24*b^2*c^2-64*a*b*c*d+35*a^2*d^2)*x*(c+d*x^2)*sqrt(a+b*x^2)/(a*b^3)-1/6*d*(6*b*c-7*a*d)*x*(c+d*x^2)^2*sqrt(a+b*x^2)/(a*b^2)],
[(c+d*x^2)^3/(a+b*x^2)^(3/2),x,5,3/8*d*(8*b^2*c^2-12*a*b*c*d+5*a^2*d^2)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(7/2)+(b*c-a*d)*x*(c+d*x^2)^2/(a*b*sqrt(a+b*x^2))-1/8*d*(2*b*c-5*a*d)*(4*b*c-3*a*d)*x*sqrt(a+b*x^2)/(a*b^3)-1/4*d*(4*b*c-5*a*d)*x*(c+d*x^2)*sqrt(a+b*x^2)/(a*b^2)],
[(c+d*x^2)^2/(a+b*x^2)^(3/2),x,4,1/2*d*(4*b*c-3*a*d)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(5/2)+(b*c-a*d)^2*x/(a*b^2*sqrt(a+b*x^2))+1/2*d^2*x*sqrt(a+b*x^2)/b^2,1/2*d*(4*b*c-3*a*d)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(5/2)+(b*c-a*d)*x*(c+d*x^2)/(a*b*sqrt(a+b*x^2))-1/2*d*(2*b*c-3*a*d)*x*sqrt(a+b*x^2)/(a*b^2)],
[(c+d*x^2)/(a+b*x^2)^(3/2),x,3,d*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(3/2)+(b*c-a*d)*x/(a*b*sqrt(a+b*x^2))],
[1/(a+b*x^2)^(3/2),x,1,x/(a*sqrt(a+b*x^2))],
[1/((a+b*x^2)^(3/2)*(c+d*x^2)),x,3,-d*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))/((b*c-a*d)^(3/2)*sqrt(c))+b*x/(a*(b*c-a*d)*sqrt(a+b*x^2))],
[1/((a+b*x^2)^(3/2)*(c+d*x^2)^2),x,5,-1/2*d*(4*b*c-a*d)*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))/(c^(3/2)*(b*c-a*d)^(5/2))+1/2*b*(2*b*c+a*d)*x/(a*c*(b*c-a*d)^2*sqrt(a+b*x^2))-1/2*d*x/(c*(b*c-a*d)*(c+d*x^2)*sqrt(a+b*x^2))],
[1/((a+b*x^2)^(3/2)*(c+d*x^2)^3),x,6,-3/8*d*(8*b^2*c^2-4*a*b*c*d+a^2*d^2)*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))/(c^(5/2)*(b*c-a*d)^(7/2))-1/4*d*x/(c*(b*c-a*d)*(c+d*x^2)^2*sqrt(a+b*x^2))+1/4*b*(4*b*c+a*d)*x/(a*c*(b*c-a*d)^2*(c+d*x^2)*sqrt(a+b*x^2))+1/8*d*(4*b*c-a*d)*(2*b*c+3*a*d)*x*sqrt(a+b*x^2)/(a*c^2*(b*c-a*d)^3*(c+d*x^2))],
[(c+d*x^2)^4/(a+b*x^2)^(5/2),x,6,1/3*(b*c-a*d)*x*(c+d*x^2)^3/(a*b*(a+b*x^2)^(3/2))+1/8*d^2*(48*b^2*c^2-80*a*b*c*d+35*a^2*d^2)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(9/2)+1/3*(b*c-a*d)*(2*b*c+7*a*d)*x*(c+d*x^2)^2/(a^2*b^2*sqrt(a+b*x^2))-1/24*d*(16*b^3*c^3+40*a*b^2*c^2*d-170*a^2*b*c*d^2+105*a^3*d^3)*x*sqrt(a+b*x^2)/(a^2*b^4)-1/12*d*(8*b^2*c^2+24*a*b*c*d-35*a^2*d^2)*x*(c+d*x^2)*sqrt(a+b*x^2)/(a^2*b^3)],
[(c+d*x^2)^3/(a+b*x^2)^(5/2),x,5,1/3*(b*c-a*d)*x*(c+d*x^2)^2/(a*b*(a+b*x^2)^(3/2))+1/2*d^2*(6*b*c-5*a*d)*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(7/2)+1/3*(b*c-a*d)*(2*b*c+5*a*d)*x*(c+d*x^2)/(a^2*b^2*sqrt(a+b*x^2))-1/6*d*(4*b^2*c^2+8*a*b*c*d-15*a^2*d^2)*x*sqrt(a+b*x^2)/(a^2*b^3)],
[(c+d*x^2)^2/(a+b*x^2)^(5/2),x,4,1/3*(b*c-a*d)*x*(c+d*x^2)/(a*b*(a+b*x^2)^(3/2))+d^2*arctanh(x*sqrt(b)/sqrt(a+b*x^2))/b^(5/2)+1/3*(b*c-a*d)*(2*b*c+3*a*d)*x/(a^2*b^2*sqrt(a+b*x^2))],
[(c+d*x^2)/(a+b*x^2)^(5/2),x,2,1/3*x*(c+d*x^2)/(a*(a+b*x^2)^(3/2))+2/3*c*x/(a^2*sqrt(a+b*x^2))],
[1/(a+b*x^2)^(5/2),x,2,1/3*x/(a*(a+b*x^2)^(3/2))+2/3*x/(a^2*sqrt(a+b*x^2))],
[1/((a+b*x^2)^(5/2)*(c+d*x^2)),x,5,1/3*b*x/(a*(b*c-a*d)*(a+b*x^2)^(3/2))+d^2*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))/((b*c-a*d)^(5/2)*sqrt(c))+1/3*b*(2*b*c-5*a*d)*x/(a^2*(b*c-a*d)^2*sqrt(a+b*x^2))],
[1/((a+b*x^2)^(5/2)*(c+d*x^2)^2),x,6,1/6*b*(2*b*c+3*a*d)*x/(a*c*(b*c-a*d)^2*(a+b*x^2)^(3/2))-1/2*d*x/(c*(b*c-a*d)*(a+b*x^2)^(3/2)*(c+d*x^2))+1/2*d^2*(6*b*c-a*d)*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))/(c^(3/2)*(b*c-a*d)^(7/2))+1/6*b*(4*b^2*c^2-16*a*b*c*d-3*a^2*d^2)*x/(a^2*c*(b*c-a*d)^3*sqrt(a+b*x^2))],
[1/((a+b*x^2)^(5/2)*(c+d*x^2)^3),x,7,-1/4*d*x/(c*(b*c-a*d)*(a+b*x^2)^(3/2)*(c+d*x^2)^2)+1/12*b*(4*b*c+3*a*d)*x/(a*c*(b*c-a*d)^2*(a+b*x^2)^(3/2)*(c+d*x^2))+1/8*d^2*(48*b^2*c^2-16*a*b*c*d+3*a^2*d^2)*arctanh(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)))/(c^(5/2)*(b*c-a*d)^(9/2))+1/12*b*(8*b^2*c^2-40*a*b*c*d-3*a^2*d^2)*x/(a^2*c*(b*c-a*d)^3*(c+d*x^2)*sqrt(a+b*x^2))+1/24*d*(16*b^3*c^3-88*a*b^2*c^2*d-42*a^2*b*c*d^2+9*a^3*d^3)*x*sqrt(a+b*x^2)/(a^2*c^2*(b*c-a*d)^4*(c+d*x^2))],
[(a+b*x^2)^3/(c+d*x^2)^(11/2),x,5,-1/9*d*x*(a+b*x^2)^4/(c*(b*c-a*d)*(c+d*x^2)^(9/2))+1/63*(9*b*c-8*a*d)*x*(a+b*x^2)^3/(c^2*(b*c-a*d)*(c+d*x^2)^(7/2))+2/105*a*(9*b*c-8*a*d)*x*(a+b*x^2)^2/(c^3*(b*c-a*d)*(c+d*x^2)^(5/2))+8/315*a^2*(9*b*c-8*a*d)*x*(a+b*x^2)/(c^4*(b*c-a*d)*(c+d*x^2)^(3/2))+16/315*a^3*(9*b*c-8*a*d)*x/(c^5*(b*c-a*d)*sqrt(c+d*x^2))],
[(a+b*x^2)^2/(c+d*x^2)^(9/2),x,4,-1/7*d*x*(a+b*x^2)^3/(c*(b*c-a*d)*(c+d*x^2)^(7/2))+1/35*(7*b*c-6*a*d)*x*(a+b*x^2)^2/(c^2*(b*c-a*d)*(c+d*x^2)^(5/2))+4/105*a*(7*b*c-6*a*d)*x*(a+b*x^2)/(c^3*(b*c-a*d)*(c+d*x^2)^(3/2))+8/105*a^2*(7*b*c-6*a*d)*x/(c^4*(b*c-a*d)*sqrt(c+d*x^2))],
[(a+b*x^2)/(c+d*x^2)^(7/2),x,3,-1/5*(b*c-a*d)*x/(c*d*(c+d*x^2)^(5/2))+1/15*(b*c+4*a*d)*x/(c^2*d*(c+d*x^2)^(3/2))+2/15*(b*c+4*a*d)*x/(c^3*d*sqrt(c+d*x^2))],
[1/(c+d*x^2)^(5/2),x,2,1/3*x/(c*(c+d*x^2)^(3/2))+2/3*x/(c^2*sqrt(c+d*x^2))],
[1/((a+b*x^2)*(c+d*x^2)^(3/2)),x,3,b*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/((b*c-a*d)^(3/2)*sqrt(a))-d*x/(c*(b*c-a*d)*sqrt(c+d*x^2))],
[1/((a+b*x^2)^2*(c+d*x^2)^(1/2)),x,3,1/2*(b*c-2*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(3/2)*(b*c-a*d)^(3/2))+1/2*b*x*sqrt(c+d*x^2)/(a*(b*c-a*d)*(a+b*x^2))],
[(c+d*x^2)^(1/2)/(a+b*x^2)^3,x,4,1/4*b*x*(c+d*x^2)^(3/2)/(a*(b*c-a*d)*(a+b*x^2)^2)+1/8*c*(3*b*c-4*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(5/2)*(b*c-a*d)^(3/2))+1/8*(3*b*c-4*a*d)*x*sqrt(c+d*x^2)/(a^2*(b*c-a*d)*(a+b*x^2))],
[(c+d*x^2)^(3/2)/(a+b*x^2)^4,x,5,1/24*(5*b*c-6*a*d)*x*(c+d*x^2)^(3/2)/(a^2*(b*c-a*d)*(a+b*x^2)^2)+1/6*b*x*(c+d*x^2)^(5/2)/(a*(b*c-a*d)*(a+b*x^2)^3)+1/16*c^2*(5*b*c-6*a*d)*arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(a^(7/2)*(b*c-a*d)^(3/2))+1/16*c*(5*b*c-6*a*d)*x*sqrt(c+d*x^2)/(a^3*(b*c-a*d)*(a+b*x^2))],
[1/((b*c/d+b*x^2)*sqrt(c+d*x^2)),x,2,d*x/(b*c*sqrt(c+d*x^2))],
[1/((1+x^2)*sqrt(1-x^2)),x,2,arctan(x*sqrt(2)/sqrt(1-x^2))/sqrt(2)],
[1/((a+b*x^2)*sqrt(c+d*x^2)),x,2,arctan(x*sqrt(b*c-a*d)/(sqrt(a)*sqrt(c+d*x^2)))/(sqrt(a)*sqrt(b*c-a*d))],
[(-1+x^2)/(1+x^2)^(3/2),x,2,arcsinh(x)-2*x/sqrt(1+x^2)],

