# Maple integration test file: "1 Algebraic functions\1.1 Binomial products\1.1.2 Quadratic\1.1.2.5 (a+b x^2)^p (c+d x^2)^q (e+f x^2)^r.txt"

lst:=[

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

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

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

# q>0
[(a+b*x^2)*(c+d*x^2)*(e+f*x^2)^4,x,2,a*c*e^4*x+1/3*e^3*(b*c*e+a*d*e+4*a*c*f)*x^3+1/5*e^2*(2*a*f*(2*d*e+3*c*f)+b*e*(d*e+4*c*f))*x^5+2/7*e*f*(a*f*(3*d*e+2*c*f)+b*e*(2*d*e+3*c*f))*x^7+1/9*f^2*(a*f*(4*d*e+c*f)+2*b*e*(3*d*e+2*c*f))*x^9+1/11*f^3*(4*b*d*e+b*c*f+a*d*f)*x^11+1/13*b*d*f^4*x^13],
[(a+b*x^2)*(c+d*x^2)*(e+f*x^2)^3,x,2,a*c*e^3*x+1/3*e^2*(b*c*e+a*d*e+3*a*c*f)*x^3+1/5*e*(3*a*f*(d*e+c*f)+b*e*(d*e+3*c*f))*x^5+1/7*f*(3*b*e*(d*e+c*f)+a*f*(3*d*e+c*f))*x^7+1/9*f^2*(3*b*d*e+b*c*f+a*d*f)*x^9+1/11*b*d*f^3*x^11],
[(a+b*x^2)*(c+d*x^2)*(e+f*x^2)^2,x,2,a*c*e^2*x+1/3*e*(b*c*e+a*d*e+2*a*c*f)*x^3+1/5*(a*f*(2*d*e+c*f)+b*e*(d*e+2*c*f))*x^5+1/7*f*(2*b*d*e+b*c*f+a*d*f)*x^7+1/9*b*d*f^2*x^9],
[(a+b*x^2)*(c+d*x^2)*(e+f*x^2),x,2,a*c*e*x+1/3*(b*c*e+a*d*e+a*c*f)*x^3+1/5*(b*d*e+b*c*f+a*d*f)*x^5+1/7*b*d*f*x^7],
[(a+b*x^2)*(c+d*x^2)/(e+f*x^2),x,3,-1/3*(3*b*d*e-3*b*c*f-2*a*d*f)*x/f^2+1/3*d*x*(a+b*x^2)/f+(b*e-a*f)*(d*e-c*f)*arctan(x*sqrt(f)/sqrt(e))/(f^(5/2)*sqrt(e))],
[(a+b*x^2)*(c+d*x^2)/(e+f*x^2)^2,x,3,1/2*b*(3*d*e-c*f)*x/(e*f^2)-1/2*(d*e-c*f)*x*(a+b*x^2)/(e*f*(e+f*x^2))-1/2*(b*e*(3*d*e-c*f)-a*f*(d*e+c*f))*arctan(x*sqrt(f)/sqrt(e))/(e^(3/2)*f^(5/2))],
[(a+b*x^2)*(c+d*x^2)/(e+f*x^2)^3,x,3,-1/4*(d*e-c*f)*x*(a+b*x^2)/(e*f*(e+f*x^2)^2)-1/8*(b*e*(3*d*e+c*f)-a*f*(d*e+3*c*f))*x/(e^2*f^2*(e+f*x^2))+1/8*(b*e*(3*d*e+c*f)+a*f*(d*e+3*c*f))*arctan(x*sqrt(f)/sqrt(e))/(e^(5/2)*f^(5/2))],
[(a+b*x^2)*(c+d*x^2)/(e+f*x^2)^4,x,4,-1/6*(d*e-c*f)*x*(a+b*x^2)/(e*f*(e+f*x^2)^3)-1/24*(3*b*e*(d*e+c*f)-a*f*(d*e+5*c*f))*x/(e^2*f^2*(e+f*x^2)^2)+1/16*(b*e*(d*e+c*f)+a*f*(d*e+5*c*f))*x/(e^3*f^2*(e+f*x^2))+1/16*(b*e*(d*e+c*f)+a*f*(d*e+5*c*f))*arctan(x*sqrt(f)/sqrt(e))/(e^(7/2)*f^(5/2))],
[(a+b*x^2)*(c+d*x^2)^2*(e+f*x^2)^3,x,2,a*c^2*e^3*x+1/3*c*e^2*(b*c*e+2*a*d*e+3*a*c*f)*x^3+1/5*e*(b*c*e*(2*d*e+3*c*f)+a*(d^2*e^2+6*c*d*e*f+3*c^2*f^2))*x^5+1/7*(a*f*(3*d^2*e^2+6*c*d*e*f+c^2*f^2)+b*e*(d^2*e^2+6*c*d*e*f+3*c^2*f^2))*x^7+1/9*f*(a*d*f*(3*d*e+2*c*f)+b*(3*d^2*e^2+6*c*d*e*f+c^2*f^2))*x^9+1/11*d*f^2*(3*b*d*e+2*b*c*f+a*d*f)*x^11+1/13*b*d^2*f^3*x^13],
[(a+b*x^2)*(c+d*x^2)^2*(e+f*x^2)^2,x,2,a*c^2*e^2*x+1/3*c*e*(b*c*e+2*a*(d*e+c*f))*x^3+1/5*(2*b*c*e*(d*e+c*f)+a*(d^2*e^2+4*c*d*e*f+c^2*f^2))*x^5+1/7*(2*a*d*f*(d*e+c*f)+b*(d^2*e^2+4*c*d*e*f+c^2*f^2))*x^7+1/9*d*f*(a*d*f+2*b*(d*e+c*f))*x^9+1/11*b*d^2*f^2*x^11],
[(a+b*x^2)*(c+d*x^2)^2*(e+f*x^2),x,2,a*c^2*e*x+1/3*c*(b*c*e+2*a*d*e+a*c*f)*x^3+1/5*(b*c*(2*d*e+c*f)+a*d*(d*e+2*c*f))*x^5+1/7*d*(b*d*e+2*b*c*f+a*d*f)*x^7+1/9*b*d^2*f*x^9],
[(a+b*x^2)*(c+d*x^2)^2/(e+f*x^2),x,4,-1/15*(5*a*d*f*(3*d*e-5*c*f)-b*(15*d^2*e^2-25*c*d*e*f+8*c^2*f^2))*x/f^3-1/15*(5*b*d*e-4*b*c*f-5*a*d*f)*x*(c+d*x^2)/f^2+1/5*b*x*(c+d*x^2)^2/f-(b*e-a*f)*(d*e-c*f)^2*arctan(x*sqrt(f)/sqrt(e))/(f^(7/2)*sqrt(e))],
[(a+b*x^2)*(c+d*x^2)^2/(e+f*x^2)^2,x,4,-1/6*d*(b*e*(15*d*e-13*c*f)-3*a*f*(3*d*e-c*f))*x/(e*f^3)+1/6*d*(5*b*e-3*a*f)*x*(c+d*x^2)/(e*f^2)-1/2*(b*e-a*f)*x*(c+d*x^2)^2/(e*f*(e+f*x^2))+1/2*(d*e-c*f)*(b*e*(5*d*e-c*f)-a*f*(3*d*e+c*f))*arctan(x*sqrt(f)/sqrt(e))/(e^(3/2)*f^(7/2))],