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

# Integrands of the form (a+b x^2)^(p/3) (c+d x^2)^q when b c+3 a d=0

# p>0
[(a-b*x^2)^(2/3)*(3*a+b*x^2)^3,x,8,18144/1235*a^3*x*(a-b*x^2)^(2/3)-23544/6175*a^2*x*(a-b*x^2)^(5/3)-378/475*a*x*(a-b*x^2)^(5/3)*(3*a+b*x^2)-3/25*x*(a-b*x^2)^(5/3)*(3*a+b*x^2)^2-72576/1235*a^4*x/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))+24192/1235*3^(3/4)*a^(13/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))-36288/1235*3^(1/4)*a^(13/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(a-b*x^2)^(2/3)*(3*a+b*x^2)^2,x,7,7776/1729*a^2*x*(a-b*x^2)^(2/3)-252/247*a*x*(a-b*x^2)^(5/3)-3/19*x*(a-b*x^2)^(5/3)*(3*a+b*x^2)-31104/1729*a^3*x/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))+10368/1729*3^(3/4)*a^(10/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))-15552/1729*3^(1/4)*a^(10/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(a-b*x^2)^(2/3)*(3*a+b*x^2),x,6,18/13*a*x*(a-b*x^2)^(2/3)-3/13*x*(a-b*x^2)^(5/3)-72/13*a^2*x/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))+24/13*3^(3/4)*a^(7/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))-36/13*3^(1/4)*a^(7/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(a-b*x^2)^(2/3)/(3*a+b*x^2),x,6,3*x/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))+2^(1/3)*a^(1/6)*arctanh(x*sqrt(b)/(a^(1/6)*(a^(1/3)+2^(1/3)*(a-b*x^2)^(1/3))))/sqrt(b)-1/3*2^(1/3)*a^(1/6)*arctanh(x*sqrt(b)/sqrt(a))/sqrt(b)+2^(1/3)*a^(1/6)*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*(a-b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))/(sqrt(3)*sqrt(b))+2^(1/3)*a^(1/6)*arctan(sqrt(3)*sqrt(a)/(x*sqrt(b)))/(sqrt(3)*sqrt(b))-3^(3/4)*a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))+3/2*3^(1/4)*a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(a-b*x^2)^(2/3)/(3*a+b*x^2)^2,x,6,1/6*x*(a-b*x^2)^(2/3)/(a*(3*a+b*x^2))-1/6*x/(a*(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))))+1/3*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(3^(1/4)*a^(2/3)*b*x*sqrt(2)*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))-1/4*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(3^(3/4)*a^(2/3)*b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(a-b*x^2)^(2/3)/(3*a+b*x^2)^3,x,8,1/12*x*(a-b*x^2)^(2/3)/(a*(3*a+b*x^2)^2)+1/36*x*(a-b*x^2)^(2/3)/(a^2*(3*a+b*x^2))-1/36*x/(a^2*(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))))+1/72*arctanh(x*sqrt(b)/(a^(1/6)*(a^(1/3)+2^(1/3)*(a-b*x^2)^(1/3))))/(2^(2/3)*a^(11/6)*sqrt(b))-1/216*arctanh(x*sqrt(b)/sqrt(a))/(2^(2/3)*a^(11/6)*sqrt(b))+1/72*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*(a-b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))/(2^(2/3)*a^(11/6)*sqrt(3)*sqrt(b))+1/72*arctan(sqrt(3)*sqrt(a)/(x*sqrt(b)))/(2^(2/3)*a^(11/6)*sqrt(3)*sqrt(b))+1/18*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(3^(1/4)*a^(5/3)*b*x*sqrt(2)*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))-1/24*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(3^(3/4)*a^(5/3)*b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(a-b*x^2)^(2/3)/(3*a+b*x^2)^4,x,9,1/18*x*(a-b*x^2)^(2/3)/(a*(3*a+b*x^2)^3)+1/54*x*(a-b*x^2)^(2/3)/(a^2*(3*a+b*x^2)^2)+1/144*x*(a-b*x^2)^(2/3)/(a^3*(3*a+b*x^2))-1/144*x/(a^3*(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))))+7/1296*arctanh(x*sqrt(b)/(a^(1/6)*(a^(1/3)+2^(1/3)*(a-b*x^2)^(1/3))))/(2^(2/3)*a^(17/6)*sqrt(b))-7/3888*arctanh(x*sqrt(b)/sqrt(a))/(2^(2/3)*a^(17/6)*sqrt(b))+7/1296*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*(a-b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))/(2^(2/3)*a^(17/6)*sqrt(3)*sqrt(b))+7/1296*arctan(sqrt(3)*sqrt(a)/(x*sqrt(b)))/(2^(2/3)*a^(17/6)*sqrt(3)*sqrt(b))+1/72*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(3^(1/4)*a^(8/3)*b*x*sqrt(2)*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))-1/96*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(3^(3/4)*a^(8/3)*b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(a-b*x^2)^(5/3)*(3*a+b*x^2)^3,x,9,2809728/267995*a^4*x*(a-b*x^2)^(2/3)+1404864/191425*a^3*x*(a-b*x^2)^(5/3)-33264/14725*a^2*x*(a-b*x^2)^(8/3)-432/775*a*x*(a-b*x^2)^(8/3)*(3*a+b*x^2)-3/31*x*(a-b*x^2)^(8/3)*(3*a+b*x^2)^2-11238912/267995*a^5*x/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))+3746304/267995*3^(3/4)*a^(16/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))-5619456/267995*3^(1/4)*a^(16/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(a-b*x^2)^(5/3)*(3*a+b*x^2)^2,x,8,28512/8645*a^3*x*(a-b*x^2)^(2/3)+14256/6175*a^2*x*(a-b*x^2)^(5/3)-306/475*a*x*(a-b*x^2)^(8/3)-3/25*x*(a-b*x^2)^(8/3)*(3*a+b*x^2)-114048/8645*a^4*x/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))+38016/8645*3^(3/4)*a^(13/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))-57024/8645*3^(1/4)*a^(13/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(a-b*x^2)^(5/3)*(3*a+b*x^2),x,7,1800/1729*a^2*x*(a-b*x^2)^(2/3)+180/247*a*x*(a-b*x^2)^(5/3)-3/19*x*(a-b*x^2)^(8/3)-7200/1729*a^3*x/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))+2400/1729*3^(3/4)*a^(10/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))-3600/1729*3^(1/4)*a^(10/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(a-b*x^2)^(5/3)/(3*a+b*x^2),x,7,-3/7*x*(a-b*x^2)^(2/3)+96/7*a*x/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))+4*2^(1/3)*a^(7/6)*arctanh(x*sqrt(b)/(a^(1/6)*(a^(1/3)+2^(1/3)*(a-b*x^2)^(1/3))))/sqrt(b)-4/3*2^(1/3)*a^(7/6)*arctanh(x*sqrt(b)/sqrt(a))/sqrt(b)+4*2^(1/3)*a^(7/6)*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*(a-b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))/(sqrt(3)*sqrt(b))+4*2^(1/3)*a^(7/6)*arctan(sqrt(3)*sqrt(a)/(x*sqrt(b)))/(sqrt(3)*sqrt(b))-32/7*3^(3/4)*a^(4/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))+48/7*3^(1/4)*a^(4/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(a-b*x^2)^(5/3)/(3*a+b*x^2)^2,x,7,2/3*x*(a-b*x^2)^(2/3)/(3*a+b*x^2)-11/3*x/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))-2^(1/3)*a^(1/6)*arctanh(x*sqrt(b)/(a^(1/6)*(a^(1/3)+2^(1/3)*(a-b*x^2)^(1/3))))/sqrt(b)+1/3*2^(1/3)*a^(1/6)*arctanh(x*sqrt(b)/sqrt(a))/sqrt(b)-2^(1/3)*a^(1/6)*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*(a-b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))/(sqrt(3)*sqrt(b))-2^(1/3)*a^(1/6)*arctan(sqrt(3)*sqrt(a)/(x*sqrt(b)))/(sqrt(3)*sqrt(b))+11/3*a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(3^(1/4)*b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))-11/2*a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(3^(3/4)*b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(a-b*x^2)^(5/3)/(3*a+b*x^2)^3,x,9,1/3*x*(a-b*x^2)^(2/3)/(3*a+b*x^2)^2-1/18*x*(a-b*x^2)^(2/3)/(a*(3*a+b*x^2))+1/18*x/(a*(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))))+1/18*arctanh(x*sqrt(b)/(a^(1/6)*(a^(1/3)+2^(1/3)*(a-b*x^2)^(1/3))))/(2^(2/3)*a^(5/6)*sqrt(b))-1/54*arctanh(x*sqrt(b)/sqrt(a))/(2^(2/3)*a^(5/6)*sqrt(b))+1/18*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*(a-b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))/(2^(2/3)*a^(5/6)*sqrt(3)*sqrt(b))+1/18*arctan(sqrt(3)*sqrt(a)/(x*sqrt(b)))/(2^(2/3)*a^(5/6)*sqrt(3)*sqrt(b))-1/9*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(3^(1/4)*a^(2/3)*b*x*sqrt(2)*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))+1/12*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(3^(3/4)*a^(2/3)*b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],