[(a+b*x^2)*(c+d*x^2)^2/(e+f*x^2)^3,x,4,1/8*d*(b*e*(15*d*e-c*f)-3*a*f*(d*e+c*f))*x/(e^2*f^3)-1/4*(b*e-a*f)*x*(c+d*x^2)^2/(e*f*(e+f*x^2)^2)-1/8*(b*e*(5*d*e-c*f)-a*f*(d*e+3*c*f))*x*(c+d*x^2)/(e^2*f^2*(e+f*x^2))-1/8*(b*e*(15*d^2*e^2-6*c*d*e*f-c^2*f^2)-a*f*(3*d^2*e^2+2*c*d*e*f+3*c^2*f^2))*arctan(x*sqrt(f)/sqrt(e))/(e^(5/2)*f^(7/2))],
[(a+b*x^2)*(c+d*x^2)^2/(e+f*x^2)^4,x,4,-1/6*(b*e-a*f)*x*(c+d*x^2)^2/(e*f*(e+f*x^2)^3)-1/24*(d*e*(5*b*e+a*f)-c*f*(b*e+5*a*f))*x*(c+d*x^2)/(e^2*f^2*(e+f*x^2)^2)-1/48*(a*f*(3*d^2*e^2+4*c*d*e*f-15*c^2*f^2)+b*e*(15*d^2*e^2-4*c*d*e*f-3*c^2*f^2))*x/(e^3*f^3*(e+f*x^2))+1/16*(b*e*(5*d^2*e^2+2*c*d*e*f+c^2*f^2)+a*f*(d^2*e^2+2*c*d*e*f+5*c^2*f^2))*arctan(x*sqrt(f)/sqrt(e))/(e^(7/2)*f^(7/2))],
[(a+b*x^2)*(c+d*x^2)^3*(e+f*x^2)^3,x,2,a*c^3*e^3*x+1/3*c^2*e^2*(b*c*e+3*a*(d*e+c*f))*x^3+3/5*c*e*(b*c*e*(d*e+c*f)+a*(d^2*e^2+3*c*d*e*f+c^2*f^2))*x^5+1/7*(3*b*c*e*(d^2*e^2+3*c*d*e*f+c^2*f^2)+a*(d^3*e^3+9*c*d^2*e^2*f+9*c^2*d*e*f^2+c^3*f^3))*x^7+1/9*(3*a*d*f*(d^2*e^2+3*c*d*e*f+c^2*f^2)+b*(d^3*e^3+9*c*d^2*e^2*f+9*c^2*d*e*f^2+c^3*f^3))*x^9+3/11*d*f*(a*d*f*(d*e+c*f)+b*(d^2*e^2+3*c*d*e*f+c^2*f^2))*x^11+1/13*d^2*f^2*(a*d*f+3*b*(d*e+c*f))*x^13+1/15*b*d^3*f^3*x^15],
[(a+b*x^2)*(c+d*x^2)^3*(e+f*x^2)^2,x,2,a*c^3*e^2*x+1/3*c^2*e*(b*c*e+3*a*d*e+2*a*c*f)*x^3+1/5*c*(b*c*e*(3*d*e+2*c*f)+a*(3*d^2*e^2+6*c*d*e*f+c^2*f^2))*x^5+1/7*(b*c*(3*d^2*e^2+6*c*d*e*f+c^2*f^2)+a*d*(d^2*e^2+6*c*d*e*f+3*c^2*f^2))*x^7+1/9*d*(a*d*f*(2*d*e+3*c*f)+b*(d^2*e^2+6*c*d*e*f+3*c^2*f^2))*x^9+1/11*d^2*f*(2*b*d*e+3*b*c*f+a*d*f)*x^11+1/13*b*d^3*f^2*x^13],
[(a+b*x^2)*(c+d*x^2)^3*(e+f*x^2),x,2,a*c^3*e*x+1/3*c^2*(b*c*e+3*a*d*e+a*c*f)*x^3+1/5*c*(3*a*d*(d*e+c*f)+b*c*(3*d*e+c*f))*x^5+1/7*d*(3*b*c*(d*e+c*f)+a*d*(d*e+3*c*f))*x^7+1/9*d^2*(b*d*e+3*b*c*f+a*d*f)*x^9+1/11*b*d^3*f*x^11],
[(a+b*x^2)*(c+d*x^2)^3/(e+f*x^2),x,5,1/105*(7*a*d*f*(15*d^2*e^2-40*c*d*e*f+33*c^2*f^2)-b*(105*d^3*e^3-280*c*d^2*e^2*f+231*c^2*d*e*f^2-48*c^3*f^3))*x/f^4-1/105*(7*a*d*f*(5*d*e-9*c*f)-b*(35*d^2*e^2-63*c*d*e*f+24*c^2*f^2))*x*(c+d*x^2)/f^3-1/35*(7*b*d*e-6*b*c*f-7*a*d*f)*x*(c+d*x^2)^2/f^2+1/7*b*x*(c+d*x^2)^3/f+(b*e-a*f)*(d*e-c*f)^3*arctan(x*sqrt(f)/sqrt(e))/(f^(9/2)*sqrt(e))],
[(a+b*x^2)*(c+d*x^2)^3/(e+f*x^2)^2,x,5,-1/30*d*(5*a*f*(15*d^2*e^2-22*c*d*e*f+3*c^2*f^2)-b*e*(105*d^2*e^2-190*c*d*e*f+81*c^2*f^2))*x/(e*f^4)-1/30*d*(b*e*(35*d*e-33*c*f)-5*a*f*(5*d*e-3*c*f))*x*(c+d*x^2)/(e*f^3)+1/10*d*(7*b*e-5*a*f)*x*(c+d*x^2)^2/(e*f^2)-1/2*(b*e-a*f)*x*(c+d*x^2)^3/(e*f*(e+f*x^2))-1/2*(d*e-c*f)^2*(b*e*(7*d*e-c*f)-a*f*(5*d*e+c*f))*arctan(x*sqrt(f)/sqrt(e))/(e^(3/2)*f^(9/2))],
[(a+b*x^2)*(c+d*x^2)^3/(e+f*x^2)^3,x,5,1/24*d*(3*a*f*(15*d^2*e^2-4*c*d*e*f-3*c^2*f^2)-b*e*(105*d^2*e^2-100*c*d*e*f+3*c^2*f^2))*x/(e^2*f^4)+1/24*d*(b*e*(35*d*e-3*c*f)-3*a*f*(5*d*e+3*c*f))*x*(c+d*x^2)/(e^2*f^3)-1/4*(b*e-a*f)*x*(c+d*x^2)^3/(e*f*(e+f*x^2)^2)-1/8*(b*e*(7*d*e-c*f)-3*a*f*(d*e+c*f))*x*(c+d*x^2)^2/(e^2*f^2*(e+f*x^2))+1/8*(d*e-c*f)*(b*e*(35*d^2*e^2-10*c*d*e*f-c^2*f^2)-3*a*f*(5*d^2*e^2+2*c*d*e*f+c^2*f^2))*arctan(x*sqrt(f)/sqrt(e))/(e^(5/2)*f^(9/2))],
[(a+b*x^2)*(c+d*x^2)^3/(e+f*x^2)^4,x,5,1/48*d*(b*e*(105*d^2*e^2-10*c*d*e*f-3*c^2*f^2)-a*f*(15*d^2*e^2+14*c*d*e*f+15*c^2*f^2))*x/(e^3*f^4)-1/6*(b*e-a*f)*x*(c+d*x^2)^3/(e*f*(e+f*x^2)^3)-1/24*(b*e*(7*d*e-c*f)-a*f*(d*e+5*c*f))*x*(c+d*x^2)^2/(e^2*f^2*(e+f*x^2)^2)-1/48*(b*e*(35*d^2*e^2-8*c*d*e*f-3*c^2*f^2)-a*f*(5*d^2*e^2+4*c*d*e*f+15*c^2*f^2))*x*(c+d*x^2)/(e^3*f^3*(e+f*x^2))-1/16*(b*e*(35*d^3*e^3-15*c*d^2*e^2*f-3*c^2*d*e*f^2-c^3*f^3)-a*f*(5*d^3*e^3+3*c*d^2*e^2*f+3*c^2*d*e*f^2+5*c^3*f^3))*arctan(x*sqrt(f)/sqrt(e))/(e^(7/2)*f^(9/2))],