# p<0
[(3*a+b*x^2)^4/(a-b*x^2)^(1/3),x,8,-1552608/43225*a^3*x*(a-b*x^2)^(2/3)-36288/6175*a^2*x*(a-b*x^2)^(2/3)*(3*a+b*x^2)-18/19*a*x*(a-b*x^2)^(2/3)*(3*a+b*x^2)^2-3/25*x*(a-b*x^2)^(2/3)*(3*a+b*x^2)^3-3794688/8645*a^4*x/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))+1264896/8645*3^(3/4)*a^(13/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))-1897344/8645*3^(1/4)*a^(13/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(3*a+b*x^2)^3/(a-b*x^2)^(1/3),x,7,-15768/1729*a^2*x*(a-b*x^2)^(2/3)-324/247*a*x*(a-b*x^2)^(2/3)*(3*a+b*x^2)-3/19*x*(a-b*x^2)^(2/3)*(3*a+b*x^2)^2-215136/1729*a^3*x/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))+71712/1729*3^(3/4)*a^(10/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))-107568/1729*3^(1/4)*a^(10/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(3*a+b*x^2)^2/(a-b*x^2)^(1/3),x,6,-198/91*a*x*(a-b*x^2)^(2/3)-3/13*x*(a-b*x^2)^(2/3)*(3*a+b*x^2)-3240/91*a^2*x/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))+1080/91*3^(3/4)*a^(7/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))-1620/91*3^(1/4)*a^(7/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(3*a+b*x^2)/(a-b*x^2)^(1/3),x,5,-3/7*x*(a-b*x^2)^(2/3)-72/7*a*x/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))+24/7*3^(3/4)*a^(4/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))-36/7*3^(1/4)*a^(4/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[1/((a-b*x^2)^(1/3)*(3*a+b*x^2)),x,1,1/2*arctanh(x*sqrt(b)/(a^(1/6)*(a^(1/3)+2^(1/3)*(a-b*x^2)^(1/3))))/(2^(2/3)*a^(5/6)*sqrt(b))-1/6*arctanh(x*sqrt(b)/sqrt(a))/(2^(2/3)*a^(5/6)*sqrt(b))+1/2*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*(a-b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))/(2^(2/3)*a^(5/6)*sqrt(3)*sqrt(b))+1/2*arctan(sqrt(3)*sqrt(a)/(x*sqrt(b)))/(2^(2/3)*a^(5/6)*sqrt(3)*sqrt(b))],
[1/((a-b*x^2)^(1/3)*(3*a+b*x^2)^2),x,7,1/24*x*(a-b*x^2)^(2/3)/(a^2*(3*a+b*x^2))-1/24*x/(a^2*(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))))+1/8*arctanh(x*sqrt(b)/(a^(1/6)*(a^(1/3)+2^(1/3)*(a-b*x^2)^(1/3))))/(2^(2/3)*a^(11/6)*sqrt(b))-1/24*arctanh(x*sqrt(b)/sqrt(a))/(2^(2/3)*a^(11/6)*sqrt(b))+1/8*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*(a-b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))/(2^(2/3)*a^(11/6)*sqrt(3)*sqrt(b))+1/8*arctan(sqrt(3)*sqrt(a)/(x*sqrt(b)))/(2^(2/3)*a^(11/6)*sqrt(3)*sqrt(b))+1/12*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(3^(1/4)*a^(5/3)*b*x*sqrt(2)*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))-1/16*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(3^(3/4)*a^(5/3)*b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[1/((a-b*x^2)^(1/3)*(3*a+b*x^2)^3),x,8,1/48*x*(a-b*x^2)^(2/3)/(a^2*(3*a+b*x^2)^2)+5/288*x*(a-b*x^2)^(2/3)/(a^3*(3*a+b*x^2))-5/288*x/(a^3*(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))))+5/144*arctanh(x*sqrt(b)/(a^(1/6)*(a^(1/3)+2^(1/3)*(a-b*x^2)^(1/3))))/(2^(2/3)*a^(17/6)*sqrt(b))-5/432*arctanh(x*sqrt(b)/sqrt(a))/(2^(2/3)*a^(17/6)*sqrt(b))+5/144*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*(a-b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))/(2^(2/3)*a^(17/6)*sqrt(3)*sqrt(b))+5/144*arctan(sqrt(3)*sqrt(a)/(x*sqrt(b)))/(2^(2/3)*a^(17/6)*sqrt(3)*sqrt(b))+5/144*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(3^(1/4)*a^(8/3)*b*x*sqrt(2)*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))-5/192*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(3^(3/4)*a^(8/3)*b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(3*a+b*x^2)^3/(a-b*x^2)^(4/3),x,7,2538/91*a*x*(a-b*x^2)^(2/3)+81/13*x*(a-b*x^2)^(2/3)*(3*a+b*x^2)+6*x*(3*a+b*x^2)^2/(a-b*x^2)^(1/3)+20088/91*a^2*x/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))-6696/91*3^(3/4)*a^(7/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))+10044/91*3^(1/4)*a^(7/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(3*a+b*x^2)^2/(a-b*x^2)^(4/3),x,6,45/7*x*(a-b*x^2)^(2/3)+6*x*(3*a+b*x^2)/(a-b*x^2)^(1/3)+324/7*a*x/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))-108/7*3^(3/4)*a^(4/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))+162/7*3^(1/4)*a^(4/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(3*a+b*x^2)/(a-b*x^2)^(4/3),x,5,6*x/(a-b*x^2)^(1/3)+9*x/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))-3*3^(3/4)*a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))+9/2*3^(1/4)*a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[1/((a-b*x^2)^(4/3)*(3*a+b*x^2)),x,7,3/8*x/(a^2*(a-b*x^2)^(1/3))+3/8*x/(a^2*(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))))+1/8*arctanh(x*sqrt(b)/(a^(1/6)*(a^(1/3)+2^(1/3)*(a-b*x^2)^(1/3))))/(2^(2/3)*a^(11/6)*sqrt(b))-1/24*arctanh(x*sqrt(b)/sqrt(a))/(2^(2/3)*a^(11/6)*sqrt(b))+1/8*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*(a-b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))/(2^(2/3)*a^(11/6)*sqrt(3)*sqrt(b))+1/8*arctan(sqrt(3)*sqrt(a)/(x*sqrt(b)))/(2^(2/3)*a^(11/6)*sqrt(3)*sqrt(b))-1/4*3^(3/4)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(a^(5/3)*b*x*sqrt(2)*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))+3/16*3^(1/4)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(a^(5/3)*b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[1/((a-b*x^2)^(4/3)*(3*a+b*x^2)^2),x,8,1/12*x/(a^3*(a-b*x^2)^(1/3))+1/24*x/(a^2*(a-b*x^2)^(1/3)*(3*a+b*x^2))+1/12*x/(a^3*(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))))+1/16*arctanh(x*sqrt(b)/(a^(1/6)*(a^(1/3)+2^(1/3)*(a-b*x^2)^(1/3))))/(2^(2/3)*a^(17/6)*sqrt(b))-1/48*arctanh(x*sqrt(b)/sqrt(a))/(2^(2/3)*a^(17/6)*sqrt(b))+1/16*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*(a-b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))/(2^(2/3)*a^(17/6)*sqrt(3)*sqrt(b))+1/16*arctan(sqrt(3)*sqrt(a)/(x*sqrt(b)))/(2^(2/3)*a^(17/6)*sqrt(3)*sqrt(b))-1/6*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(3^(1/4)*a^(8/3)*b*x*sqrt(2)*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))+1/8*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(3^(3/4)*a^(8/3)*b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[1/((a-b*x^2)^(4/3)*(3*a+b*x^2)^3),x,9,1/48*x/(a^2*(a-b*x^2)^(1/3)*(3*a+b*x^2)^2)+17/192*x/(a^3*(a-b*x^2)^(1/3)*(3*a+b*x^2))-19/1152*x*(a-b*x^2)^(2/3)/(a^4*(3*a+b*x^2))+19/1152*x/(a^4*(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))))+7/288*arctanh(x*sqrt(b)/(a^(1/6)*(a^(1/3)+2^(1/3)*(a-b*x^2)^(1/3))))/(2^(2/3)*a^(23/6)*sqrt(b))-7/864*arctanh(x*sqrt(b)/sqrt(a))/(2^(2/3)*a^(23/6)*sqrt(b))+7/288*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*(a-b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))/(2^(2/3)*a^(23/6)*sqrt(3)*sqrt(b))+7/288*arctan(sqrt(3)*sqrt(a)/(x*sqrt(b)))/(2^(2/3)*a^(23/6)*sqrt(3)*sqrt(b))-19/576*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(3^(1/4)*a^(11/3)*b*x*sqrt(2)*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))+19/768*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(3^(3/4)*a^(11/3)*b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(3*a+b*x^2)^4/(a-b*x^2)^(7/3),x,8,-3240/91*a*x*(a-b*x^2)^(2/3)-81/13*x*(a-b*x^2)^(2/3)*(3*a+b*x^2)-9/2*x*(3*a+b*x^2)^2/(a-b*x^2)^(1/3)+3/2*x*(3*a+b*x^2)^3/(a-b*x^2)^(4/3)-36936/91*a^2*x/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))+12312/91*3^(3/4)*a^(7/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))-18468/91*3^(1/4)*a^(7/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(3*a+b*x^2)^3/(a-b*x^2)^(7/3),x,7,-27/14*x*(a-b*x^2)^(2/3)+3/2*x*(3*a+b*x^2)^2/(a-b*x^2)^(4/3)-324/7*a*x/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))+108/7*3^(3/4)*a^(4/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt(2)*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))-162/7*3^(1/4)*a^(4/3)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[(3*a+b*x^2)^2/(a-b*x^2)^(7/3),x,2,9/2*x/(a-b*x^2)^(1/3)+3/2*x*(3*a+b*x^2)/(a-b*x^2)^(4/3)],