# q<0

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

# p>0
[(a+b*x^2)*(c+d*x^2)^(3/2)*(e+f*x^2)^(1/2),x,7,-1/105*(7*a*d*f*(2*d^2*e^2-7*c*d*e*f-3*c^2*f^2)-b*(8*d^3*e^3-19*c*d^2*e^2*f+9*c^2*d*e*f^2-6*c^3*f^3))*x*sqrt(c+d*x^2)/(d^2*f^2*sqrt(e+f*x^2))-1/105*e^(3/2)*(7*a*d*f*(d*e-9*c*f)-b*(4*d^2*e^2-9*c*d*e*f-3*c^2*f^2))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(d*f^(5/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/105*(7*a*d*f*(2*d^2*e^2-7*c*d*e*f-3*c^2*f^2)-b*(8*d^3*e^3-19*c*d^2*e^2*f+9*c^2*d*e*f^2-6*c^3*f^3))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(d^2*f^(5/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/35*(b*d*e-2*b*c*f+7*a*d*f)*x*(c+d*x^2)^(3/2)*sqrt(e+f*x^2)/(d*f)+1/7*b*x*(c+d*x^2)^(5/2)*sqrt(e+f*x^2)/d+1/105*(7*a*d*f*(d*e+3*c*f)-b*(4*d^2*e^2-6*c*d*e*f+6*c^2*f^2))*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/(d*f^2)],
[(a+b*x^2)*(c+d*x^2)^(1/2)*(e+f*x^2)^(1/2),x,6,1/15*(5*a*d*f*(d*e+c*f)-2*b*(d^2*e^2-c*d*e*f+c^2*f^2))*x*sqrt(c+d*x^2)/(d^2*f*sqrt(e+f*x^2))-1/15*e^(3/2)*(b*d*e+b*c*f-10*a*d*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(d*f^(3/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/15*(5*a*d*f*(d*e+c*f)-2*b*(d^2*e^2-c*d*e*f+c^2*f^2))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(d^2*f^(3/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/5*b*x*(c+d*x^2)^(3/2)*sqrt(e+f*x^2)/d+1/15*(b*d*e-2*b*c*f+5*a*d*f)*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/(d*f)],
[(a+b*x^2)*(e+f*x^2)^(1/2)/(c+d*x^2)^(1/2),x,5,1/3*(b*d*e-2*b*c*f+3*a*d*f)*x*sqrt(c+d*x^2)/(d^2*sqrt(e+f*x^2))-1/3*(b*c-3*a*d)*e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(c*d*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/3*(b*d*e-2*b*c*f+3*a*d*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(d^2*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/3*b*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/d],
[(a+b*x^2)*(e+f*x^2)^(1/2)/(c+d*x^2)^(3/2),x,5,(2*b*c-a*d)*f*x*sqrt(c+d*x^2)/(c*d^2*sqrt(e+f*x^2))+b*e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(c*d*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-(2*b*c-a*d)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(f)*sqrt(c+d*x^2)/(c*d^2*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-(b*c-a*d)*x*sqrt(e+f*x^2)/(c*d*sqrt(c+d*x^2))],
[(a+b*x^2)*(e+f*x^2)^(1/2)/(c+d*x^2)^(5/2),x,4,1/3*(b*c-a*d)*e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(f)*sqrt(c+d*x^2)/(c^2*d*(d*e-c*f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/3*(b*c-a*d)*x*sqrt(e+f*x^2)/(c*d*(c+d*x^2)^(3/2))+1/3*(d*(b*c+2*a*d)*e-c*(2*b*c+a*d)*f)*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-c*f/(d*e)))*sqrt(e+f*x^2)/(c^(3/2)*d^(3/2)*(d*e-c*f)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],
[(a+b*x^2)*(e+f*x^2)^(1/2)/(c+d*x^2)^(7/2),x,5,-1/15*e^(3/2)*(2*a*d*(2*d*e-3*c*f)+b*c*(d*e+c*f))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(f)*sqrt(c+d*x^2)/(c^3*d*(d*e-c*f)^2*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/5*(b*c-a*d)*x*sqrt(e+f*x^2)/(c*d*(c+d*x^2)^(5/2))+1/15*(a*d*(4*d*e-3*c*f)+b*c*(d*e-2*c*f))*x*sqrt(e+f*x^2)/(c^2*d*(d*e-c*f)*(c+d*x^2)^(3/2))+1/15*(2*b*c*(d^2*e^2-c*d*e*f+c^2*f^2)+a*d*(8*d^2*e^2-13*c*d*e*f+3*c^2*f^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-c*f/(d*e)))*sqrt(e+f*x^2)/(c^(5/2)*d^(3/2)*(d*e-c*f)^2*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],
[(a+b*x^2)*(c+d*x^2)^(1/2)*(e+f*x^2)^(3/2),x,7,1/7*b*x*(c+d*x^2)^(3/2)*(e+f*x^2)^(3/2)/d+1/105*(7*a*d*f*(3*d^2*e^2+7*c*d*e*f-2*c^2*f^2)-b*(6*d^3*e^3-9*c*d^2*e^2*f+19*c^2*d*e*f^2-8*c^3*f^3))*x*sqrt(c+d*x^2)/(d^3*f*sqrt(e+f*x^2))+1/105*e^(3/2)*(7*a*d*f*(9*d*e-c*f)-b*(3*d^2*e^2+9*c*d*e*f-4*c^2*f^2))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(d^2*f^(3/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/105*(7*a*d*f*(3*d^2*e^2+7*c*d*e*f-2*c^2*f^2)-b*(6*d^3*e^3-9*c*d^2*e^2*f+19*c^2*d*e*f^2-8*c^3*f^3))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(d^3*f^(3/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/35*(3*b*d*e-4*b*c*f+7*a*d*f)*x*(c+d*x^2)^(3/2)*sqrt(e+f*x^2)/d^2+1/105*(14*a*d*f*(3*d*e-c*f)+b*(3*d^2*e^2-15*c*d*e*f+8*c^2*f^2))*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/(d^2*f)],
[(a+b*x^2)*(e+f*x^2)^(3/2)/(c+d*x^2)^(1/2),x,6,1/5*b*x*(e+f*x^2)^(3/2)*sqrt(c+d*x^2)/d+1/15*(10*a*d*f*(2*d*e-c*f)+b*(3*d^2*e^2-13*c*d*e*f+8*c^2*f^2))*x*sqrt(c+d*x^2)/(d^3*sqrt(e+f*x^2))+1/15*e^(3/2)*(5*a*d*(3*d*e-c*f)-b*(6*c*d*e-4*c^2*f))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(c*d^2*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/15*(10*a*d*f*(2*d*e-c*f)+b*(3*d^2*e^2-13*c*d*e*f+8*c^2*f^2))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(d^3*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/15*(3*b*d*e-4*b*c*f+5*a*d*f)*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/d^2],
[(a+b*x^2)*(e+f*x^2)^(3/2)/(c+d*x^2)^(3/2),x,6,-(b*c-a*d)*x*(e+f*x^2)^(3/2)/(c*d*sqrt(c+d*x^2))+1/3*f*(b*c*(7*d*e-8*c*f)-3*a*d*(d*e-2*c*f))*x*sqrt(c+d*x^2)/(c*d^3*sqrt(e+f*x^2))+1/3*e^(3/2)*(3*b*d*e-4*b*c*f+3*a*d*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(c*d^2*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/3*(b*c*(7*d*e-8*c*f)-3*a*d*(d*e-2*c*f))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(f)*sqrt(c+d*x^2)/(c*d^3*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/3*(4*b*c-3*a*d)*f*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/(c*d^2)],
[(a+b*x^2)*(e+f*x^2)^(3/2)/(c+d*x^2)^(5/2),x,6,-1/3*(b*c-a*d)*x*(e+f*x^2)^(3/2)/(c*d*(c+d*x^2)^(3/2))-1/3*f*(b*c*(d*e-8*c*f)+2*a*d*(d*e+c*f))*x*sqrt(c+d*x^2)/(c^2*d^3*sqrt(e+f*x^2))+1/3*(4*b*c-a*d)*e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(f)*sqrt(c+d*x^2)/(c^2*d^2*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/3*(b*c*(d*e-8*c*f)+2*a*d*(d*e+c*f))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(f)*sqrt(c+d*x^2)/(c^2*d^3*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/3*(b*c*(d*e-4*c*f)+a*d*(2*d*e+c*f))*x*sqrt(e+f*x^2)/(c^2*d^2*sqrt(c+d*x^2))],
[(a+b*x^2)*(e+f*x^2)^(3/2)/(c+d*x^2)^(7/2),x,5,-1/5*(b*c-a*d)*x*(e+f*x^2)^(3/2)/(c*d*(c+d*x^2)^(5/2))-1/15*e^(3/2)*(b*c*(d*e-4*c*f)+a*d*(4*d*e-c*f))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(f)*sqrt(c+d*x^2)/(c^3*d^2*(d*e-c*f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/15*(d*(b*c+4*a*d)*e-c*(4*b*c+a*d)*f)*x*sqrt(e+f*x^2)/(c^2*d^2*(c+d*x^2)^(3/2))+1/15*(b*c*(2*d^2*e^2+3*c*d*e*f-8*c^2*f^2)+a*d*(8*d^2*e^2-3*c*d*e*f-2*c^2*f^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-c*f/(d*e)))*sqrt(e+f*x^2)/(c^(5/2)*d^(5/2)*(d*e-c*f)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],
[(a+b*x^2)*(e+f*x^2)^(3/2)/(c+d*x^2)^(9/2),x,6,-1/7*(b*c-a*d)*x*(e+f*x^2)^(3/2)/(c*d*(c+d*x^2)^(7/2))-1/105*e^(3/2)*(3*a*d*(8*d^2*e^2-11*c*d*e*f+c^2*f^2)+2*b*c*(2*d^2*e^2-c*d*e*f+2*c^2*f^2))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(f)*sqrt(c+d*x^2)/(c^4*d^2*(d*e-c*f)^2*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/35*(d*(b*c+6*a*d)*e-c*(4*b*c+3*a*d)*f)*x*sqrt(e+f*x^2)/(c^2*d^2*(c+d*x^2)^(5/2))+1/105*(b*c*(4*d^2*e^2+c*d*e*f-8*c^2*f^2)+3*a*d*(8*d^2*e^2-5*c*d*e*f-2*c^2*f^2))*x*sqrt(e+f*x^2)/(c^3*d^2*(d*e-c*f)*(c+d*x^2)^(3/2))+1/105*(6*a*d*(8*d^3*e^3-12*c*d^2*e^2*f+2*c^2*d*e*f^2+c^3*f^3)+b*c*(8*d^3*e^3-5*c*d^2*e^2*f-5*c^2*d*e*f^2+8*c^3*f^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-c*f/(d*e)))*sqrt(e+f*x^2)/(c^(7/2)*d^(5/2)*(d*e-c*f)^2*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],

# q<0
[(a+b*x^2)*(c+d*x^2)^(5/2)/(e+f*x^2)^(1/2),x,7,1/105*(7*a*d*f*(8*d^2*e^2-23*c*d*e*f+23*c^2*f^2)-b*(48*d^3*e^3-128*c*d^2*e^2*f+103*c^2*d*e*f^2-15*c^3*f^3))*x*sqrt(c+d*x^2)/(d*f^3*sqrt(e+f*x^2))-1/105*(7*a*d*f*(8*d^2*e^2-23*c*d*e*f+23*c^2*f^2)-b*(48*d^3*e^3-128*c*d^2*e^2*f+103*c^2*d*e*f^2-15*c^3*f^3))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(d*f^(7/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/105*(7*a*f*(4*d^2*e^2-11*c*d*e*f+15*c^2*f^2)-b*e*(24*d^2*e^2-61*c*d*e*f+45*c^2*f^2))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(f^(7/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/35*(6*b*d*e-5*b*c*f-7*a*d*f)*x*(c+d*x^2)^(3/2)*sqrt(e+f*x^2)/f^2+1/7*b*x*(c+d*x^2)^(5/2)*sqrt(e+f*x^2)/f-1/105*(28*a*d*f*(d*e-2*c*f)-b*(24*d^2*e^2-43*c*d*e*f+15*c^2*f^2))*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/f^3],
[(a+b*x^2)*(c+d*x^2)^(3/2)/(e+f*x^2)^(1/2),x,6,-1/15*(10*a*d*f*(d*e-2*c*f)-b*(8*d^2*e^2-13*c*d*e*f+3*c^2*f^2))*x*sqrt(c+d*x^2)/(d*f^2*sqrt(e+f*x^2))+1/15*(10*a*d*f*(d*e-2*c*f)-b*(8*d^2*e^2-13*c*d*e*f+3*c^2*f^2))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(d*f^(5/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/15*(5*a*f*(d*e-3*c*f)-b*(4*d*e^2-6*c*e*f))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(f^(5/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/5*b*x*(c+d*x^2)^(3/2)*sqrt(e+f*x^2)/f-1/15*(4*b*d*e-3*b*c*f-5*a*d*f)*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/f^2],
[(a+b*x^2)*(c+d*x^2)^(1/2)/(e+f*x^2)^(1/2),x,5,-1/3*(2*b*d*e-b*c*f-3*a*d*f)*x*sqrt(c+d*x^2)/(d*f*sqrt(e+f*x^2))+1/3*(2*b*d*e-b*c*f-3*a*d*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(d*f^(3/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/3*(b*e-3*a*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(f^(3/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/3*b*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/f],
[(a+b*x^2)/((c+d*x^2)^(1/2)*(e+f*x^2)^(1/2)),x,4,b*x*sqrt(c+d*x^2)/(d*sqrt(e+f*x^2))-b*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(d*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+a*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(c*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))],
[(a+b*x^2)/((c+d*x^2)^(3/2)*(e+f*x^2)^(1/2)),x,3,(b*e-a*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(c*(d*e-c*f)*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*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-c*f/(d*e)))*sqrt(e+f*x^2)/((d*e-c*f)*sqrt(c)*sqrt(d)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],
[(a+b*x^2)/((c+d*x^2)^(5/2)*(e+f*x^2)^(1/2)),x,4,-1/3*(2*b*c*e+a*d*e-3*a*c*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(f)*sqrt(c+d*x^2)/(c^2*(d*e-c*f)^2*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/3*(b*c-a*d)*x*sqrt(e+f*x^2)/(c*(d*e-c*f)*(c+d*x^2)^(3/2))+1/3*(2*a*d*(d*e-2*c*f)+b*c*(d*e+c*f))*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-c*f/(d*e)))*sqrt(e+f*x^2)/(c^(3/2)*(d*e-c*f)^2*sqrt(d)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],
[(a+b*x^2)/((c+d*x^2)^(7/2)*(e+f*x^2)^(1/2)),x,5,-1/15*(b*c*e*(d*e-9*c*f)+a*(4*d^2*e^2-11*c*d*e*f+15*c^2*f^2))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(f)*sqrt(c+d*x^2)/(c^3*(d*e-c*f)^3*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/5*(b*c-a*d)*x*sqrt(e+f*x^2)/(c*(d*e-c*f)*(c+d*x^2)^(5/2))+1/15*(4*a*d*(d*e-2*c*f)+b*c*(d*e+3*c*f))*x*sqrt(e+f*x^2)/(c^2*(d*e-c*f)^2*(c+d*x^2)^(3/2))+1/15*(b*c*(2*d^2*e^2-7*c*d*e*f-3*c^2*f^2)+a*d*(8*d^2*e^2-23*c*d*e*f+23*c^2*f^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-c*f/(d*e)))*sqrt(e+f*x^2)/(c^(5/2)*(d*e-c*f)^3*sqrt(d)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],
[(a+b*x^2)*(c+d*x^2)^(5/2)/(e+f*x^2)^(3/2),x,7,-(b*e-a*f)*x*(c+d*x^2)^(5/2)/(e*f*sqrt(e+f*x^2))-1/15*(5*a*f*(8*d^2*e^2-13*c*d*e*f+3*c^2*f^2)-2*b*e*(24*d^2*e^2-44*c*d*e*f+19*c^2*f^2))*x*sqrt(c+d*x^2)/(e*f^3*sqrt(e+f*x^2))+1/15*(5*a*f*(8*d^2*e^2-13*c*d*e*f+3*c^2*f^2)-2*b*e*(24*d^2*e^2-44*c*d*e*f+19*c^2*f^2))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(f^(7/2)*sqrt(e)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/15*(10*a*d*f*(2*d*e-3*c*f)-b*(24*d^2*e^2-41*c*d*e*f+15*c^2*f^2))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(f^(7/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/5*d*(6*b*e-5*a*f)*x*(c+d*x^2)^(3/2)*sqrt(e+f*x^2)/(e*f^2)-1/15*d*(b*e*(24*d*e-23*c*f)-5*a*f*(4*d*e-3*c*f))*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/(e*f^3)],
[(a+b*x^2)*(c+d*x^2)^(3/2)/(e+f*x^2)^(3/2),x,6,-(b*e-a*f)*x*(c+d*x^2)^(3/2)/(e*f*sqrt(e+f*x^2))-1/3*(b*e*(8*d*e-7*c*f)-3*a*f*(2*d*e-c*f))*x*sqrt(c+d*x^2)/(e*f^2*sqrt(e+f*x^2))+1/3*(b*e*(8*d*e-7*c*f)-3*a*f*(2*d*e-c*f))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(f^(5/2)*sqrt(e)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/3*(4*b*d*e-3*b*c*f-3*a*d*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(f^(5/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/3*d*(4*b*e-3*a*f)*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/(e*f^2)],
[(a+b*x^2)*(c+d*x^2)^(1/2)/(e+f*x^2)^(3/2),x,5,-(b*e-a*f)*x*sqrt(c+d*x^2)/(e*f*sqrt(e+f*x^2))+(2*b*e-a*f)*x*sqrt(c+d*x^2)/(e*f*sqrt(e+f*x^2))-(2*b*e-a*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(f^(3/2)*sqrt(e)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+b*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(f^(3/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))],
[(a+b*x^2)/((c+d*x^2)^(1/2)*(e+f*x^2)^(3/2)),x,3,(b*e-a*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/((d*e-c*f)*sqrt(e)*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-(b*c-a*d)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(c*(d*e-c*f)*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))],
[(a+b*x^2)/((c+d*x^2)^(3/2)*(e+f*x^2)^(3/2)),x,4,-(b*c-a*d)*x/(c*(d*e-c*f)*sqrt(c+d*x^2)*sqrt(e+f*x^2))+(b*d*e+b*c*f-2*a*d*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(c*(d*e-c*f)^2*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-(2*b*c*e-a*d*e-a*c*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(f)*sqrt(c+d*x^2)/(c*(d*e-c*f)^2*sqrt(e)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))],
[(a+b*x^2)/((c+d*x^2)^(5/2)*(e+f*x^2)^(3/2)),x,5,-1/3*(b*c-a*d)*x/(c*(d*e-c*f)*(c+d*x^2)^(3/2)*sqrt(e+f*x^2))+1/3*(2*a*d*(d*e-3*c*f)+b*c*(d*e+3*c*f))*x/(c^2*(d*e-c*f)^2*sqrt(c+d*x^2)*sqrt(e+f*x^2))+1/3*(b*c*e*(d*e+7*c*f)+a*(2*d^2*e^2-7*c*d*e*f-3*c^2*f^2))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(f)*sqrt(c+d*x^2)/(c^2*(d*e-c*f)^3*sqrt(e)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/3*(a*d*(d*e-9*c*f)+b*c*(5*d*e+3*c*f))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(f)*sqrt(c+d*x^2)/(c^2*(d*e-c*f)^3*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))],
[(e+f*x^2)/((c+d*x^2)^(3/2)*sqrt(a+b*x^2)),x,3,-(d*e-c*f)*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)/((b*c-a*d)*sqrt(c)*sqrt(d)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+(b*e-a*f)*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))],
[(e+f*x^2)/((c+d*x^2)^(3/2)*sqrt(a-b*x^2)),x,8,(d*e-c*f)*x*sqrt(a-b*x^2)/(c*(b*c+a*d)*sqrt(c+d*x^2))+(d*e-c*f)*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)/(c*d*(b*c+a*d)*sqrt(a-b*x^2)*sqrt(1+d*x^2/c))+f*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))],
[(e+f*x^2)/((c-d*x^2)^(3/2)*sqrt(a+b*x^2)),x,8,(d*e+c*f)*x*sqrt(a+b*x^2)/(c*(b*c+a*d)*sqrt(c-d*x^2))-(d*e+c*f)*EllipticE(x*sqrt(d)/sqrt(c),sqrt(-b*c/(a*d)))*sqrt(a+b*x^2)*sqrt(1-d*x^2/c)/((b*c+a*d)*sqrt(c)*sqrt(d)*sqrt(1+b*x^2/a)*sqrt(c-d*x^2))+e*EllipticF(x*sqrt(d)/sqrt(c),sqrt(-b*c/(a*d)))*sqrt(1+b*x^2/a)*sqrt(1-d*x^2/c)/(sqrt(c)*sqrt(d)*sqrt(a+b*x^2)*sqrt(c-d*x^2))],
[(e+f*x^2)/((c-d*x^2)^(3/2)*sqrt(a-b*x^2)),x,8,-(d*e+c*f)*x*sqrt(a-b*x^2)/(c*(b*c-a*d)*sqrt(c-d*x^2))+(d*e+c*f)*EllipticE(x*sqrt(d)/sqrt(c),sqrt(b*c/(a*d)))*sqrt(a-b*x^2)*sqrt(1-d*x^2/c)/((b*c-a*d)*sqrt(c)*sqrt(d)*sqrt(1-b*x^2/a)*sqrt(c-d*x^2))+e*EllipticF(x*sqrt(d)/sqrt(c),sqrt(b*c/(a*d)))*sqrt(1-b*x^2/a)*sqrt(1-d*x^2/c)/(sqrt(c)*sqrt(d)*sqrt(a-b*x^2)*sqrt(c-d*x^2))],
[(a+b*x^2)/(sqrt(2+d*x^2)*sqrt(3+f*x^2)),x,4,b*x*sqrt(2+d*x^2)/(d*sqrt(3+f*x^2))+a*sqrt(cos(arctan(x*sqrt(f)/sqrt(3)))^2)/cos(arctan(x*sqrt(f)/sqrt(3)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(3))),sqrt(1-3/2*d/f))*sqrt(2+d*x^2)/(sqrt(2)*sqrt(f)*sqrt((2+d*x^2)/(3+f*x^2))*sqrt(3+f*x^2))-b*sqrt(cos(arctan(x*sqrt(f)/sqrt(3)))^2)/cos(arctan(x*sqrt(f)/sqrt(3)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(3))),sqrt(1-3/2*d/f))*sqrt(2)*sqrt(2+d*x^2)/(d*sqrt(f)*sqrt((2+d*x^2)/(3+f*x^2))*sqrt(3+f*x^2))],
[(a+b*x^2)*sqrt(2+d*x^2)/sqrt(3+f*x^2),x,5,-1/3*(6*b*d-2*b*f-3*a*d*f)*x*sqrt(2+d*x^2)/(d*f*sqrt(3+f*x^2))+1/3*(6*b*d-2*b*f-3*a*d*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(3)))^2)/cos(arctan(x*sqrt(f)/sqrt(3)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(3))),sqrt(1-3/2*d/f))*sqrt(2)*sqrt(2+d*x^2)/(d*f^(3/2)*sqrt((2+d*x^2)/(3+f*x^2))*sqrt(3+f*x^2))-(b-a*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(3)))^2)/cos(arctan(x*sqrt(f)/sqrt(3)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(3))),sqrt(1-3/2*d/f))*sqrt(2)*sqrt(2+d*x^2)/(f^(3/2)*sqrt((2+d*x^2)/(3+f*x^2))*sqrt(3+f*x^2))+1/3*b*x*sqrt(2+d*x^2)*sqrt(3+f*x^2)/f],
[(a+b*x^2)*sqrt(2+d*x^2)*sqrt(3+f*x^2),x,6,1/15*(5*a*d*f*(3*d+2*f)-2*b*(9*d^2-6*d*f+4*f^2))*x*sqrt(2+d*x^2)/(d^2*f*sqrt(3+f*x^2))-1/15*(5*a*d*f*(3*d+2*f)-2*b*(9*d^2-6*d*f+4*f^2))*sqrt(cos(arctan(x*sqrt(f)/sqrt(3)))^2)/cos(arctan(x*sqrt(f)/sqrt(3)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(3))),sqrt(1-3/2*d/f))*sqrt(2)*sqrt(2+d*x^2)/(d^2*f^(3/2)*sqrt((2+d*x^2)/(3+f*x^2))*sqrt(3+f*x^2))-1/5*(3*b*d+2*b*f-10*a*d*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(3)))^2)/cos(arctan(x*sqrt(f)/sqrt(3)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(3))),sqrt(1-3/2*d/f))*sqrt(2)*sqrt(2+d*x^2)/(d*f^(3/2)*sqrt((2+d*x^2)/(3+f*x^2))*sqrt(3+f*x^2))+1/5*b*x*(2+d*x^2)^(3/2)*sqrt(3+f*x^2)/d+1/15*(3*b*d-4*b*f+5*a*d*f)*x*sqrt(2+d*x^2)*sqrt(3+f*x^2)/(d*f)],
[(-b+2*c*x^2-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,2,-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(b-sqrt(b^2-4*a*c))/(sqrt(2)*sqrt(c))],
[(b+2*c*x^2-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,5,x*(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(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))))*(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))))*(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))))],