[(3*a+b*x^2)/(a-b*x^2)^(7/3),x,6,3/2*x/(a-b*x^2)^(4/3)+9/4*x/(a*(a-b*x^2)^(1/3))+9/4*x/(a*(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))))-3/2*3^(3/4)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(a^(2/3)*b*x*sqrt(2)*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))+9/8*3^(1/4)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(a^(2/3)*b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[1/((a-b*x^2)^(7/3)*(3*a+b*x^2)),x,8,3/32*x/(a^2*(a-b*x^2)^(4/3))+21/64*x/(a^3*(a-b*x^2)^(1/3))+21/64*x/(a^3*(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))))+1/32*arctanh(x*sqrt(b)/(a^(1/6)*(a^(1/3)+2^(1/3)*(a-b*x^2)^(1/3))))/(2^(2/3)*a^(17/6)*sqrt(b))-1/96*arctanh(x*sqrt(b)/sqrt(a))/(2^(2/3)*a^(17/6)*sqrt(b))+1/32*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*(a-b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))/(2^(2/3)*a^(17/6)*sqrt(3)*sqrt(b))+1/32*arctan(sqrt(3)*sqrt(a)/(x*sqrt(b)))/(2^(2/3)*a^(17/6)*sqrt(3)*sqrt(b))-7/32*3^(3/4)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(a^(8/3)*b*x*sqrt(2)*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))+21/128*3^(1/4)*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(a^(8/3)*b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[1/((a-b*x^2)^(7/3)*(3*a+b*x^2)^2),x,9,5/384*x/(a^3*(a-b*x^2)^(4/3))+79/768*x/(a^4*(a-b*x^2)^(1/3))+1/24*x/(a^2*(a-b*x^2)^(4/3)*(3*a+b*x^2))+79/768*x/(a^4*(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))))+3/128*arctanh(x*sqrt(b)/(a^(1/6)*(a^(1/3)+2^(1/3)*(a-b*x^2)^(1/3))))/(2^(2/3)*a^(23/6)*sqrt(b))-1/128*arctanh(x*sqrt(b)/sqrt(a))/(2^(2/3)*a^(23/6)*sqrt(b))+1/128*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*(a-b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))*sqrt(3)/(2^(2/3)*a^(23/6)*sqrt(b))+1/128*arctan(sqrt(3)*sqrt(a)/(x*sqrt(b)))*sqrt(3)/(2^(2/3)*a^(23/6)*sqrt(b))-79/384*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticF((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)/(3^(1/4)*a^(11/3)*b*x*sqrt(2)*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))+79/512*(a^(1/3)-(a-b*x^2)^(1/3))*EllipticE((-(a-b*x^2)^(1/3)+a^(1/3)*(1+sqrt(3)))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3))),sqrt(-7+4*sqrt(3)))*sqrt((a^(2/3)+a^(1/3)*(a-b*x^2)^(1/3)+(a-b*x^2)^(2/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2)*sqrt(2+sqrt(3))/(3^(3/4)*a^(11/3)*b*x*sqrt(-a^(1/3)*(a^(1/3)-(a-b*x^2)^(1/3))/(-(a-b*x^2)^(1/3)+a^(1/3)*(1-sqrt(3)))^2))],
[1/((-3*a-b*x^2)*(-a+b*x^2)^(1/3)),x,1,1/6*arctanh(x*sqrt(b)/sqrt(a))/(2^(2/3)*(-a)^(1/3)*sqrt(a)*sqrt(b))-1/2*arctanh((-a)^(1/3)*x*sqrt(b)/(((-a)^(1/3)+2^(1/3)*(-a+b*x^2)^(1/3))*sqrt(a)))/(2^(2/3)*(-a)^(1/3)*sqrt(a)*sqrt(b))-1/2*arctan(sqrt(3)*sqrt(a)/(x*sqrt(b)))/(2^(2/3)*(-a)^(1/3)*sqrt(3)*sqrt(a)*sqrt(b))-1/2*arctan(((-a)^(1/3)-2^(1/3)*(-a+b*x^2)^(1/3))*sqrt(3)*sqrt(a)/((-a)^(1/3)*x*sqrt(b)))/(2^(2/3)*(-a)^(1/3)*sqrt(3)*sqrt(a)*sqrt(b))],
[1/((3*a-b*x^2)*(a+b*x^2)^(1/3)),x,1,1/2*arctan(x*sqrt(b)/(a^(1/6)*(a^(1/3)+2^(1/3)*(a+b*x^2)^(1/3))))/(2^(2/3)*a^(5/6)*sqrt(b))-1/6*arctan(x*sqrt(b)/sqrt(a))/(2^(2/3)*a^(5/6)*sqrt(b))-1/2*arctanh(a^(1/6)*(a^(1/3)-2^(1/3)*(a+b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))/(2^(2/3)*a^(5/6)*sqrt(3)*sqrt(b))-1/2*arctanh(sqrt(3)*sqrt(a)/(x*sqrt(b)))/(2^(2/3)*a^(5/6)*sqrt(3)*sqrt(b))],
[1/((c-d*x^2)*(c+3*d*x^2)^(1/3)),x,1,-1/2*arctanh(c^(1/6)*(c^(1/3)-2^(1/3)*(c+3*d*x^2)^(1/3))/(x*sqrt(d)))/(2^(2/3)*c^(5/6)*sqrt(d))-1/2*arctanh(sqrt(c)/(x*sqrt(d)))/(2^(2/3)*c^(5/6)*sqrt(d))-1/2*arctan(x*sqrt(3)*sqrt(d)/sqrt(c))/(2^(2/3)*c^(5/6)*sqrt(3)*sqrt(d))+1/2*arctan(x*sqrt(3)*sqrt(d)/(c^(1/6)*(c^(1/3)+2^(1/3)*(c+3*d*x^2)^(1/3))))*sqrt(3)/(2^(2/3)*c^(5/6)*sqrt(d))],
[1/((a-b*x^2)^(1/3)*(3*a+b*x^2)),x,1,1/2*arctanh(x*sqrt(b)/(a^(1/6)*(a^(1/3)+2^(1/3)*(a-b*x^2)^(1/3))))/(2^(2/3)*a^(5/6)*sqrt(b))-1/6*arctanh(x*sqrt(b)/sqrt(a))/(2^(2/3)*a^(5/6)*sqrt(b))+1/2*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*(a-b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))/(2^(2/3)*a^(5/6)*sqrt(3)*sqrt(b))+1/2*arctan(sqrt(3)*sqrt(a)/(x*sqrt(b)))/(2^(2/3)*a^(5/6)*sqrt(3)*sqrt(b))],
[1/((c-3*d*x^2)^(1/3)*(c+d*x^2)),x,1,1/2*arctan(c^(1/6)*(c^(1/3)-2^(1/3)*(c-3*d*x^2)^(1/3))/(x*sqrt(d)))/(2^(2/3)*c^(5/6)*sqrt(d))+1/2*arctan(sqrt(c)/(x*sqrt(d)))/(2^(2/3)*c^(5/6)*sqrt(d))-1/2*arctanh(x*sqrt(3)*sqrt(d)/sqrt(c))/(2^(2/3)*c^(5/6)*sqrt(3)*sqrt(d))+1/2*arctanh(x*sqrt(3)*sqrt(d)/(c^(1/6)*(c^(1/3)+2^(1/3)*(c-3*d*x^2)^(1/3))))*sqrt(3)/(2^(2/3)*c^(5/6)*sqrt(d))],
[1/((1-x^2)^(1/3)*(3+x^2)),x,1,-1/6*arctanh(x)/2^(2/3)+1/2*arctanh(x/(1+2^(1/3)*(1-x^2)^(1/3)))/2^(2/3)+1/2*arctan(sqrt(3)/x)/(2^(2/3)*sqrt(3))+1/2*arctan((1-2^(1/3)*(1-x^2)^(1/3))*sqrt(3)/x)/(2^(2/3)*sqrt(3))],
[1/((3-x^2)*(1+x^2)^(1/3)),x,1,-1/6*arctan(x)/2^(2/3)+1/2*arctan(x/(1+2^(1/3)*(1+x^2)^(1/3)))/2^(2/3)-1/2*arctanh(sqrt(3)/x)/(2^(2/3)*sqrt(3))-1/2*arctanh((1-2^(1/3)*(1+x^2)^(1/3))*sqrt(3)/x)/(2^(2/3)*sqrt(3))],
[(3-x)/((1-x^2)^(1/3)*(3+x^2)),x,1,-1/2*log(3+x^2)/2^(2/3)+3/2*log(2^(1/3)*(1-x)^(1/3)+(1+x)^(2/3))/2^(2/3)-arctan(1/sqrt(3)-2^(2/3)*(1+x)^(2/3)/((1-x)^(1/3)*sqrt(3)))*sqrt(3)/2^(2/3)],
[(3+x)/((1-x^2)^(1/3)*(3+x^2)),x,1,1/2*log(3+x^2)/2^(2/3)-3/2*log((1-x)^(2/3)+2^(1/3)*(1+x)^(1/3))/2^(2/3)+arctan(1/sqrt(3)-2^(2/3)*(1-x)^(2/3)/((1+x)^(1/3)*sqrt(3)))*sqrt(3)/2^(2/3)],

# Integrands of the form (a+b x^2)^(p/3) (c+d x^2)^q when b c-9 a d=0
[1/((a+b*x^2)^(1/3)*(9*a*d/b+d*x^2)),x,1,1/12*arctan(1/3*(a^(1/3)-(a+b*x^2)^(1/3))^2/(a^(1/6)*x*sqrt(b)))*sqrt(b)/(a^(5/6)*d)+1/12*arctan(1/3*x*sqrt(b)/sqrt(a))*sqrt(b)/(a^(5/6)*d)-1/4*arctanh(a^(1/6)*(a^(1/3)-(a+b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))*sqrt(b)/(a^(5/6)*d*sqrt(3))],
[1/((a-b*x^2)^(1/3)*(-9*a*d/b+d*x^2)),x,1,1/12*arctanh(1/3*(a^(1/3)-(a-b*x^2)^(1/3))^2/(a^(1/6)*x*sqrt(b)))*sqrt(b)/(a^(5/6)*d)-1/12*arctanh(1/3*x*sqrt(b)/sqrt(a))*sqrt(b)/(a^(5/6)*d)-1/4*arctan(a^(1/6)*(a^(1/3)-(a-b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))*sqrt(b)/(a^(5/6)*d*sqrt(3))],
[1/((-a+b*x^2)^(1/3)*(-9*a*d/b+d*x^2)),x,1,-1/12*arctanh(1/3*(a^(1/3)+(-a+b*x^2)^(1/3))^2/(a^(1/6)*x*sqrt(b)))*sqrt(b)/(a^(5/6)*d)+1/12*arctanh(1/3*x*sqrt(b)/sqrt(a))*sqrt(b)/(a^(5/6)*d)+1/4*arctan(a^(1/6)*(a^(1/3)+(-a+b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))*sqrt(b)/(a^(5/6)*d*sqrt(3))],
[1/((-a-b*x^2)^(1/3)*(9*a*d/b+d*x^2)),x,1,-1/12*arctan(1/3*(a^(1/3)+(-a-b*x^2)^(1/3))^2/(a^(1/6)*x*sqrt(b)))*sqrt(b)/(a^(5/6)*d)-1/12*arctan(1/3*x*sqrt(b)/sqrt(a))*sqrt(b)/(a^(5/6)*d)+1/4*arctanh(a^(1/6)*(a^(1/3)+(-a-b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))*sqrt(b)/(a^(5/6)*d*sqrt(3))],
[1/((2+b*x^2)^(1/3)*(18*d/b+d*x^2)),x,1,1/12*arctan(1/3*(2^(1/3)-(2+b*x^2)^(1/3))^2/(2^(1/6)*x*sqrt(b)))*sqrt(b)/(2^(5/6)*d)+1/12*arctan(1/3*x*sqrt(b)/sqrt(2))*sqrt(b)/(2^(5/6)*d)-1/4*arctanh(2^(1/6)*(2^(1/3)-(2+b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))*sqrt(b)/(2^(5/6)*d*sqrt(3))],
[1/((-2+b*x^2)^(1/3)*(-18*d/b+d*x^2)),x,1,-1/12*arctanh(1/3*(2^(1/3)+(-2+b*x^2)^(1/3))^2/(2^(1/6)*x*sqrt(b)))*sqrt(b)/(2^(5/6)*d)+1/12*arctanh(1/3*x*sqrt(b)/sqrt(2))*sqrt(b)/(2^(5/6)*d)+1/4*arctan(2^(1/6)*(2^(1/3)+(-2+b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))*sqrt(b)/(2^(5/6)*d*sqrt(3))],
[1/((2+3*x^2)^(1/3)*(6*d+d*x^2)),x,1,-1/4*arctanh(2^(1/6)*(2^(1/3)-(2+3*x^2)^(1/3))/x)/(2^(5/6)*d)+1/4*arctan(1/3*(2^(1/3)-(2+3*x^2)^(1/3))^2/(2^(1/6)*x*sqrt(3)))/(2^(5/6)*d*sqrt(3))+1/4*arctan(x/sqrt(6))/(2^(5/6)*d*sqrt(3))],
[1/((2-3*x^2)^(1/3)*(-6*d+d*x^2)),x,1,-1/4*arctan(2^(1/6)*(2^(1/3)-(2-3*x^2)^(1/3))/x)/(2^(5/6)*d)+1/4*arctanh(1/3*(2^(1/3)-(2-3*x^2)^(1/3))^2/(2^(1/6)*x*sqrt(3)))/(2^(5/6)*d*sqrt(3))-1/4*arctanh(x/sqrt(6))/(2^(5/6)*d*sqrt(3))],
[1/((-2+3*x^2)^(1/3)*(-6*d+d*x^2)),x,1,1/4*arctan(2^(1/6)*(2^(1/3)+(-2+3*x^2)^(1/3))/x)/(2^(5/6)*d)-1/4*arctanh(1/3*(2^(1/3)+(-2+3*x^2)^(1/3))^2/(2^(1/6)*x*sqrt(3)))/(2^(5/6)*d*sqrt(3))+1/4*arctanh(x/sqrt(6))/(2^(5/6)*d*sqrt(3))],
[1/((-2-3*x^2)^(1/3)*(6*d+d*x^2)),x,1,1/4*arctanh(2^(1/6)*(2^(1/3)+(-2-3*x^2)^(1/3))/x)/(2^(5/6)*d)-1/4*arctan(1/3*(2^(1/3)+(-2-3*x^2)^(1/3))^2/(2^(1/6)*x*sqrt(3)))/(2^(5/6)*d*sqrt(3))-1/4*arctan(x/sqrt(6))/(2^(5/6)*d*sqrt(3))],
[1/((1+x^2)^(1/3)*(9+x^2)),x,1,1/12*arctan(1/3*x)+1/12*arctan(1/3*(1-(1+x^2)^(1/3))^2/x)-1/4*arctanh((1-(1+x^2)^(1/3))*sqrt(3)/x)/sqrt(3)],
[1/((1+b*x^2)^(1/3)*(9+b*x^2)),x,1,1/12*arctan(1/3*(1-(1+b*x^2)^(1/3))^2/(x*sqrt(b)))/sqrt(b)+1/12*arctan(1/3*x*sqrt(b))/sqrt(b)-1/4*arctanh((1-(1+b*x^2)^(1/3))*sqrt(3)/(x*sqrt(b)))/(sqrt(3)*sqrt(b))],
[1/((1-x^2)^(1/3)*(9-x^2)),x,1,1/12*arctanh(1/3*x)-1/12*arctanh(1/3*(1-(1-x^2)^(1/3))^2/x)+1/4*arctan((1-(1-x^2)^(1/3))*sqrt(3)/x)/sqrt(3)],

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

# Integrands of the form (a+b x^2)^(p/2) (c+d x^2)^(q/2) when b c-a d=0
[sqrt(-1+c^2*x^2)/(d-c^2*d*x^2)^(5/2),x,3,1/2*x*sqrt(-1+c^2*x^2)/(d*(d-c^2*d*x^2)^(3/2))+1/2*arctanh(c*x)*sqrt(-1+c^2*x^2)/(c*d^2*sqrt(d-c^2*d*x^2)),1/2*x*sqrt(-1+c^2*x^2)/(d^2*(1-c^2*x^2)*sqrt(d-c^2*d*x^2))+1/2*arctanh(c*x)*sqrt(-1+c^2*x^2)/(c*d^2*sqrt(d-c^2*d*x^2))],
[1/((-1+c^2*x^2)^(3/2)*sqrt(d-c^2*d*x^2)),x,3,1/2*d*x*sqrt(-1+c^2*x^2)/(d-c^2*d*x^2)^(3/2)+1/2*arctanh(c*x)*sqrt(-1+c^2*x^2)/(c*sqrt(d-c^2*d*x^2)),1/2*x*(d-c^2*d*x^2)^(3/2)/(d^2*(1-c^2*x^2)*(-1+c^2*x^2)^(3/2))+1/2*(d-c^2*d*x^2)^(3/2)*arctanh(c*x)/(c*d^2*(-1+c^2*x^2)^(3/2))],
[1/((-1+c^2*x^2)^(1/2)*(d-c^2*d*x^2)^(3/2)),x,3,-1/2*x*sqrt(-1+c^2*x^2)/(d-c^2*d*x^2)^(3/2)-1/2*arctanh(c*x)*sqrt(-1+c^2*x^2)/(c*d*sqrt(d-c^2*d*x^2)),1/2*x*sqrt(d-c^2*d*x^2)/(d^2*(1-c^2*x^2)*sqrt(-1+c^2*x^2))+1/2*arctanh(c*x)*sqrt(d-c^2*d*x^2)/(c*d^2*sqrt(-1+c^2*x^2))],

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

# q>0
[(a+b*x^2)^(3/2)*sqrt(c+d*x^2),x,6,1/5*b*x*(c+d*x^2)^(3/2)*sqrt(a+b*x^2)/d+1/15*(7*a*c-2*b*c^2/d+3*a^2*d/b)*x*sqrt(a+b*x^2)/sqrt(c+d*x^2)-1/15*c^(3/2)*(b*c-9*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(a+b*x^2)/(d^(3/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/15*(2*b^2*c^2-7*a*b*c*d-3*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(a+b*x^2)/(b*d^(3/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-2/15*(b*c-3*a*d)*x*sqrt(a+b*x^2)*sqrt(c+d*x^2)/d],
[(a+b*x^2)^(1/2)*sqrt(c+d*x^2),x,5,1/3*(b*c+a*d)*x*sqrt(a+b*x^2)/(b*sqrt(c+d*x^2))+2/3*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(a+b*x^2)/(sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-1/3*(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(a+b*x^2)/(b*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/3*x*sqrt(a+b*x^2)*sqrt(c+d*x^2)],
[sqrt(c+d*x^2)/(a+b*x^2)^(1/2),x,4,d*x*sqrt(a+b*x^2)/(b*sqrt(c+d*x^2))+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(a+b*x^2)/(a*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(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(d)*sqrt(a+b*x^2)/(b*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))],
[sqrt(c+d*x^2)/(a+b*x^2)^(3/2),x,1,sqrt(cos(arctan(x*sqrt(b)/sqrt(a)))^2)/cos(arctan(x*sqrt(b)/sqrt(a)))*EllipticE(sin(arctan(x*sqrt(b)/sqrt(a))),sqrt(1-a*d/(b*c)))*sqrt(c+d*x^2)/(sqrt(a)*sqrt(b)*sqrt(a+b*x^2)*sqrt(a*(c+d*x^2)/(c*(a+b*x^2))))],
[sqrt(c+d*x^2)/(a+b*x^2)^(5/2),x,4,-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(d)*sqrt(a+b*x^2)/(a^2*(b*c-a*d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/3*x*sqrt(c+d*x^2)/(a*(a+b*x^2)^(3/2))+1/3*(2*b*c-a*d)*sqrt(cos(arctan(x*sqrt(b)/sqrt(a)))^2)/cos(arctan(x*sqrt(b)/sqrt(a)))*EllipticE(sin(arctan(x*sqrt(b)/sqrt(a))),sqrt(1-a*d/(b*c)))*sqrt(c+d*x^2)/(a^(3/2)*(b*c-a*d)*sqrt(b)*sqrt(a+b*x^2)*sqrt(a*(c+d*x^2)/(c*(a+b*x^2))))],
[sqrt(c+d*x^2)/(a+b*x^2)^(7/2),x,5,-2/15*c^(3/2)*(2*b*c-3*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(a+b*x^2)/(a^3*(b*c-a*d)^2*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/5*x*sqrt(c+d*x^2)/(a*(a+b*x^2)^(5/2))+1/15*(4*b*c-3*a*d)*x*sqrt(c+d*x^2)/(a^2*(b*c-a*d)*(a+b*x^2)^(3/2))+1/15*(8*b^2*c^2-13*a*b*c*d+3*a^2*d^2)*sqrt(cos(arctan(x*sqrt(b)/sqrt(a)))^2)/cos(arctan(x*sqrt(b)/sqrt(a)))*EllipticE(sin(arctan(x*sqrt(b)/sqrt(a))),sqrt(1-a*d/(b*c)))*sqrt(c+d*x^2)/(a^(5/2)*(b*c-a*d)^2*sqrt(b)*sqrt(a+b*x^2)*sqrt(a*(c+d*x^2)/(c*(a+b*x^2))))],
[(a+b*x^2)^(3/2)*(c+d*x^2)^(3/2),x,7,-2/35*(b*c+a*d)*(b^2*c^2-6*a*b*c*d+a^2*d^2)*x*sqrt(a+b*x^2)/(b^2*d*sqrt(c+d*x^2))-1/35*c^(3/2)*(b^2*c^2-18*a*b*c*d+a^2*d^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(a+b*x^2)/(b*d^(3/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+2/35*(b*c+a*d)*(b^2*c^2-6*a*b*c*d+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(a+b*x^2)/(b^2*d^(3/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+2/35*(4*b*c-a*d)*x*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/b+1/7*d*x*(a+b*x^2)^(5/2)*sqrt(c+d*x^2)/b+1/35*(9*a*c+b*c^2/d-2*a^2*d/b)*x*sqrt(a+b*x^2)*sqrt(c+d*x^2)],
[(a+b*x^2)^(1/2)*(c+d*x^2)^(3/2),x,6,1/15*(3*b^2*c^2+7*a*b*c*d-2*a^2*d^2)*x*sqrt(a+b*x^2)/(b^2*sqrt(c+d*x^2))+1/15*c^(3/2)*(9*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(a+b*x^2)/(b*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-1/15*(3*b^2*c^2+7*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(a+b*x^2)/(b^2*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/5*d*x*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/b+2/15*(3*b*c-a*d)*x*sqrt(a+b*x^2)*sqrt(c+d*x^2)/b],
[(c+d*x^2)^(3/2)/(a+b*x^2)^(1/2),x,5,2/3*d*(2*b*c-a*d)*x*sqrt(a+b*x^2)/(b^2*sqrt(c+d*x^2))+1/3*c^(3/2)*(3*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(a+b*x^2)/(a*b*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-2/3*(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(d)*sqrt(a+b*x^2)/(b^2*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/3*d*x*sqrt(a+b*x^2)*sqrt(c+d*x^2)/b],
[(c+d*x^2)^(3/2)/(a+b*x^2)^(3/2),x,5,-d*(b*c-2*a*d)*x*sqrt(a+b*x^2)/(a*b^2*sqrt(c+d*x^2))+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(d)*sqrt(a+b*x^2)/(a*b*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+(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(c)*sqrt(d)*sqrt(a+b*x^2)/(a*b^2*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+(b*c-a*d)*x*sqrt(c+d*x^2)/(a*b*sqrt(a+b*x^2))],
[(c+d*x^2)^(3/2)/(a+b*x^2)^(5/2),x,4,-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(d)*sqrt(a+b*x^2)/(a^2*b*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/3*(b*c-a*d)*x*sqrt(c+d*x^2)/(a*b*(a+b*x^2)^(3/2))+2/3*(b*c+a*d)*sqrt(cos(arctan(x*sqrt(b)/sqrt(a)))^2)/cos(arctan(x*sqrt(b)/sqrt(a)))*EllipticE(sin(arctan(x*sqrt(b)/sqrt(a))),sqrt(1-a*d/(b*c)))*sqrt(c+d*x^2)/(a^(3/2)*b^(3/2)*sqrt(a+b*x^2)*sqrt(a*(c+d*x^2)/(c*(a+b*x^2))))],
[(c+d*x^2)^(3/2)/(a+b*x^2)^(7/2),x,5,-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(d)*sqrt(a+b*x^2)/(a^3*b*(b*c-a*d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/5*(b*c-a*d)*x*sqrt(c+d*x^2)/(a*b*(a+b*x^2)^(5/2))+2/15*(2*b*c+a*d)*x*sqrt(c+d*x^2)/(a^2*b*(a+b*x^2)^(3/2))+1/15*(8*b^2*c^2-3*a*b*c*d-2*a^2*d^2)*sqrt(cos(arctan(x*sqrt(b)/sqrt(a)))^2)/cos(arctan(x*sqrt(b)/sqrt(a)))*EllipticE(sin(arctan(x*sqrt(b)/sqrt(a))),sqrt(1-a*d/(b*c)))*sqrt(c+d*x^2)/(a^(5/2)*b^(3/2)*(b*c-a*d)*sqrt(a+b*x^2)*sqrt(a*(c+d*x^2)/(c*(a+b*x^2))))],
[sqrt(2+b*x^2)*sqrt(3+d*x^2),x,5,1/3*(3*b+2*d)*x*sqrt(2+b*x^2)/(b*sqrt(3+d*x^2))-1/3*(3*b+2*d)*sqrt(cos(arctan(x*sqrt(d)/sqrt(3)))^2)/cos(arctan(x*sqrt(d)/sqrt(3)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(3))),sqrt(1-3/2*b/d))*sqrt(2)*sqrt(2+b*x^2)/(b*sqrt(d)*sqrt((2+b*x^2)/(3+d*x^2))*sqrt(3+d*x^2))+2*sqrt(cos(arctan(x*sqrt(d)/sqrt(3)))^2)/cos(arctan(x*sqrt(d)/sqrt(3)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(3))),sqrt(1-3/2*b/d))*sqrt(2)*sqrt(2+b*x^2)/(sqrt(d)*sqrt((2+b*x^2)/(3+d*x^2))*sqrt(3+d*x^2))+1/3*x*sqrt(2+b*x^2)*sqrt(3+d*x^2)],
[sqrt(3-6*x^2)*sqrt(2+4*x^2),x,3,2*EllipticF(x*sqrt(2),I)/sqrt(3)+x*sqrt(2/3)*sqrt(1-4*x^4)],
[sqrt(2+4*x^2)*sqrt(3+6*x^2),x,2,2*x^3*sqrt(2/3)+x*sqrt(6)],
[sqrt(2+b*x^2)/sqrt(3+d*x^2),x,4,x*sqrt(2+b*x^2)/sqrt(3+d*x^2)-sqrt(cos(arctan(x*sqrt(d)/sqrt(3)))^2)/cos(arctan(x*sqrt(d)/sqrt(3)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(3))),sqrt(1-3/2*b/d))*sqrt(2)*sqrt(2+b*x^2)/(sqrt(d)*sqrt((2+b*x^2)/(3+d*x^2))*sqrt(3+d*x^2))+sqrt(cos(arctan(x*sqrt(d)/sqrt(3)))^2)/cos(arctan(x*sqrt(d)/sqrt(3)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(3))),sqrt(1-3/2*b/d))*sqrt(2)*sqrt(2+b*x^2)/(sqrt(d)*sqrt((2+b*x^2)/(3+d*x^2))*sqrt(3+d*x^2))],
[sqrt(4-x^2)/sqrt(c+d*x^2),x,5,-EllipticE(1/2*x,sqrt(-4*d/c))*sqrt(c+d*x^2)/(d*sqrt(1+d*x^2/c))+(c+4*d)*EllipticF(1/2*x,sqrt(-4*d/c))*sqrt(1+d*x^2/c)/(d*sqrt(c+d*x^2))],
[sqrt(4+x^2)/sqrt(c+d*x^2),x,4,x*sqrt(c+d*x^2)/(d*sqrt(4+x^2))-sqrt(cos(arctan(1/2*x))^2)/cos(arctan(1/2*x))*EllipticE(sin(arctan(1/2*x)),sqrt(1-4*d/c))*sqrt(c+d*x^2)/(d*sqrt(4+x^2)*sqrt((c+d*x^2)/(c*(4+x^2))))+4*sqrt(cos(arctan(1/2*x))^2)/cos(arctan(1/2*x))*EllipticF(sin(arctan(1/2*x)),sqrt(1-4*d/c))*sqrt(c+d*x^2)/(c*sqrt(4+x^2)*sqrt((c+d*x^2)/(c*(4+x^2))))],
[sqrt(1-x^2)/sqrt(2-3*x^2),x,1,EllipticE(x*sqrt(3/2),sqrt(2/3))/sqrt(3)],
[sqrt(4-x^2)/sqrt(2-3*x^2),x,1,2*EllipticE(x*sqrt(3/2),sqrt(1/6))/sqrt(3)],
[sqrt(1-4*x^2)/sqrt(2-3*x^2),x,1,EllipticE(x*sqrt(3/2),sqrt(8/3))/sqrt(3)],
[sqrt(1+x^2)/sqrt(1-x^2),x,1,EllipticE(x,I)],
[sqrt(1+x^2)/sqrt(2-3*x^2),x,1,EllipticE(x*sqrt(3/2),sqrt(-2/3))/sqrt(3)],
[sqrt(4+x^2)/sqrt(2-3*x^2),x,1,2*EllipticE(x*sqrt(3/2),sqrt(-1/6))/sqrt(3)],
[sqrt(1+4*x^2)/sqrt(2-3*x^2),x,1,EllipticE(x*sqrt(3/2),sqrt(-8/3))/sqrt(3)],
[sqrt(1-x^2)/sqrt(1+x^2),x,4,-EllipticE(x,I)+2*EllipticF(x,I)],
[sqrt(1-x^2)/sqrt(2+3*x^2),x,3,5/3*EllipticF(x,sqrt(-3/2))/sqrt(2)-1/3*EllipticE(x,sqrt(-3/2))*sqrt(2)],
[sqrt(4-x^2)/sqrt(2+3*x^2),x,3,-1/3*EllipticE(1/2*x,sqrt(-6))*sqrt(2)+7/3*EllipticF(1/2*x,sqrt(-6))*sqrt(2)],
[sqrt(1-4*x^2)/sqrt(2+3*x^2),x,3,11/6*EllipticF(2*x,sqrt(-3/8))/sqrt(2)-2/3*EllipticE(2*x,sqrt(-3/8))*sqrt(2)],
[sqrt(1+x^2)/sqrt(2+3*x^2),x,4,1/3*x*sqrt(2+3*x^2)/sqrt(1+x^2)+sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(-1/2))*sqrt(2+3*x^2)/(sqrt(2)*sqrt(1+x^2)*sqrt((2+3*x^2)/(1+x^2)))-1/3*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(-1/2))*sqrt(2)*sqrt(2+3*x^2)/(sqrt(1+x^2)*sqrt((2+3*x^2)/(1+x^2)))],
[sqrt(4+x^2)/sqrt(2+3*x^2),x,4,1/3*x*sqrt(2+3*x^2)/sqrt(4+x^2)-1/3*sqrt(cos(arctan(1/2*x))^2)/cos(arctan(1/2*x))*EllipticE(sin(arctan(1/2*x)),sqrt(-5))*sqrt(2)*sqrt(2+3*x^2)/(sqrt(4+x^2)*sqrt((2+3*x^2)/(4+x^2)))+2*sqrt(cos(arctan(1/2*x))^2)/cos(arctan(1/2*x))*EllipticF(sin(arctan(1/2*x)),sqrt(-5))*sqrt(2)*sqrt(2+3*x^2)/(sqrt(4+x^2)*sqrt((2+3*x^2)/(4+x^2)))],
[sqrt(1+4*x^2)/sqrt(2+3*x^2),x,4,4/3*x*sqrt(2+3*x^2)/sqrt(1+4*x^2)+1/2*sqrt(cos(arctan(2*x))^2)/cos(arctan(2*x))*EllipticF(sin(arctan(2*x)),sqrt(5/8))*sqrt(2+3*x^2)/(sqrt(2)*sqrt((2+3*x^2)/(1+4*x^2))*sqrt(1+4*x^2))-2/3*sqrt(cos(arctan(2*x))^2)/cos(arctan(2*x))*EllipticE(sin(arctan(2*x)),sqrt(5/8))*sqrt(2)*sqrt(2+3*x^2)/(sqrt((2+3*x^2)/(1+4*x^2))*sqrt(1+4*x^2))],
[sqrt(1-x^2)/sqrt(-1+2*x^2),x,2,EllipticE(x*sqrt(2),sqrt(1/2))*sqrt(1-2*x^2)/(sqrt(2)*sqrt(-1+2*x^2))],