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

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

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

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

# q<0
[(a+b*x^2)^3/((c+d*x^2)*sqrt(e+f*x^2)),x,14,1/2*b*(b*c-a*d)*(b*e-2*a*f)*arctanh(x*sqrt(f)/sqrt(e+f*x^2))/(d^2*f^(3/2))+1/8*b*(3*b^2*e^2-8*a*b*e*f+8*a^2*f^2)*arctanh(x*sqrt(f)/sqrt(e+f*x^2))/(d*f^(5/2))+b*(b*c-a*d)^2*arctanh(x*sqrt(f)/sqrt(e+f*x^2))/(d^3*sqrt(f))-(b*c-a*d)^3*arctan(x*sqrt(d*e-c*f)/(sqrt(c)*sqrt(e+f*x^2)))/(d^3*sqrt(c)*sqrt(d*e-c*f))-1/2*b^2*(b*c-a*d)*x*sqrt(e+f*x^2)/(d^2*f)-3/8*b^2*(b*e-2*a*f)*x*sqrt(e+f*x^2)/(d*f^2)+1/4*b^2*x*(a+b*x^2)*sqrt(e+f*x^2)/(d*f)],
[(a+b*x^2)^2/((c+d*x^2)*sqrt(e+f*x^2)),x,9,-1/2*b*(b*e-2*a*f)*arctanh(x*sqrt(f)/sqrt(e+f*x^2))/(d*f^(3/2))-b*(b*c-a*d)*arctanh(x*sqrt(f)/sqrt(e+f*x^2))/(d^2*sqrt(f))+(b*c-a*d)^2*arctan(x*sqrt(d*e-c*f)/(sqrt(c)*sqrt(e+f*x^2)))/(d^2*sqrt(c)*sqrt(d*e-c*f))+1/2*b^2*x*sqrt(e+f*x^2)/(d*f)],
[(a+b*x^2)/((c+d*x^2)*sqrt(e+f*x^2)),x,5,b*arctanh(x*sqrt(f)/sqrt(e+f*x^2))/(d*sqrt(f))-(b*c-a*d)*arctan(x*sqrt(d*e-c*f)/(sqrt(c)*sqrt(e+f*x^2)))/(d*sqrt(c)*sqrt(d*e-c*f))],
[1/((c+d*x^2)*sqrt(e+f*x^2)),x,2,arctan(x*sqrt(d*e-c*f)/(sqrt(c)*sqrt(e+f*x^2)))/(sqrt(c)*sqrt(d*e-c*f))],
[1/((a+b*x^2)*(c+d*x^2)*sqrt(e+f*x^2)),x,5,b*arctan(x*sqrt(b*e-a*f)/(sqrt(a)*sqrt(e+f*x^2)))/((b*c-a*d)*sqrt(a)*sqrt(b*e-a*f))-d*arctan(x*sqrt(d*e-c*f)/(sqrt(c)*sqrt(e+f*x^2)))/((b*c-a*d)*sqrt(c)*sqrt(d*e-c*f))],
[1/((a+b*x^2)^2*(c+d*x^2)*sqrt(e+f*x^2)),x,7,1/2*b*(b^2*c*e-3*a*b*d*e-2*a*b*c*f+4*a^2*d*f)*arctan(x*sqrt(b*e-a*f)/(sqrt(a)*sqrt(e+f*x^2)))/(a^(3/2)*(b*c-a*d)^2*(b*e-a*f)^(3/2))+d^2*arctan(x*sqrt(d*e-c*f)/(sqrt(c)*sqrt(e+f*x^2)))/((b*c-a*d)^2*sqrt(c)*sqrt(d*e-c*f))+1/2*b^2*x*sqrt(e+f*x^2)/(a*(b*c-a*d)*(b*e-a*f)*(a+b*x^2))],