# q<0
[(a+b*x^2)^(7/2)/sqrt(c+d*x^2),x,7,-8/105*(b*c-2*a*d)*(6*b^2*c^2-11*a*b*c*d+11*a^2*d^2)*x*sqrt(a+b*x^2)/(d^3*sqrt(c+d*x^2))+8/105*(b*c-2*a*d)*(6*b^2*c^2-11*a*b*c*d+11*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(a+b*x^2)/(d^(7/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-1/105*(3*b*c-7*a*d)*(8*b^2*c^2-11*a*b*c*d+15*a^2*d^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(a+b*x^2)/(d^(7/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-6/35*b*(b*c-2*a*d)*x*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/d^2+1/7*b*x*(a+b*x^2)^(5/2)*sqrt(c+d*x^2)/d+1/105*b*(24*b^2*c^2-71*a*b*c*d+71*a^2*d^2)*x*sqrt(a+b*x^2)*sqrt(c+d*x^2)/d^3],
[(a+b*x^2)^(5/2)/sqrt(c+d*x^2),x,6,1/15*(8*b^2*c^2-23*a*b*c*d+23*a^2*d^2)*x*sqrt(a+b*x^2)/(d^2*sqrt(c+d*x^2))-1/15*(8*b^2*c^2-23*a*b*c*d+23*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(a+b*x^2)/(d^(5/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/15*(4*b^2*c^2-11*a*b*c*d+15*a^2*d^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(a+b*x^2)/(d^(5/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/5*b*x*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/d-4/15*b*(b*c-2*a*d)*x*sqrt(a+b*x^2)*sqrt(c+d*x^2)/d^2],
[(a+b*x^2)^(3/2)/sqrt(c+d*x^2),x,5,-2/3*(b*c-2*a*d)*x*sqrt(a+b*x^2)/(d*sqrt(c+d*x^2))+2/3*(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(c)*sqrt(a+b*x^2)/(d^(3/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-1/3*(b*c-3*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(c)*sqrt(a+b*x^2)/(d^(3/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/3*b*x*sqrt(a+b*x^2)*sqrt(c+d*x^2)/d],
[(a+b*x^2)^(1/2)/sqrt(c+d*x^2),x,4,x*sqrt(a+b*x^2)/sqrt(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(a+b*x^2)/(sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(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(a+b*x^2)/(sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))],
[1/((a+b*x^2)^(1/2)*sqrt(c+d*x^2)),x,1,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(a+b*x^2)/(a*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))],
[1/((a+b*x^2)^(3/2)*sqrt(c+d*x^2)),x,6,-d*x*sqrt(a+b*x^2)/(a*(b*c-a*d)*sqrt(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(d)*sqrt(a+b*x^2)/(a*(b*c-a*d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(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(d)*sqrt(a+b*x^2)/(a*(b*c-a*d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+b*x*sqrt(c+d*x^2)/(a*(b*c-a*d)*sqrt(a+b*x^2))],
[1/((a+b*x^2)^(5/2)*sqrt(c+d*x^2)),x,4,-1/3*(b*c-3*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(c)*sqrt(d)*sqrt(a+b*x^2)/(a^2*(b*c-a*d)^2*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/3*b*x*sqrt(c+d*x^2)/(a*(b*c-a*d)*(a+b*x^2)^(3/2))+2/3*(b*c-2*a*d)*sqrt(cos(arctan(x*sqrt(b)/sqrt(a)))^2)/cos(arctan(x*sqrt(b)/sqrt(a)))*EllipticE(sin(arctan(x*sqrt(b)/sqrt(a))),sqrt(1-a*d/(b*c)))*sqrt(b)*sqrt(c+d*x^2)/(a^(3/2)*(b*c-a*d)^2*sqrt(a+b*x^2)*sqrt(a*(c+d*x^2)/(c*(a+b*x^2))))],
[1/((a+b*x^2)^(7/2)*sqrt(c+d*x^2)),x,5,-1/15*(4*b^2*c^2-11*a*b*c*d+15*a^2*d^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)*sqrt(a+b*x^2)/(a^3*(b*c-a*d)^3*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/5*b*x*sqrt(c+d*x^2)/(a*(b*c-a*d)*(a+b*x^2)^(5/2))+4/15*b*(b*c-2*a*d)*x*sqrt(c+d*x^2)/(a^2*(b*c-a*d)^2*(a+b*x^2)^(3/2))+1/15*(8*b^2*c^2-23*a*b*c*d+23*a^2*d^2)*sqrt(cos(arctan(x*sqrt(b)/sqrt(a)))^2)/cos(arctan(x*sqrt(b)/sqrt(a)))*EllipticE(sin(arctan(x*sqrt(b)/sqrt(a))),sqrt(1-a*d/(b*c)))*sqrt(b)*sqrt(c+d*x^2)/(a^(5/2)*(b*c-a*d)^3*sqrt(a+b*x^2)*sqrt(a*(c+d*x^2)/(c*(a+b*x^2))))],
[(a+b*x^2)^(7/2)/(c+d*x^2)^(3/2),x,7,-(b*c-a*d)*x*(a+b*x^2)^(5/2)/(c*d*sqrt(c+d*x^2))+1/15*(48*b^3*c^3-128*a*b^2*c^2*d+103*a^2*b*c*d^2-15*a^3*d^3)*x*sqrt(a+b*x^2)/(c*d^3*sqrt(c+d*x^2))-1/15*(48*b^3*c^3-128*a*b^2*c^2*d+103*a^2*b*c*d^2-15*a^3*d^3)*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(a+b*x^2)/(d^(7/2)*sqrt(c)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/15*b*(24*b^2*c^2-61*a*b*c*d+45*a^2*d^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(a+b*x^2)/(d^(7/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/5*b*(6*b*c-5*a*d)*x*(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/(c*d^2)-1/15*b*(24*b^2*c^2-43*a*b*c*d+15*a^2*d^2)*x*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(c*d^3)],
[(a+b*x^2)^(5/2)/(c+d*x^2)^(3/2),x,6,-(b*c-a*d)*x*(a+b*x^2)^(3/2)/(c*d*sqrt(c+d*x^2))-1/3*(8*b^2*c^2-13*a*b*c*d+3*a^2*d^2)*x*sqrt(a+b*x^2)/(c*d^2*sqrt(c+d*x^2))+1/3*(8*b^2*c^2-13*a*b*c*d+3*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(a+b*x^2)/(d^(5/2)*sqrt(c)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-2/3*b*(2*b*c-3*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(c)*sqrt(a+b*x^2)/(d^(5/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+1/3*b*(4*b*c-3*a*d)*x*sqrt(a+b*x^2)*sqrt(c+d*x^2)/(c*d^2)],
[(a+b*x^2)^(3/2)/(c+d*x^2)^(3/2),x,5,-(b*c-a*d)*x*sqrt(a+b*x^2)/(c*d*sqrt(c+d*x^2))+(2*b*c-a*d)*x*sqrt(a+b*x^2)/(c*d*sqrt(c+d*x^2))-(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(a+b*x^2)/(d^(3/2)*sqrt(c)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+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(a+b*x^2)/(d^(3/2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))],
[(a+b*x^2)^(1/2)/(c+d*x^2)^(3/2),x,1,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(a+b*x^2)/(sqrt(c)*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))],
[1/((a+b*x^2)^(1/2)*(c+d*x^2)^(3/2)),x,6,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(a+b*x^2)/(a*(b*c-a*d)*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(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(a+b*x^2)/((b*c-a*d)*sqrt(c)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))],
[1/((a+b*x^2)^(3/2)*(c+d*x^2)^(3/2)),x,4,b*x/(a*(b*c-a*d)*sqrt(a+b*x^2)*sqrt(c+d*x^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(d)*sqrt(a+b*x^2)/(a*(b*c-a*d)^2*sqrt(c)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-2*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(d)*sqrt(a+b*x^2)/(a*(b*c-a*d)^2*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))],
[1/((a+b*x^2)^(5/2)*(c+d*x^2)^(3/2)),x,5,1/3*b*x/(a*(b*c-a*d)*(a+b*x^2)^(3/2)*sqrt(c+d*x^2))+2/3*b*(b*c-3*a*d)*x/(a^2*(b*c-a*d)^2*sqrt(a+b*x^2)*sqrt(c+d*x^2))+1/3*(2*b^2*c^2-7*a*b*c*d-3*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(a+b*x^2)/(a^2*(b*c-a*d)^3*sqrt(c)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-1/3*b*(b*c-9*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(c)*sqrt(d)*sqrt(a+b*x^2)/(a^2*(b*c-a*d)^3*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))],
[1/(sqrt(a+b*x^2)*sqrt(c+d*x^2)),x,1,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(a+b*x^2)/(a*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))],
[1/(sqrt(a-b*x^2)*sqrt(c+d*x^2)),x,3,EllipticF(x*sqrt(b)/sqrt(a),sqrt(-a*d/(b*c)))*sqrt(a)*sqrt(1-b*x^2/a)*sqrt(1+d*x^2/c)/(sqrt(b)*sqrt(a-b*x^2)*sqrt(c+d*x^2))],
[1/(sqrt(a+b*x^2)*sqrt(c-d*x^2)),x,3,EllipticF(x*sqrt(d)/sqrt(c),sqrt(-b*c/(a*d)))*sqrt(c)*sqrt(1+b*x^2/a)*sqrt(1-d*x^2/c)/(sqrt(d)*sqrt(a+b*x^2)*sqrt(c-d*x^2))],
[1/(sqrt(a-b*x^2)*sqrt(c-d*x^2)),x,3,EllipticF(x*sqrt(d)/sqrt(c),sqrt(b*c/(a*d)))*sqrt(c)*sqrt(1-b*x^2/a)*sqrt(1-d*x^2/c)/(sqrt(d)*sqrt(a-b*x^2)*sqrt(c-d*x^2))],
[1/(sqrt(1-x^2)*sqrt(2+5*x^2)),x,1,EllipticF(x,sqrt(-5/2))/sqrt(2)],
[1/(sqrt(1-x^2)*sqrt(2+4*x^2)),x,1,EllipticF(x,sqrt(-2))/sqrt(2)],
[1/(sqrt(1-x^2)*sqrt(2+3*x^2)),x,1,EllipticF(x,sqrt(-3/2))/sqrt(2)],
[1/(sqrt(1-x^2)*sqrt(2+2*x^2)),x,2,EllipticF(x,I)/sqrt(2)],
[1/(sqrt(1-x^2)*sqrt(2+x^2)),x,1,EllipticF(x,sqrt(-1/2))/sqrt(2)],
[1/(sqrt(1-x^2)*sqrt(2-x^2)),x,1,EllipticF(x,sqrt(1/2))/sqrt(2)],
[1/(sqrt(2-2*x^2)*sqrt(1-x^2)),x,2,arctanh(x)/sqrt(2)],
[1/(sqrt(2-3*x^2)*sqrt(1-x^2)),x,1,EllipticF(x,sqrt(3/2))/sqrt(2)],
[1/(sqrt(2-4*x^2)*sqrt(1-x^2)),x,1,EllipticF(x,sqrt(2))/sqrt(2)],
[1/(sqrt(2-5*x^2)*sqrt(1-x^2)),x,1,EllipticF(x,sqrt(5/2))/sqrt(2)],
[1/(sqrt(1+x^2)*sqrt(2+5*x^2)),x,1,sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(-3/2))*sqrt(2+5*x^2)/(sqrt(2)*sqrt(1+x^2)*sqrt((2+5*x^2)/(1+x^2)))],
[1/(sqrt(1+x^2)*sqrt(2+4*x^2)),x,1,sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),I)*sqrt(1+2*x^2)/(sqrt(2)*sqrt(1+x^2)*sqrt((1+2*x^2)/(1+x^2)))],
[1/(sqrt(1+x^2)*sqrt(2+3*x^2)),x,1,sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(-1/2))*sqrt(2+3*x^2)/(sqrt(2)*sqrt(1+x^2)*sqrt((2+3*x^2)/(1+x^2)))],
[1/(sqrt(1+x^2)*sqrt(2+2*x^2)),x,2,arctan(x)/sqrt(2)],
[1/(sqrt(1+x^2)*sqrt(2+x^2)),x,1,sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt(2+x^2)/(sqrt(2)*sqrt(1+x^2)*sqrt((2+x^2)/(1+x^2)))],
[1/(sqrt(2-x^2)*sqrt(1+x^2)),x,1,EllipticF(x/sqrt(2),sqrt(-2))],
[1/(sqrt(2-2*x^2)*sqrt(1+x^2)),x,2,EllipticF(x,I)/sqrt(2)],
[1/(sqrt(2-3*x^2)*sqrt(1+x^2)),x,1,EllipticF(x*sqrt(3/2),sqrt(-2/3))/sqrt(3)],
[1/(sqrt(2-4*x^2)*sqrt(1+x^2)),x,1,1/2*EllipticF(x*sqrt(2),sqrt(-1/2))],
[1/(sqrt(2-5*x^2)*sqrt(1+x^2)),x,1,EllipticF(x*sqrt(5/2),sqrt(-2/5))/sqrt(5)],
[1/(sqrt(-1+x^2)*sqrt(2+5*x^2)),x,2,EllipticF(x,sqrt(-5/2))*sqrt(1-x^2)/(sqrt(2)*sqrt(-1+x^2))],
[1/(sqrt(-1+x^2)*sqrt(2+4*x^2)),x,2,EllipticF(x,sqrt(-2))*sqrt(1-x^2)/(sqrt(2)*sqrt(-1+x^2))],
[1/(sqrt(-1+x^2)*sqrt(2+3*x^2)),x,2,EllipticF(x,sqrt(-3/2))*sqrt(1-x^2)/(sqrt(2)*sqrt(-1+x^2))],
[1/(sqrt(-1+x^2)*sqrt(2+2*x^2)),x,2,1/2*EllipticF(x*sqrt(2)/sqrt(-1+x^2),sqrt(1/2))],
[1/(sqrt(-1+x^2)*sqrt(2+x^2)),x,2,EllipticF(x,sqrt(-1/2))*sqrt(1-x^2)/(sqrt(2)*sqrt(-1+x^2))],
[1/(sqrt(2-x^2)*sqrt(-1+x^2)),x,1,-sqrt(cos(arccos(x/sqrt(2)))^2)/cos(arccos(x/sqrt(2)))*EllipticF(sin(arccos(x/sqrt(2))),sqrt(2))],
[1/(sqrt(2-2*x^2)*sqrt(-1+x^2)),x,2,-arctanh(x)*sqrt(-1+x^2)/(sqrt(2)*sqrt(1-x^2))],
[1/(sqrt(2-3*x^2)*sqrt(-1+x^2)),x,2,EllipticF(x,sqrt(3/2))*sqrt(1-x^2)/(sqrt(2)*sqrt(-1+x^2))],
[1/(sqrt(2-4*x^2)*sqrt(-1+x^2)),x,2,EllipticF(x,sqrt(2))*sqrt(1-x^2)/(sqrt(2)*sqrt(-1+x^2))],
[1/(sqrt(2-5*x^2)*sqrt(-1+x^2)),x,2,EllipticF(x,sqrt(5/2))*sqrt(1-x^2)/(sqrt(2)*sqrt(-1+x^2))],
[1/(sqrt(-1-x^2)*sqrt(2+5*x^2)),x,1,sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(-3/2))*sqrt(2+5*x^2)/(sqrt(2)*sqrt(-1-x^2)*sqrt((2+5*x^2)/(1+x^2)))],
[1/(sqrt(-1-x^2)*sqrt(2+4*x^2)),x,1,sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),I)*sqrt(1+2*x^2)/(sqrt(2)*sqrt(-1-x^2)*sqrt((1+2*x^2)/(1+x^2)))],
[1/(sqrt(-1-x^2)*sqrt(2+3*x^2)),x,1,sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(-1/2))*sqrt(2+3*x^2)/(sqrt(2)*sqrt(-1-x^2)*sqrt((2+3*x^2)/(1+x^2)))],
[1/(sqrt(-1-x^2)*sqrt(2+2*x^2)),x,2,arctan(x)*sqrt(1+x^2)/(sqrt(2)*sqrt(-1-x^2))],
[1/(sqrt(-1-x^2)*sqrt(2+x^2)),x,1,sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt(2+x^2)/(sqrt(2)*sqrt(-1-x^2)*sqrt((2+x^2)/(1+x^2)))],
[1/(sqrt(-1-x^2)*sqrt(2-x^2)),x,2,EllipticF(x/sqrt(2),sqrt(-2))*sqrt(1+x^2)/sqrt(-1-x^2)],
[1/(sqrt(2-2*x^2)*sqrt(-1-x^2)),x,2,-x^2*sqrt(cos(arccsc(x))^2)/cos(arccsc(x))*EllipticF(sin(arccsc(x)),I)*sqrt(1+(-1)/x^4)/(sqrt(2-2*x^2)*sqrt(-1-x^2)),1/2*EllipticF(x*sqrt(2)/sqrt(-1+x^2),sqrt(1/2))*sqrt(-1+x^2)*sqrt(1+x^2)/(sqrt(-1-x^2)*sqrt(1-x^2))],
[1/(sqrt(2-3*x^2)*sqrt(-1-x^2)),x,2,EllipticF(x*sqrt(3/2),sqrt(-2/3))*sqrt(1+x^2)/(sqrt(3)*sqrt(-1-x^2))],
[1/(sqrt(2-4*x^2)*sqrt(-1-x^2)),x,2,1/2*EllipticF(x*sqrt(2),sqrt(-1/2))*sqrt(1+x^2)/sqrt(-1-x^2)],
[1/(sqrt(2-5*x^2)*sqrt(-1-x^2)),x,2,EllipticF(x*sqrt(5/2),sqrt(-2/5))*sqrt(1+x^2)/(sqrt(5)*sqrt(-1-x^2))],
[sqrt(a+b*x^2)/sqrt(c-d*x^2),x,3,EllipticE(x*sqrt(d)/sqrt(c),sqrt(-b*c/(a*d)))*sqrt(c)*sqrt(a+b*x^2)*sqrt(1-d*x^2/c)/(sqrt(d)*sqrt(1+b*x^2/a)*sqrt(c-d*x^2))],