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

# q>0
[(c+d*x^2)^(5/2)*(e+f*x^2)^(1/2)/(a+b*x^2),x,14,1/5*d^2*x*(e+f*x^2)^(3/2)*sqrt(c+d*x^2)/(b*f)+1/15*d*(7*c*e-2*d*e^2/f+3*c^2*f/d)*x*sqrt(c+d*x^2)/(b*sqrt(e+f*x^2))+1/3*(b*c-a*d)*(b*d*e+4*b*c*f-3*a*d*f)*x*sqrt(c+d*x^2)/(b^3*sqrt(e+f*x^2))+1/15*d*e^(3/2)*(-40*a*b*c*d*f+15*a^2*d^2*f+b^2*c*(-d*e+34*c*f))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(b^3*c*f^(3/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/15*(15*a^2*d^2*f^2-5*a*b*d*f*(d*e+7*c*f)+b^2*(-2*d^2*e^2+12*c*d*e*f+23*c^2*f^2))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(b^3*f^(3/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+(b*c-a*d)^3*e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticPi(sin(arctan(x*sqrt(f)/sqrt(e))),1-b*e/(a*f),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(a*b^3*c*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/3*d*(b*c-a*d)*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/b^2-2/15*d*(d*e-3*c*f)*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/(b*f),1/5*d^2*x*(e+f*x^2)^(3/2)*sqrt(c+d*x^2)/(b*f)+1/15*d*(7*c*e-2*d*e^2/f+3*c^2*f/d)*x*sqrt(c+d*x^2)/(b*sqrt(e+f*x^2))+1/3*(b*c-a*d)*(b*d*e+4*b*c*f-3*a*d*f)*x*sqrt(c+d*x^2)/(b^3*sqrt(e+f*x^2))-1/15*d*e^(3/2)*(d*e-9*c*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(b*f^(3/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/15*(2*d^2*e^2-7*c*d*e*f-3*c^2*f^2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(b*f^(3/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/3*d*(5*b*c-3*a*d)*(b*c-a*d)*e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(b^3*c*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+(b*c-a*d)^3*e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticPi(sin(arctan(x*sqrt(f)/sqrt(e))),1-b*e/(a*f),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(a*b^3*c*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/3*(b*c-a*d)*(b*d*e+4*b*c*f-3*a*d*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(b^3*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/3*d*(b*c-a*d)*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/b^2-2/15*d*(d*e-3*c*f)*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/(b*f)],
[(c+d*x^2)^(3/2)*(e+f*x^2)^(1/2)/(a+b*x^2),x,7,1/3*(b*d*e+4*b*c*f-3*a*d*f)*x*sqrt(c+d*x^2)/(b^2*sqrt(e+f*x^2))+1/3*d*(5*b*c-3*a*d)*e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(b^2*c*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+(b*c-a*d)^2*e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticPi(sin(arctan(x*sqrt(f)/sqrt(e))),1-b*e/(a*f),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(a*b^2*c*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/3*(b*d*e+4*b*c*f-3*a*d*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(b^2*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/3*d*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/b],
[(c+d*x^2)^(1/2)*(e+f*x^2)^(1/2)/(a+b*x^2),x,6,f*x*sqrt(c+d*x^2)/(b*sqrt(e+f*x^2))+d*e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(b*c*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+(b*c-a*d)*e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticPi(sin(arctan(x*sqrt(f)/sqrt(e))),1-b*e/(a*f),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(a*b*c*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(f)*sqrt(c+d*x^2)/(b*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))],
[(e+f*x^2)^(1/2)/((a+b*x^2)*(c+d*x^2)^(1/2)),x,1,e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticPi(sin(arctan(x*sqrt(f)/sqrt(e))),1-b*e/(a*f),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(a*c*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))],
[(e+f*x^2)^(1/2)/((a+b*x^2)*(c+d*x^2)^(3/2)),x,3,b*e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticPi(sin(arctan(x*sqrt(f)/sqrt(e))),1-b*e/(a*f),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(a*c*(b*c-a*d)*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*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-c*f/(d*e)))*sqrt(d)*sqrt(e+f*x^2)/((b*c-a*d)*sqrt(c)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],
[(e+f*x^2)^(1/2)/((a+b*x^2)*(c+d*x^2)^(5/2)),x,6,b^2*e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticPi(sin(arctan(x*sqrt(f)/sqrt(e))),1-b*e/(a*f),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(a*c*(b*c-a*d)^2*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/3*d*e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(f)*sqrt(c+d*x^2)/(c^2*(b*c-a*d)*(d*e-c*f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/3*d*x*sqrt(e+f*x^2)/(c*(b*c-a*d)*(c+d*x^2)^(3/2))-1/3*(b*c*(5*d*e-4*c*f)-a*d*(2*d*e-c*f))*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-c*f/(d*e)))*sqrt(d)*sqrt(e+f*x^2)/(c^(3/2)*(b*c-a*d)^2*(d*e-c*f)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],
[(e+f*x^2)^(1/2)/((a+b*x^2)*(c+d*x^2)^(7/2)),x,9,b^3*e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticPi(sin(arctan(x*sqrt(f)/sqrt(e))),1-b*e/(a*f),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(a*c*(b*c-a*d)^3*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/15*d*e^(3/2)*(b*c*(9*d*e-11*c*f)-2*a*d*(2*d*e-3*c*f))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(f)*sqrt(c+d*x^2)/(c^3*(b*c-a*d)^2*(d*e-c*f)^2*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/5*d*x*sqrt(e+f*x^2)/(c*(b*c-a*d)*(c+d*x^2)^(5/2))-1/15*d*(b*c*(9*d*e-8*c*f)-a*d*(4*d*e-3*c*f))*x*sqrt(e+f*x^2)/(c^2*(b*c-a*d)^2*(d*e-c*f)*(c+d*x^2)^(3/2))+1/15*(a*d*(8*d^2*e^2-13*c*d*e*f+3*c^2*f^2)-2*b*c*(9*d^2*e^2-14*c*d*e*f+4*c^2*f^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-c*f/(d*e)))*sqrt(d)*sqrt(e+f*x^2)/(c^(5/2)*(b*c-a*d)^2*(d*e-c*f)^2*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))-b^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-c*f/(d*e)))*sqrt(d)*sqrt(e+f*x^2)/((b*c-a*d)^3*sqrt(c)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],

#  {(e + f*x^2)^(3/2)/(a + b*x^2)*(c + d*x^2)^(5/2), x, 22, ((b*c - a*d)^2*f*(4*b*d*e + b*c*f - 3*a*d*f)*x*Sqrt[c + d*x^2])/(3*b^4*d*Sqrt[e + f*x^2]) + ((b*c - a*d)*(3*d^2*e^2 + 7*c*d*e*f - 2*c^2*f^2)*x*Sqrt[c + d*x^2])/(15*b^2*d*Sqrt[e + f*x^2]) - (2*(d*e + c*f)*(d^2*e^2 - 6*c*d*e*f + c^2*f^2)*x*Sqrt[c + d*x^2])/(35*b*d*f*Sqrt[e + f*x^2]) + ((b*c - a*d)^2*f*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*b^3) + (2*(b*c - a*d)*(3*d*e - c*f)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(15*b^2) + (d*(9*c*e + (d*e^2)/f - (2*c^2*f)/d)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(35*b) + ((b*c - a*d)*f*x*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(5*b^2) + (2*(4*d*e - c*f)*x*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(35*b) + (f*x*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])/(7*b) - ((b*c - a*d)^2*Sqrt[e]*Sqrt[f]*(4*b*d*e + b*c*f - 3*a*d*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b^4*d*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - ((b*c - a*d)*Sqrt[e]*(3*d^2*e^2 + 7*c*d*e*f - 2*c^2*f^2)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*b^2*d*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (2*Sqrt[e]*(d*e + c*f)*(d^2*e^2 - 6*c*d*e*f + c^2*f^2)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(35*b*d*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + ((b*c - a*d)^2*Sqrt[e]*Sqrt[f]*(5*b*e - 3*a*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*b^4*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + ((b*c - a*d)*e^(3/2)*(9*d*e - c*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*b^2*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (e^(3/2)*(d^2*e^2 - 18*c*d*e*f + c^2*f^2)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(35*b*f^(3/2)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (c^(3/2)*(b*c - a*d)^2*(b*e - a*f)^2*Sqrt[e + f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(a*b^4*Sqrt[d]*e*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])} 
[(c+d*x^2)^(3/2)*(e+f*x^2)^(3/2)/(a+b*x^2),x,14,(b*c-a*d)^2*f^2*x*sqrt(c+d*x^2)/(b^3*d*sqrt(e+f*x^2))+2/3*(b*c-a*d)*f*(2*d*e-c*f)*x*sqrt(c+d*x^2)/(b^2*d*sqrt(e+f*x^2))+1/15*(3*d^2*e^2+7*c*d*e*f-2*c^2*f^2)*x*sqrt(c+d*x^2)/(b*d*sqrt(e+f*x^2))+1/15*e^(3/2)*(15*a^2*d^2*f+3*b^2*c*(8*d*e+3*c*f)-5*a*b*d*(3*d*e+5*c*f))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(b^3*c*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+(b*c-a*d)^2*e^(3/2)*(b*e-a*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticPi(sin(arctan(x*sqrt(f)/sqrt(e))),1-b*e/(a*f),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(a*b^3*c*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/15*(15*a^2*d^2*f^2-20*a*b*d*f*(d*e+c*f)+3*b^2*(d^2*e^2+9*c*d*e*f+c^2*f^2))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(b^3*d*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/5*f*x*(c+d*x^2)^(3/2)*sqrt(e+f*x^2)/b+1/3*(b*c-a*d)*f*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/b^2+2/15*(3*d*e-c*f)*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/b,1/3*(b*c-a*d)*f*(4*b*d*e+b*c*f-3*a*d*f)*x*sqrt(c+d*x^2)/(b^3*d*sqrt(e+f*x^2))+1/15*(3*d^2*e^2+7*c*d*e*f-2*c^2*f^2)*x*sqrt(c+d*x^2)/(b*d*sqrt(e+f*x^2))+1/15*e^(3/2)*(9*d*e-c*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(b*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/15*(3*d^2*e^2+7*c*d*e*f-2*c^2*f^2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(b*d*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/3*(b*c-a*d)*(4*b*d*e+b*c*f-3*a*d*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(f)*sqrt(c+d*x^2)/(b^3*d*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/3*(b*c-a*d)*(5*b*e-3*a*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(f)*sqrt(c+d*x^2)/(b^3*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/5*f*x*(c+d*x^2)^(3/2)*sqrt(e+f*x^2)/b+1/3*(b*c-a*d)*f*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/b^2+2/15*(3*d*e-c*f)*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/b+c^(3/2)*(b*c-a*d)*(b*e-a*f)^2*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticPi(sin(arctan(x*sqrt(d)/sqrt(c))),1-b*c/(a*d),sqrt(1-c*f/(d*e)))*sqrt(e+f*x^2)/(a*b^3*e*sqrt(d)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],
[(c+d*x^2)^(1/2)*(e+f*x^2)^(3/2)/(a+b*x^2),x,7,1/3*f*(4*b*d*e+b*c*f-3*a*d*f)*x*sqrt(c+d*x^2)/(b^2*d*sqrt(e+f*x^2))-1/3*(4*b*d*e+b*c*f-3*a*d*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(f)*sqrt(c+d*x^2)/(b^2*d*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/3*(5*b*e-3*a*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(f)*sqrt(c+d*x^2)/(b^2*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/3*f*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/b+c^(3/2)*(b*e-a*f)^2*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticPi(sin(arctan(x*sqrt(d)/sqrt(c))),1-b*c/(a*d),sqrt(1-c*f/(d*e)))*sqrt(e+f*x^2)/(a*b^2*e*sqrt(d)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],
[(e+f*x^2)^(3/2)/((a+b*x^2)*(c+d*x^2)^(1/2)),x,6,f^2*x*sqrt(c+d*x^2)/(b*d*sqrt(e+f*x^2))-f^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(b*d*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+e^(3/2)*(b*e-a*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticPi(sin(arctan(x*sqrt(f)/sqrt(e))),1-b*e/(a*f),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(a*b*c*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(f)*sqrt(c+d*x^2)/(b*c*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))],
[(e+f*x^2)^(3/2)/((a+b*x^2)*(c+d*x^2)^(3/2)),x,3,e^(3/2)*(b*e-a*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticPi(sin(arctan(x*sqrt(f)/sqrt(e))),1-b*e/(a*f),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(a*c*(b*c-a*d)*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-(d*e-c*f)*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-c*f/(d*e)))*sqrt(e+f*x^2)/((b*c-a*d)*sqrt(c)*sqrt(d)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],
[(e+f*x^2)^(3/2)/((a+b*x^2)*(c+d*x^2)^(5/2)),x,6,b*e^(3/2)*(b*e-a*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticPi(sin(arctan(x*sqrt(f)/sqrt(e))),1-b*e/(a*f),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(a*c*(b*c-a*d)^2*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/3*e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(f)*sqrt(c+d*x^2)/(c^2*(b*c-a*d)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/3*(d*e-c*f)*x*sqrt(e+f*x^2)/(c*(b*c-a*d)*(c+d*x^2)^(3/2))-1/3*(b*c*(5*d*e-c*f)-2*a*d*(d*e+c*f))*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-c*f/(d*e)))*sqrt(e+f*x^2)/(c^(3/2)*(b*c-a*d)^2*sqrt(d)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],
[(e+f*x^2)^(3/2)/((a+b*x^2)*(c+d*x^2)^(7/2)),x,9,b^2*e^(3/2)*(b*e-a*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticPi(sin(arctan(x*sqrt(f)/sqrt(e))),1-b*e/(a*f),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(a*c*(b*c-a*d)^3*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/15*e^(3/2)*(3*b*c*(3*d*e-2*c*f)-a*d*(4*d*e-c*f))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(f)*sqrt(c+d*x^2)/(c^3*(b*c-a*d)^2*(d*e-c*f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/5*(d*e-c*f)*x*sqrt(e+f*x^2)/(c*(b*c-a*d)*(c+d*x^2)^(5/2))-1/15*(3*b*c*(3*d*e-c*f)-2*a*d*(2*d*e+c*f))*x*sqrt(e+f*x^2)/(c^2*(b*c-a*d)^2*(c+d*x^2)^(3/2))+1/15*(a*d*(8*d^2*e^2-3*c*d*e*f-2*c^2*f^2)-3*b*c*(6*d^2*e^2-6*c*d*e*f+c^2*f^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-c*f/(d*e)))*sqrt(e+f*x^2)/(c^(5/2)*(b*c-a*d)^2*(d*e-c*f)*sqrt(d)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))-b*(b*e-a*f)*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-c*f/(d*e)))*sqrt(d)*sqrt(e+f*x^2)/((b*c-a*d)^3*sqrt(c)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],

# q<0
[(c+d*x^2)^(5/2)/((a+b*x^2)*(e+f*x^2)^(1/2)),x,12,d*(b*c-a*d)*x*sqrt(c+d*x^2)/(b^2*sqrt(e+f*x^2))-2/3*d*(d*e-2*c*f)*x*sqrt(c+d*x^2)/(b*f*sqrt(e+f*x^2))+2/3*d*(d*e-2*c*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(b*f^(3/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/3*d*(d*e-3*c*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(b*f^(3/2)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-d*(b*c-a*d)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(b^2*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+d*(b*c-a*d)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(b^2*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/3*d^2*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/(b*f)+c^(3/2)*(b*c-a*d)^2*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticPi(sin(arctan(x*sqrt(d)/sqrt(c))),1-b*c/(a*d),sqrt(1-c*f/(d*e)))*sqrt(e+f*x^2)/(a*b^2*e*sqrt(d)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],
[(c+d*x^2)^(3/2)/((a+b*x^2)*(e+f*x^2)^(1/2)),x,6,d*x*sqrt(c+d*x^2)/(b*sqrt(e+f*x^2))-d*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(b*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+d*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(b*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+c^(3/2)*(b*c-a*d)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticPi(sin(arctan(x*sqrt(d)/sqrt(c))),1-b*c/(a*d),sqrt(1-c*f/(d*e)))*sqrt(e+f*x^2)/(a*b*e*sqrt(d)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],
[(c+d*x^2)^(1/2)/((a+b*x^2)*(e+f*x^2)^(1/2)),x,1,c^(3/2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticPi(sin(arctan(x*sqrt(d)/sqrt(c))),1-b*c/(a*d),sqrt(1-c*f/(d*e)))*sqrt(e+f*x^2)/(a*e*sqrt(d)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],
[1/((a+b*x^2)*(c+d*x^2)^(1/2)*(e+f*x^2)^(1/2)),x,3,EllipticPi(x*sqrt(d)/sqrt(-c),b*c/(a*d),sqrt(c*f/(d*e)))*sqrt(-c)*sqrt(1+d*x^2/c)*sqrt(1+f*x^2/e)/(a*sqrt(d)*sqrt(c+d*x^2)*sqrt(e+f*x^2))],
[1/((a+b*x^2)*(c+d*x^2)^(3/2)*(e+f*x^2)^(1/2)),x,5,-d*(b*d*e-2*b*c*f+a*d*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(c*(b*c-a*d)^2*(d*e-c*f)*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-d^(3/2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticE(sin(arctan(x*sqrt(d)/sqrt(c))),sqrt(1-c*f/(d*e)))*sqrt(e+f*x^2)/((b*c-a*d)*(d*e-c*f)*sqrt(c)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))+b^2*c^(3/2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticPi(sin(arctan(x*sqrt(d)/sqrt(c))),1-b*c/(a*d),sqrt(1-c*f/(d*e)))*sqrt(e+f*x^2)/(a*(b*c-a*d)^2*e*sqrt(d)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],
[1/((a+b*x^2)*(c+d*x^2)^(5/2)*(e+f*x^2)^(1/2)),x,8,-1/3*d*(a*d*(d*e-3*c*f)-2*b*c*(2*d*e-3*c*f))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(f)*sqrt(c+d*x^2)/(c^2*(b*c-a*d)^2*(d*e-c*f)^2*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/3*d^2*x*sqrt(e+f*x^2)/(c*(b*c-a*d)*(d*e-c*f)*(c+d*x^2)^(3/2))-1/3*d^(3/2)*(b*c*(5*d*e-7*c*f)-2*a*d*(d*e-2*c*f))*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-c*f/(d*e)))*sqrt(e+f*x^2)/(c^(3/2)*(b*c-a*d)^2*(d*e-c*f)^2*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))+b^2*EllipticPi(x*sqrt(d)/sqrt(-c),b*c/(a*d),sqrt(c*f/(d*e)))*sqrt(-c)*sqrt(1+d*x^2/c)*sqrt(1+f*x^2/e)/(a*(b*c-a*d)^2*sqrt(d)*sqrt(c+d*x^2)*sqrt(e+f*x^2))],
[(c+d*x^2)^(5/2)/((a+b*x^2)*(e+f*x^2)^(3/2)),x,14,(d*e-c*f)*x*(c+d*x^2)^(3/2)/(e*(b*e-a*f)*sqrt(e+f*x^2))+1/3*(b*c-a*d)*(b*d*e+4*b*c*f-3*a*d*f)*x*sqrt(c+d*x^2)/(b*(b*e-a*f)^2*sqrt(e+f*x^2))+1/3*(b*e*(6*d^2*e^2-7*c*d*e*f-c^2*f^2)-a*f*(8*d^2*e^2-13*c*d*e*f+3*c^2*f^2))*x*sqrt(c+d*x^2)/(e*f*(b*e-a*f)^2*sqrt(e+f*x^2))-1/3*(b*e*(6*d^2*e^2-7*c*d*e*f-c^2*f^2)-a*f*(8*d^2*e^2-13*c*d*e*f+3*c^2*f^2))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(f^(3/2)*(b*e-a*f)^2*sqrt(e)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/3*(2*a*d*f*(2*d*e-3*c*f)-b*(3*d^2*e^2-2*c*d*e*f-3*c^2*f^2))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(f^(3/2)*(b*e-a*f)^2*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/3*d*(5*b*c-3*a*d)*(b*c-a*d)*e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(b*c*(b*e-a*f)^2*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+(b*c-a*d)^3*e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticPi(sin(arctan(x*sqrt(f)/sqrt(e))),1-b*e/(a*f),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(a*b*c*(b*e-a*f)^2*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/3*(b*c-a*d)*(b*d*e+4*b*c*f-3*a*d*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(b*(b*e-a*f)^2*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/3*d*(b*c-a*d)*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/(b*e-a*f)^2+1/3*d*(a*f*(4*d*e-3*c*f)-b*e*(3*d*e-2*c*f))*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/(e*f*(b*e-a*f)^2)],
[(c+d*x^2)^(3/2)/((a+b*x^2)*(e+f*x^2)^(3/2)),x,3,(d*e-c*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/((b*e-a*f)*sqrt(e)*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+c^(3/2)*(b*c-a*d)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticPi(sin(arctan(x*sqrt(d)/sqrt(c))),1-b*c/(a*d),sqrt(1-c*f/(d*e)))*sqrt(e+f*x^2)/(a*e*(b*e-a*f)*sqrt(d)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],
[(c+d*x^2)^(1/2)/((a+b*x^2)*(e+f*x^2)^(3/2)),x,3,-sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(f)*sqrt(c+d*x^2)/((b*e-a*f)*sqrt(e)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+b*c^(3/2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticPi(sin(arctan(x*sqrt(d)/sqrt(c))),1-b*c/(a*d),sqrt(1-c*f/(d*e)))*sqrt(e+f*x^2)/(a*e*(b*e-a*f)*sqrt(d)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],
[1/((a+b*x^2)*(c+d*x^2)^(1/2)*(e+f*x^2)^(3/2)),x,5,f^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/((b*e-a*f)*(d*e-c*f)*sqrt(e)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+b^2*e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticPi(sin(arctan(x*sqrt(f)/sqrt(e))),1-b*e/(a*f),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(a*c*(b*e-a*f)^2*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-(2*b*d*e-b*c*f-a*d*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(f)*sqrt(c+d*x^2)/(c*(b*e-a*f)^2*(d*e-c*f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))],
[1/((a+b*x^2)*(c+d*x^2)^(3/2)*(e+f*x^2)^(3/2)),x,8,-d^2*x/(c*(b*c-a*d)*(d*e-c*f)*sqrt(c+d*x^2)*sqrt(e+f*x^2))-d^2*(b*d*e-3*b*c*f+2*a*d*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(c+d*x^2)/(c*(b*c-a*d)^2*(d*e-c*f)^2*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-b^2*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(f)*sqrt(c+d*x^2)/((b*c-a*d)^2*(b*e-a*f)*sqrt(e)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-d*(2*b*c^2*f-a*d*(d*e+c*f))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(f)*sqrt(c+d*x^2)/(c*(b*c-a*d)^2*(d*e-c*f)^2*sqrt(e)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+b^3*c^(3/2)*sqrt(cos(arctan(x*sqrt(d)/sqrt(c)))^2)/cos(arctan(x*sqrt(d)/sqrt(c)))*EllipticPi(sin(arctan(x*sqrt(d)/sqrt(c))),1-b*c/(a*d),sqrt(1-c*f/(d*e)))*sqrt(e+f*x^2)/(a*(b*c-a*d)^2*e*(b*e-a*f)*sqrt(d)*sqrt(c+d*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2))))],
[1/((a+b*x^2)*(c+d*x^2)^(5/2)*(e+f*x^2)^(3/2)),x,11,-1/3*d^2*x/(c*(b*c-a*d)*(d*e-c*f)*(c+d*x^2)^(3/2)*sqrt(e+f*x^2))-1/3*d^2*(b*c*(5*d*e-9*c*f)-2*a*d*(d*e-3*c*f))*x/(c^2*(b*c-a*d)^2*(d*e-c*f)^2*sqrt(c+d*x^2)*sqrt(e+f*x^2))+b^2*f^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/((b*c-a*d)^2*(b*e-a*f)*(d*e-c*f)*sqrt(e)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+b^4*e^(3/2)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticPi(sin(arctan(x*sqrt(f)/sqrt(e))),1-b*e/(a*f),sqrt(1-d*e/(c*f)))*sqrt(c+d*x^2)/(a*c*(b*c-a*d)^2*(b*e-a*f)^2*sqrt(f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/3*d*(b*c*(5*d^2*e^2-7*c*d*e*f-6*c^2*f^2)-a*d*(2*d^2*e^2-7*c*d*e*f-3*c^2*f^2))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(f)*sqrt(c+d*x^2)/(c^2*(b*c-a*d)^2*(d*e-c*f)^3*sqrt(e)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-b^2*(2*b*d*e-b*c*f-a*d*f)*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(f)*sqrt(c+d*x^2)/(c*(b*c-a*d)^2*(b*e-a*f)^2*(d*e-c*f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/3*d^2*(b*c*(7*d*e-15*c*f)-a*d*(d*e-9*c*f))*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(f)*sqrt(c+d*x^2)/(c^2*(b*c-a*d)^2*(d*e-c*f)^3*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))],
[(1+x^2)^(3/2)*sqrt(2+x^2)/(a+b*x^2),x,7,-(a-2*b)*x*sqrt(2+x^2)/(b^2*sqrt(1+x^2))+1/3*x*sqrt(1+x^2)*sqrt(2+x^2)/b-1/3*(3*a-7*b)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt(2+x^2)/(b^2*sqrt(2)*sqrt(1+x^2)*sqrt((2+x^2)/(1+x^2)))+(a-2*b)*(a-b)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),1-b/a,sqrt(1/2))*sqrt(2+x^2)/(a*b^2*sqrt(2)*sqrt(1+x^2)*sqrt((2+x^2)/(1+x^2)))+(a-2*b)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt(2+x^2)/(b^2*sqrt(1+x^2)*sqrt((2+x^2)/(1+x^2))),2*(a-b)^2*sqrt(cos(arctan(x/sqrt(2)))^2)/cos(arctan(x/sqrt(2)))*EllipticPi(sin(arctan(x/sqrt(2))),1-2*b/a,I)*sqrt(1+x^2)/(a*b^2*sqrt((1+x^2)/(2+x^2))*sqrt(2+x^2))-(a-2*b)*x*sqrt(2+x^2)/(b^2*sqrt(1+x^2))+1/3*x*sqrt(1+x^2)*sqrt(2+x^2)/b+(a-2*b)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt(2+x^2)/(b^2*sqrt(1+x^2)*sqrt((2+x^2)/(1+x^2)))-1/3*(3*a-5*b)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt(2+x^2)/(b^2*sqrt(1+x^2)*sqrt((2+x^2)/(1+x^2)))],
[(1+x^2)^(1/2)*sqrt(2+x^2)/(a+b*x^2),x,6,x*sqrt(2+x^2)/(b*sqrt(1+x^2))+sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt(2+x^2)/(b*sqrt(2)*sqrt(1+x^2)*sqrt((2+x^2)/(1+x^2)))-(a-2*b)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticPi(sin(arctan(x)),1-b/a,sqrt(1/2))*sqrt(2+x^2)/(a*b*sqrt(2)*sqrt(1+x^2)*sqrt((2+x^2)/(1+x^2)))-sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt(2+x^2)/(b*sqrt(1+x^2)*sqrt((2+x^2)/(1+x^2)))],
[sqrt(2+x^2)/((1+x^2)^(1/2)*(a+b*x^2)),x,1,2*sqrt(cos(arctan(x/sqrt(2)))^2)/cos(arctan(x/sqrt(2)))*EllipticPi(sin(arctan(x/sqrt(2))),1-2*b/a,I)*sqrt(1+x^2)/(a*sqrt((1+x^2)/(2+x^2))*sqrt(2+x^2))],
[sqrt(2+x^2)/((1+x^2)^(3/2)*(a+b*x^2)),x,3,-2*b*sqrt(cos(arctan(x/sqrt(2)))^2)/cos(arctan(x/sqrt(2)))*EllipticPi(sin(arctan(x/sqrt(2))),1-2*b/a,I)*sqrt(1+x^2)/(a*(a-b)*sqrt((1+x^2)/(2+x^2))*sqrt(2+x^2))+sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt(2+x^2)/((a-b)*sqrt(1+x^2)*sqrt((2+x^2)/(1+x^2)))],
[sqrt(2+x^2)/((1+x^2)^(5/2)*(a+b*x^2)),x,6,2*b^2*sqrt(cos(arctan(x/sqrt(2)))^2)/cos(arctan(x/sqrt(2)))*EllipticPi(sin(arctan(x/sqrt(2))),1-2*b/a,I)*sqrt(1+x^2)/(a*(a-b)^2*sqrt((1+x^2)/(2+x^2))*sqrt(2+x^2))+1/3*x*sqrt(2+x^2)/((a-b)*(1+x^2)^(3/2))+(a-2*b)*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticE(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt(2+x^2)/((a-b)^2*sqrt(1+x^2)*sqrt((2+x^2)/(1+x^2)))-1/3*sqrt(cos(arctan(x))^2)/cos(arctan(x))*EllipticF(sin(arctan(x)),sqrt(1/2))*sqrt(2)*sqrt(2+x^2)/((a-b)*sqrt(1+x^2)*sqrt((2+x^2)/(1+x^2)))],
[sqrt(2+d*x^2)*sqrt(3+f*x^2)/(a+b*x^2),x,6,f*x*sqrt(2+d*x^2)/(b*sqrt(3+f*x^2))+3*d*sqrt(cos(arctan(x*sqrt(f)/sqrt(3)))^2)/cos(arctan(x*sqrt(f)/sqrt(3)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(3))),sqrt(1-3/2*d/f))*sqrt(2+d*x^2)/(b*sqrt(2)*sqrt(f)*sqrt((2+d*x^2)/(3+f*x^2))*sqrt(3+f*x^2))+3*(2*b-a*d)*sqrt(cos(arctan(x*sqrt(f)/sqrt(3)))^2)/cos(arctan(x*sqrt(f)/sqrt(3)))*EllipticPi(sin(arctan(x*sqrt(f)/sqrt(3))),1-3*b/(a*f),sqrt(1-3/2*d/f))*sqrt(2+d*x^2)/(a*b*sqrt(2)*sqrt(f)*sqrt((2+d*x^2)/(3+f*x^2))*sqrt(3+f*x^2))-sqrt(cos(arctan(x*sqrt(f)/sqrt(3)))^2)/cos(arctan(x*sqrt(f)/sqrt(3)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(3))),sqrt(1-3/2*d/f))*sqrt(2)*sqrt(f)*sqrt(2+d*x^2)/(b*sqrt((2+d*x^2)/(3+f*x^2))*sqrt(3+f*x^2))],
[sqrt(2+d*x^2)/((a+b*x^2)*sqrt(3+f*x^2)),x,1,2*sqrt(cos(arctan(x*sqrt(d)/sqrt(2)))^2