[sqrt(-a-b*x^2)/sqrt(c-d*x^2),x,3,EllipticE(x*sqrt(d)/sqrt(c),sqrt(-b*c/(a*d)))*sqrt(c)*sqrt(-a-b*x^2)*sqrt(1-d*x^2/c)/(sqrt(d)*sqrt(1+b*x^2/a)*sqrt(c-d*x^2))],
[sqrt(a+b*x^2)/sqrt(-c+d*x^2),x,3,EllipticE(x*sqrt(d)/sqrt(c),sqrt(-b*c/(a*d)))*sqrt(c)*sqrt(a+b*x^2)*sqrt(1-d*x^2/c)/(sqrt(d)*sqrt(1+b*x^2/a)*sqrt(-c+d*x^2))],
[sqrt(-a-b*x^2)/sqrt(-c+d*x^2),x,3,EllipticE(x*sqrt(d)/sqrt(c),sqrt(-b*c/(a*d)))*sqrt(c)*sqrt(-a-b*x^2)*sqrt(1-d*x^2/c)/(sqrt(d)*sqrt(1+b*x^2/a)*sqrt(-c+d*x^2))],
[sqrt(a-b*x^2)/sqrt(c-d*x^2),x,3,EllipticE(x*sqrt(d)/sqrt(c),sqrt(b*c/(a*d)))*sqrt(c)*sqrt(a-b*x^2)*sqrt(1-d*x^2/c)/(sqrt(d)*sqrt(1-b*x^2/a)*sqrt(c-d*x^2))],
[sqrt(-a+b*x^2)/sqrt(c-d*x^2),x,3,EllipticE(x*sqrt(d)/sqrt(c),sqrt(b*c/(a*d)))*sqrt(c)*sqrt(-a+b*x^2)*sqrt(1-d*x^2/c)/(sqrt(d)*sqrt(1-b*x^2/a)*sqrt(c-d*x^2))],
[sqrt(a-b*x^2)/sqrt(-c+d*x^2),x,3,EllipticE(x*sqrt(d)/sqrt(c),sqrt(b*c/(a*d)))*sqrt(c)*sqrt(a-b*x^2)*sqrt(1-d*x^2/c)/(sqrt(d)*sqrt(1-b*x^2/a)*sqrt(-c+d*x^2))],
[sqrt(-a+b*x^2)/sqrt(-c+d*x^2),x,3,EllipticE(x*sqrt(d)/sqrt(c),sqrt(b*c/(a*d)))*sqrt(c)*sqrt(-a+b*x^2)*sqrt(1-d*x^2/c)/(sqrt(d)*sqrt(1-b*x^2/a)*sqrt(-c+d*x^2))],
[sqrt(a+b*x^2)/sqrt(c+d*x^2),x,4,x*sqrt(a+b*x^2)/sqrt(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(a+b*x^2)/(sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(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(a+b*x^2)/(sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))],
[sqrt(-a-b*x^2)/sqrt(c+d*x^2),x,4,x*sqrt(-a-b*x^2)/sqrt(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(-a-b*x^2)/(sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(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(-a-b*x^2)/(sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))],
[sqrt(a+b*x^2)/sqrt(-c-d*x^2),x,4,x*sqrt(a+b*x^2)/sqrt(-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(a+b*x^2)/(sqrt(d)*sqrt(-c-d*x^2)*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(a+b*x^2)/(sqrt(d)*sqrt(-c-d*x^2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))],
[sqrt(-a-b*x^2)/sqrt(-c-d*x^2),x,4,x*sqrt(-a-b*x^2)/sqrt(-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(-a-b*x^2)/(sqrt(d)*sqrt(-c-d*x^2)*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(-a-b*x^2)/(sqrt(d)*sqrt(-c-d*x^2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))],
[sqrt(a-b*x^2)/sqrt(c+d*x^2),x,7,-EllipticE(x*sqrt(b)/sqrt(a),sqrt(-a*d/(b*c)))*sqrt(a)*sqrt(b)*sqrt(1-b*x^2/a)*sqrt(c+d*x^2)/(d*sqrt(a-b*x^2)*sqrt(1+d*x^2/c))+(b*c+a*d)*EllipticF(x*sqrt(b)/sqrt(a),sqrt(-a*d/(b*c)))*sqrt(a)*sqrt(1-b*x^2/a)*sqrt(1+d*x^2/c)/(d*sqrt(b)*sqrt(a-b*x^2)*sqrt(c+d*x^2))],
[sqrt(-a+b*x^2)/sqrt(c+d*x^2),x,7,EllipticE(x*sqrt(b)/sqrt(a),sqrt(-a*d/(b*c)))*sqrt(a)*sqrt(b)*sqrt(1-b*x^2/a)*sqrt(c+d*x^2)/(d*sqrt(-a+b*x^2)*sqrt(1+d*x^2/c))-(b*c+a*d)*EllipticF(x*sqrt(b)/sqrt(a),sqrt(-a*d/(b*c)))*sqrt(a)*sqrt(1-b*x^2/a)*sqrt(1+d*x^2/c)/(d*sqrt(b)*sqrt(-a+b*x^2)*sqrt(c+d*x^2))],
[sqrt(a-b*x^2)/sqrt(-c-d*x^2),x,7,EllipticE(x*sqrt(b)/sqrt(a),sqrt(-a*d/(b*c)))*sqrt(a)*sqrt(b)*sqrt(1-b*x^2/a)*sqrt(-c-d*x^2)/(d*sqrt(a-b*x^2)*sqrt(1+d*x^2/c))+(b*c+a*d)*EllipticF(x*sqrt(b)/sqrt(a),sqrt(-a*d/(b*c)))*sqrt(a)*sqrt(1-b*x^2/a)*sqrt(1+d*x^2/c)/(d*sqrt(b)*sqrt(a-b*x^2)*sqrt(-c-d*x^2))],
[sqrt(-a+b*x^2)/sqrt(-c-d*x^2),x,7,-EllipticE(x*sqrt(b)/sqrt(a),sqrt(-a*d/(b*c)))*sqrt(a)*sqrt(b)*sqrt(1-b*x^2/a)*sqrt(-c-d*x^2)/(d*sqrt(-a+b*x^2)*sqrt(1+d*x^2/c))-(b*c+a*d)*EllipticF(x*sqrt(b)/sqrt(a),sqrt(-a*d/(b*c)))*sqrt(a)*sqrt(1-b*x^2/a)*sqrt(1+d*x^2/c)/(d*sqrt(b)*sqrt(-a+b*x^2)*sqrt(-c-d*x^2))],
[sqrt(c+d*x^2)/sqrt(a-b*x^2),x,3,EllipticE(x*sqrt(b)/sqrt(a),sqrt(-a*d/(b*c)))*sqrt(a)*sqrt(1-b*x^2/a)*sqrt(c+d*x^2)/(sqrt(b)*sqrt(a-b*x^2)*sqrt(1+d*x^2/c))],
[sqrt(-c-d*x^2)/sqrt(a-b*x^2),x,3,EllipticE(x*sqrt(b)/sqrt(a),sqrt(-a*d/(b*c)))*sqrt(a)*sqrt(1-b*x^2/a)*sqrt(-c-d*x^2)/(sqrt(b)*sqrt(a-b*x^2)*sqrt(1+d*x^2/c))],
[sqrt(c+d*x^2)/sqrt(-a+b*x^2),x,3,EllipticE(x*sqrt(b)/sqrt(a),sqrt(-a*d/(b*c)))*sqrt(a)*sqrt(1-b*x^2/a)*sqrt(c+d*x^2)/(sqrt(b)*sqrt(-a+b*x^2)*sqrt(1+d*x^2/c))],
[sqrt(-c-d*x^2)/sqrt(-a+b*x^2),x,3,EllipticE(x*sqrt(b)/sqrt(a),sqrt(-a*d/(b*c)))*sqrt(a)*sqrt(1-b*x^2/a)*sqrt(-c-d*x^2)/(sqrt(b)*sqrt(-a+b*x^2)*sqrt(1+d*x^2/c))],
[sqrt(c-d*x^2)/sqrt(a-b*x^2),x,3,EllipticE(x*sqrt(b)/sqrt(a),sqrt(a*d/(b*c)))*sqrt(a)*sqrt(1-b*x^2/a)*sqrt(c-d*x^2)/(sqrt(b)*sqrt(a-b*x^2)*sqrt(1-d*x^2/c))],
[sqrt(-c+d*x^2)/sqrt(a-b*x^2),x,3,EllipticE(x*sqrt(b)/sqrt(a),sqrt(a*d/(b*c)))*sqrt(a)*sqrt(1-b*x^2/a)*sqrt(-c+d*x^2)/(sqrt(b)*sqrt(a-b*x^2)*sqrt(1-d*x^2/c))],
[sqrt(c-d*x^2)/sqrt(-a+b*x^2),x,3,EllipticE(x*sqrt(b)/sqrt(a),sqrt(a*d/(b*c)))*sqrt(a)*sqrt(1-b*x^2/a)*sqrt(c-d*x^2)/(sqrt(b)*sqrt(-a+b*x^2)*sqrt(1-d*x^2/c))],
[sqrt(-c+d*x^2)/sqrt(-a+b*x^2),x,3,EllipticE(x*sqrt(b)/sqrt(a),sqrt(a*d/(b*c)))*sqrt(a)*sqrt(1-b*x^2/a)*sqrt(-c+d*x^2)/(sqrt(b)*sqrt(-a+b*x^2)*sqrt(1-d*x^2/c))],
[sqrt(c+d*x^2)/sqrt(a+b*x^2),x,4,d*x*sqrt(a+b*x^2)/(b*sqrt(c+d*x^2))+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(a+b*x^2)/(a*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(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(d)*sqrt(a+b*x^2)/(b*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))],
[sqrt(-c-d*x^2)/sqrt(a+b*x^2),x,4,-d*x*sqrt(a+b*x^2)/(b*sqrt(-c-d*x^2))-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(a+b*x^2)/(a*sqrt(d)*sqrt(-c-d*x^2)*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(c)*sqrt(d)*sqrt(a+b*x^2)/(b*sqrt(-c-d*x^2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))],
[sqrt(c+d*x^2)/sqrt(-a-b*x^2),x,4,-d*x*sqrt(-a-b*x^2)/(b*sqrt(c+d*x^2))-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(-a-b*x^2)/(a*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(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(d)*sqrt(-a-b*x^2)/(b*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))],
[sqrt(-c-d*x^2)/sqrt(-a-b*x^2),x,4,d*x*sqrt(-a-b*x^2)/(b*sqrt(-c-d*x^2))+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(-a-b*x^2)/(a*sqrt(d)*sqrt(-c-d*x^2)*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(c)*sqrt(d)*sqrt(-a-b*x^2)/(b*sqrt(-c-d*x^2)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2))))],
[sqrt(c-d*x^2)/sqrt(a+b*x^2),x,7,-EllipticE(x*sqrt(d)/sqrt(c),sqrt(-b*c/(a*d)))*sqrt(c)*sqrt(d)*sqrt(a+b*x^2)*sqrt(1-d*x^2/c)/(b*sqrt(1+b*x^2/a)*sqrt(c-d*x^2))+(b*c+a*d)*EllipticF(x*sqrt(d)/sqrt(c),sqrt(-b*c/(a*d)))*sqrt(c)*sqrt(1+b*x^2/a)*sqrt(1-d*x^2/c)/(b*sqrt(d)*sqrt(a+b*x^2)*sqrt(c-d*x^2))],
[sqrt(-c+d*x^2)/sqrt(a+b*x^2),x,7,EllipticE(x*sqrt(d)/sqrt(c),sqrt(-b*c/(a*d)))*sqrt(c)*sqrt(d)*sqrt(a+b*x^2)*sqrt(1-d*x^2/c)/(b*sqrt(1+b*x^2/a)*sqrt(-c+d*x^2))-(b*c+a*d)*EllipticF(x*sqrt(d)/sqrt(c),sqrt(-b*c/(a*d)))*sqrt(c)*sqrt(1+b*x^2/a)*sqrt(1-d*x^2/c)/(b*sqrt(d)*sqrt(a+b*x^2)*sqrt(-c+d*x^2))],
[sqrt(c-d*x^2)/sqrt(-a-b*x^2),x,7,EllipticE(x*sqrt(d)/sqrt(c),sqrt(-b*c/(a*d)))*sqrt(c)*sqrt(d)*sqrt(-a-b*x^2)*sqrt(1-d*x^2/c)/(b*sqrt(1+b*x^2/a)*sqrt(c-d*x^2))+(b*c+a*d)*EllipticF(x*sqrt(d)/sqrt(c),sqrt(-b*c/(a*d)))*sqrt(c)*sqrt(1+b*x^2/a)*sqrt(1-d*x^2/c)/(b*sqrt(d)*sqrt(-a-b*x^2)*sqrt(c-d*x^2))],
[sqrt(-c+d*x^2)/sqrt(-a-b*x^2),x,7,-EllipticE(x*sqrt(d)/sqrt(c),sqrt(-b*c/(a*d)))*sqrt(c)*sqrt(d)*sqrt(-a-b*x^2)*sqrt(1-d*x^2/c)/(b*sqrt(1+b*x^2/a)*sqrt(-c+d*x^2))-(b*c+a*d)*EllipticF(x*sqrt(d)/sqrt(c),sqrt(-b*c/(a*d)))*sqrt(c)*sqrt(1+b*x^2/a)*sqrt(1-d*x^2/c)/(b*sqrt(d)*sqrt(-a-b*x^2)*sqrt(-c+d*x^2))],
[1/(sqrt(2+b*x^2)*sqrt(3+d*x^2)),x,1,sqrt(cos(arctan(x*sqrt(d)/sqrt(3)))^2)/cos(arctan(x*sqrt(d)/sqrt(3)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(3))),sqrt(1-3/2*b/d))*sqrt(2+b*x^2)/(sqrt(2)*sqrt(d)*sqrt((2+b*x^2)/(3+d*x^2))*sqrt(3+d*x^2))],
[1/(sqrt(4-x^2)*sqrt(c+d*x^2)),x,2,EllipticF(1/2*x,sqrt(-4*d/c))*sqrt(1+d*x^2/c)/sqrt(c+d*x^2)],
[1/(sqrt(4+x^2)*sqrt(c+d*x^2)),x,1,sqrt(cos(arctan(1/2*x))^2)/cos(arctan(1/2*x))*EllipticF(sin(arctan(1/2*x)),sqrt(1-4*d/c))*sqrt(c+d*x^2)/(c*sqrt(4+x^2)*sqrt((c+d*x^2)/(c*(4+x^2))))],
[1/(sqrt(1-x^2)*sqrt(-1+2*x^2)),x,1,-sqrt(cos(arccos(x))^2)/cos(arccos(x))*EllipticF(sin(arccos(x)),sqrt(2))],
[sqrt(1-c^2*x^2)/sqrt(1+c^2*x^2),x,4,-EllipticE(c*x,I)/c+2*EllipticF(c*x,I)/c],
[sqrt(2+b*x^2)/sqrt(3+d*x^2),x,4,x*sqrt(2+b*x^2)/sqrt(3+d*x^2)-sqrt(cos(arctan(x*sqrt(d)/sqrt(3)))^2)/cos(arctan(x*sqrt(d)/sqrt(3)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(3))),sqrt(1-3/2*b/d))*sqrt(2)*sqrt(2+b*x^2)/(sqrt(d)*sqrt((2+b*x^2)/(3+d*x^2))*sqrt(3+d*x^2))+sqrt(cos(arctan(x*sqrt(d)/sqrt(3)))^2)/cos(arctan(x*sqrt(d)/sqrt(3)))*EllipticF(sin(arctan(x*sqrt(d)/sqrt(3))),sqrt(1-3/2*b/d))*sqrt(2)*sqrt(2+b*x^2)/(sqrt(d)*sqrt((2+b*x^2)/(3+d*x^2))*sqrt(3+d*x^2))],
[sqrt(-1+3*x^2)/sqrt(2-3*x^2),x,1,-sqrt(cos(arccos(x*sqrt(3/2)))^2)/cos(arccos(x*sqrt(3/2)))*EllipticE(sin(arccos(x*sqrt(3/2))),sqrt(2))/sqrt(3)],
[sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))/sqrt(1-2*c*x^2/(b+sqrt(b^2-4*a*c))),x,1,EllipticE(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)),sqrt((-b-sqrt(b^2-4*a*c))/(b-sqrt(b^2-4*a*c))))*sqrt(b+sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c))],
[sqrt(1-2*c*x^2/(b-sqrt(b^2-4*a*c)))/sqrt(1-2*c*x^2/(b+sqrt(b^2-4*a*c))),x,1,EllipticE(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)),sqrt((b+sqrt(b^2-4*a*c))/(b-sqrt(b^2-4*a*c))))*sqrt(b+sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c))],
[sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))/sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))),x,4,x*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))/sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c)))-sqrt(cos(arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c))))^2)/cos(arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c))))*EllipticE(sin(arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))),sqrt(-2*sqrt(b^2-4*a*c)/(b-sqrt(b^2-4*a*c))))*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(b+sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c)*sqrt((1+2*c*x^2/(b-sqrt(b^2-4*a*c)))/(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))+sqrt(cos(arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c))))^2)/cos(arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c))))*EllipticF(sin(arctan(x*sqrt(2)*sqrt(c)/sqrt(b+sqrt(b^2-4*a*c)))),sqrt(-2*sqrt(b^2-4*a*c)/(b-sqrt(b^2-4*a*c))))*sqrt(1+2*c*x^2/(b-sqrt(b^2-4*a*c)))*sqrt(b+sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c)*sqrt((1+2*c*x^2/(b-sqrt(b^2-4*a*c)))/(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))*sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))))],
[sqrt(1-2*c*x^2/(b-sqrt(b^2-4*a*c)))/sqrt(1+2*c*x^2/(b+sqrt(b^2-4*a*c))),x,3,b*EllipticF(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)),sqrt((-b+sqrt(b^2-4*a*c))/(b+sqrt(b^2-4*a*c))))*sqrt(2)/(sqrt(c)*sqrt(b-sqrt(b^2-4*a*c)))-EllipticE(x*sqrt(2)*sqrt(c)/sqrt(b-sqrt(b^2-4*a*c)),sqrt((-b+sqrt(b^2-4*a*c))/(b+sqrt(b^2-4*a*c))))*(b+sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c)*sqrt(b-sqrt(b^2-4*a*c)))],
[(1-2*x^2)^m/sqrt(1-x^2),x,-1,-2^(-2-m)*(2-4*x^2)^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],(1-2*x^2)^2)*sqrt(x^2)/((1+m)*x)],
[1/(sqrt(-1+x^2)*sqrt(7+x^2-4*sqrt(3))),x,2,EllipticF(x,sqrt(-7-4*sqrt(3)))*sqrt(1-x^2)/(sqrt(-1+x^2)*sqrt(7-4*sqrt(3)))],
[1/(sqrt(3+x^2*(-3+sqrt(3)))*sqrt(3-3*sqrt(3)+2*x^2*sqrt(3))),x,1,-1/6*sqrt(cos(arccos(x*sqrt(1/3*(3-sqrt(3)))))^2)/cos(arccos(x*sqrt(1/3*(3-sqrt(3)))))*EllipticF(sin(arccos(x*sqrt(1/3*(3-sqrt(3))))),sqrt(1/2*(1+sqrt(3))))*sqrt(3+sqrt(3))],

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

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

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

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

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

# q>0

# q<0
[(a+b*x^2)^(7/4)/(c+d*x^2),x,13,6/5*a*b*x/(d*(a+b*x^2)^(1/4))-2*b*(b*c-a*d)*x/(d^2*(a+b*x^2)^(1/4))+2/5*b*x*(a+b*x^2)^(3/4)/d-6/5*a^(3/2)*(1+b*x^2/a)^(1/4)*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)/(d*(a+b*x^2)^(1/4))+2*(b*c-a*d)*(1+b*x^2/a)^(1/4)*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)*sqrt(b)/(d^2*(a+b*x^2)^(1/4))+a^(1/4)*(-b*c+a*d)^(3/2)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),-sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(d^(5/2)*x)-a^(1/4)*(-b*c+a*d)^(3/2)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(d^(5/2)*x)],
[(a+b*x^2)^(5/4)/(c+d*x^2),x,12,2/3*b*x*(a+b*x^2)^(1/4)/d+2/3*a^(3/2)*(1+b*x^2/a)^(3/4)*sqrt(cos(1/2*arctan(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/(d*(a+b*x^2)^(3/4))-2*(b*c-a*d)*(1+b*x^2/a)^(3/4)*sqrt(cos(1/2*arctan(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)*sqrt(b)/(d^2*(a+b*x^2)^(3/4))+a^(1/4)*(b*c-a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),-sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(d^2*x)+a^(1/4)*(b*c-a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(d^2*x)],
[(a+b*x^2)^(3/4)/(c+d*x^2),x,8,2*b*x/(d*(a+b*x^2)^(1/4))-2*(1+b*x^2/a)^(1/4)*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)*sqrt(b)/(d*(a+b*x^2)^(1/4))+a^(1/4)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),-sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*c+a*d)*sqrt(-b*x^2/a)/(d^(3/2)*x)-a^(1/4)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*c+a*d)*sqrt(-b*x^2/a)/(d^(3/2)*x)],
[(a+b*x^2)^(1/4)/(c+d*x^2),x,8,2*(1+b*x^2/a)^(3/4)*sqrt(cos(1/2*arctan(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)*sqrt(b)/(d*(a+b*x^2)^(3/4))-a^(1/4)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),-sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(d*x)-a^(1/4)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(d*x)],
[1/((a+b*x^2)^(1/4)*(c+d*x^2)),x,4,a^(1/4)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),-sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(x*sqrt(d)*sqrt(-b*c+a*d))-a^(1/4)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(x*sqrt(d)*sqrt(-b*c+a*d))],
[1/((a+b*x^2)^(3/4)*(c+d*x^2)),x,5,a^(1/4)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),-sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/((b*c-a*d)*x)+a^(1/4)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/((b*c-a*d)*x)],
[1/((a+b*x^2)^(5/4)*(c+d*x^2)),x,7,2*(1+b*x^2/a)^(1/4)*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)/((b*c-a*d)*(a+b*x^2)^(1/4)*sqrt(a))+a^(1/4)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),-sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(d)*sqrt(-b*x^2/a)/((-b*c+a*d)^(3/2)*x)-a^(1/4)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(d)*sqrt(-b*x^2/a)/((-b*c+a*d)^(3/2)*x)],
[1/((a+b*x^2)^(7/4)*(c+d*x^2)),x,9,2/3*b*x/(a*(b*c-a*d)*(a+b*x^2)^(3/4))+2/3*(1+b*x^2/a)^(3/4)*sqrt(cos(1/2*arctan(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/((b*c-a*d)*(a+b*x^2)^(3/4)*sqrt(a))-a^(1/4)*d*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),-sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/((b*c-a*d)^2*x)-a^(1/4)*d*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/((b*c-a*d)^2*x)],
[1/((a+b*x^2)^(9/4)*(c+d*x^2)),x,10,2/5*b*x/(a*(b*c-a*d)*(a+b*x^2)^(5/4))+2/5*(3*b*c-8*a*d)*(1+b*x^2/a)^(1/4)*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)/(a^(3/2)*(b*c-a*d)^2*(a+b*x^2)^(1/4))+a^(1/4)*d^(3/2)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),-sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/((-b*c+a*d)^(5/2)*x)-a^(1/4)*d^(3/2)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/((-b*c+a*d)^(5/2)*x)],
[1/((a+b*x^2)^(11/4)*(c+d*x^2)),x,10,2/7*b*x/(a*(b*c-a*d)*(a+b*x^2)^(7/4))+2/21*b*(5*b*c-12*a*d)*x/(a^2*(b*c-a*d)^2*(a+b*x^2)^(3/4))+2/21*(5*b*c-12*a*d)*(1+b*x^2/a)^(3/4)*sqrt(cos(1/2*arctan(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/(a^(3/2)*(b*c-a*d)^2*(a+b*x^2)^(3/4))+a^(1/4)*d^2*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),-sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/((b*c-a*d)^3*x)+a^(1/4)*d^2*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/((b*c-a*d)^3*x)],
[(a+b*x^2)^(7/4)/(c+d*x^2)^2,x,9,1/2*b*(5*b*c-a*d)*x/(c*d^2*(a+b*x^2)^(1/4))-1/2*(b*c-a*d)*x*(a+b*x^2)^(3/4)/(c*d*(c+d*x^2))-1/2*(5*b*c-a*d)*(1+b*x^2/a)^(1/4)*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)*sqrt(b)/(c*d^2*(a+b*x^2)^(1/4))+1/4*a^(1/4)*(5*b*c+2*a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),-sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*c+a*d)*sqrt(-b*x^2/a)/(c*d^(5/2)*x)-1/4*a^(1/4)*(5*b*c+2*a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*c+a*d)*sqrt(-b*x^2/a)/(c*d^(5/2)*x)],
[(a+b*x^2)^(5/4)/(c+d*x^2)^2,x,9,-1/2*(b*c-a*d)*x*(a+b*x^2)^(1/4)/(c*d*(c+d*x^2))+1/2*(3*b*c+a*d)*(1+b*x^2/a)^(3/4)*sqrt(cos(1/2*arctan(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)*sqrt(b)/(c*d^2*(a+b*x^2)^(3/4))-1/4*a^(1/4)*(3*b*c+2*a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),-sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(c*d^2*x)-1/4*a^(1/4)*(3*b*c+2*a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(c*d^2*x)],
[(a+b*x^2)^(3/4)/(c+d*x^2)^2,x,9,-1/2*b*x/(c*d*(a+b*x^2)^(1/4))+1/2*x*(a+b*x^2)^(3/4)/(c*(c+d*x^2))+1/2*(1+b*x^2/a)^(1/4)*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)*sqrt(b)/(c*d*(a+b*x^2)^(1/4))+1/4*a^(1/4)*(b*c+2*a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),-sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(c*d^(3/2)*x*sqrt(-b*c+a*d))-1/4*a^(1/4)*(b*c+2*a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(c*d^(3/2)*x*sqrt(-b*c+a*d))],
[(a+b*x^2)^(1/4)/(c+d*x^2)^2,x,9,1/2*x*(a+b*x^2)^(1/4)/(c*(c+d*x^2))+1/2*(1+b*x^2/a)^(3/4)*sqrt(cos(1/2*arctan(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)*sqrt(b)/(c*d*(a+b*x^2)^(3/4))-1/4*a^(1/4)*(b*c-2*a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),-sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(c*d*(b*c-a*d)*x)-1/4*a^(1/4)*(b*c-2*a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(c*d*(b*c-a*d)*x)],
[1/((a+b*x^2)^(1/4)*(c+d*x^2)^2),x,9,1/2*b*x/(c*(b*c-a*d)*(a+b*x^2)^(1/4))-1/2*d*x*(a+b*x^2)^(3/4)/(c*(b*c-a*d)*(c+d*x^2))-1/2*(1+b*x^2/a)^(1/4)*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)*sqrt(b)/(c*(b*c-a*d)*(a+b*x^2)^(1/4))-1/4*a^(1/4)*(3*b*c-2*a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),-sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(c*(-b*c+a*d)^(3/2)*x*sqrt(d))+1/4*a^(1/4)*(3*b*c-2*a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(c*(-b*c+a*d)^(3/2)*x*sqrt(d))],
[1/((a+b*x^2)^(3/4)*(c+d*x^2)^2),x,9,-1/2*d*x*(a+b*x^2)^(1/4)/(c*(b*c-a*d)*(c+d*x^2))-1/2*(1+b*x^2/a)^(3/4)*sqrt(cos(1/2*arctan(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(a)*sqrt(b)/(c*(b*c-a*d)*(a+b*x^2)^(3/4))+1/4*a^(1/4)*(5*b*c-2*a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),-sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(c*(b*c-a*d)^2*x)+1/4*a^(1/4)*(5*b*c-2*a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(c*(b*c-a*d)^2*x)],
[1/((a+b*x^2)^(5/4)*(c+d*x^2)^2),x,10,-1/2*d*x/(c*(b*c-a*d)*(a+b*x^2)^(1/4)*(c+d*x^2))+1/2*(4*b*c+a*d)*(1+b*x^2/a)^(1/4)*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*(b*c-a*d)^2*(a+b*x^2)^(1/4)*sqrt(a))-1/4*a^(1/4)*(7*b*c-2*a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),-sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(d)*sqrt(-b*x^2/a)/(c*(-b*c+a*d)^(5/2)*x)+1/4*a^(1/4)*(7*b*c-2*a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(d)*sqrt(-b*x^2/a)/(c*(-b*c+a*d)^(5/2)*x)],
[1/((a+b*x^2)^(7/4)*(c+d*x^2)^2),x,10,1/6*b*(4*b*c+3*a*d)*x/(a*c*(b*c-a*d)^2*(a+b*x^2)^(3/4))-1/2*d*x/(c*(b*c-a*d)*(a+b*x^2)^(3/4)*(c+d*x^2))+1/6*(4*b*c+3*a*d)*(1+b*x^2/a)^(3/4)*sqrt(cos(1/2*arctan(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/(c*(b*c-a*d)^2*(a+b*x^2)^(3/4)*sqrt(a))-1/4*a^(1/4)*d*(9*b*c-2*a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),-sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(c*(b*c-a*d)^3*x)-1/4*a^(1/4)*d*(9*b*c-2*a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(c*(b*c-a*d)^3*x)],
[1/((a+b*x^2)^(9/4)*(c+d*x^2)^2),x,11,1/10*b*(4*b*c+5*a*d)*x/(a*c*(b*c-a*d)^2*(a+b*x^2)^(5/4))-1/2*d*x/(c*(b*c-a*d)*(a+b*x^2)^(5/4)*(c+d*x^2))+1/10*(12*b^2*c^2-52*a*b*c*d-5*a^2*d^2)*(1+b*x^2/a)^(1/4)*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)/(a^(3/2)*c*(b*c-a*d)^3*(a+b*x^2)^(1/4))-1/4*a^(1/4)*d^(3/2)*(11*b*c-2*a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),-sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(c*(-b*c+a*d)^(7/2)*x)+1/4*a^(1/4)*d^(3/2)*(11*b*c-2*a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(c*(-b*c+a*d)^(7/2)*x)],
[1/((a+b*x^2)^(11/4)*(c+d*x^2)^2),x,11,1/14*b*(4*b*c+7*a*d)*x/(a*c*(b*c-a*d)^2*(a+b*x^2)^(7/4))+1/42*b*(20*b^2*c^2-76*a*b*c*d-21*a^2*d^2)*x/(a^2*c*(b*c-a*d)^3*(a+b*x^2)^(3/4))-1/2*d*x/(c*(b*c-a*d)*(a+b*x^2)^(7/4)*(c+d*x^2))+1/42*(20*b^2*c^2-76*a*b*c*d-21*a^2*d^2)*(1+b*x^2/a)^(3/4)*sqrt(cos(1/2*arctan(x*sqrt(b)/sqrt(a)))^2)/cos(1/2*arctan(x*sqrt(b)/sqrt(a)))*EllipticF(sin(1/2*arctan(x*sqrt(b)/sqrt(a))),sqrt(2))*sqrt(b)/(a^(3/2)*c*(b*c-a*d)^3*(a+b*x^2)^(3/4))+1/4*a^(1/4)*d^2*(13*b*c-2*a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),-sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(c*(b*c-a*d)^4*x)+1/4*a^(1/4)*d^2*(13*b*c-2*a*d)*EllipticPi((a+b*x^2)^(1/4)/a^(1/4),sqrt(a)*sqrt(d)/sqrt(-b*c+a*d),I)*sqrt(-b*x^2/a)/(c*(b*c-a*d)^4*x)],

# Integrands of the form (a+b x^2)^p (c+d x^2)^q with p symbolic
[(a+b*x^2)^p*(c+d*x^2)^q,x,3,x*(a+b*x^2)^p*(c+d*x^2)^q*AppellF1(1/2,-p,-q,3/2,-b*x^2/a,-d*x^2/c)/((1+b*x^2/a)^p*(1+d*x^2/c)^q)],
[(a+b*x^2)^p*(c+d*x^2)^3,x,5,d*(15*a^2*d^2-8*a*b*c*d*(6+p)+b^2*c^2*(57+28*p+4*p^2))*x*(a+b*x^2)^(1+p)/(b^3*(105+142*p+60*p^2+8*p^3))-d*(5*a*d-b*c*(11+2*p))*x*(a+b*x^2)^(1+p)*(c+d*x^2)/(b^2*(35+24*p+4*p^2))+d*x*(a+b*x^2)^(1+p)*(c+d*x^2)^2/(b*(7+2*p))-(15*a^3*d^3-9*a^2*b*c*d^2*(7+2*p)+3*a*b^2*c^2*d*(35+24*p+4*p^2)-b^3*c^3*(105+142*p+60*p^2+8*p^3))*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(b^3*(105+142*p+60*p^2+8*p^3)*(1+b*x^2/a)^p)],
[(a+b*x^2)^p*(c+d*x^2)^2,x,4,-d*(3*a*d-b*c*(7+2*p))*x*(a+b*x^2)^(1+p)/(b^2*(15+16*p+4*p^2))+d*x*(a+b*x^2)^(1+p)*(c+d*x^2)/(b*(5+2*p))+(3*a^2*d^2-2*a*b*c*d*(5+2*p)+b^2*c^2*(15+16*p+4*p^2))*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(b^2*(15+16*p+4*p^2)*(1+b*x^2/a)^p)],
[(a+b*x^2)^p*(c+d*x^2),x,3,d*x*(a+b*x^2)^(1+p)/(b*(3+2*p))-(a*d-b*c*(3+2*p))*x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(b*(3+2*p)*(1+b*x^2/a)^p)],
[(a+b*x^2)^p,x,2,x*(a+b*x^2)^p*hypergeom([1/2,-p],[3/2],-b*x^2/a)/(1+b*x^2/a)^p],
[(a+b*x^2)^p/(c+d*x^2),x,2,x*(a+b*x^2)^p*AppellF1(1/2,-p,1,3/2,-b*x^2/a,-d*x^2/c)/(c*(1+b*x^2/a)^p)],
[(a+b*x^2)^p/(c+d*x^2)^2,x,2,x*(a+b*x^2)^p*AppellF1(1/2,-p,2,3/2,-b*x^2/a,-d*x^2/c)/(c^2*(1+b*x^2/a)^p)],
[(a+b*x^2)^p/(c+d*x^2)^3,x,2,x*(a+b*x^2)^p*AppellF1(1/2,-p,3,3/2,-b*x^2/a,-d*x^2/c)/(c^3*(1+b*x^2/a)^p)],
[(a+b*x^2)^(-1-b*c/(2*b*c-2*a*d))*(c+d*x^2)^(-1+a*d/(2*b*c-2*a*d)),x,1,x*(c+d*x^2)^(a*d/(2*b*c-2*a*d))/(a*c*(a+b*x^2)^(b*c/(2*b*c-2*a*d)))]]:
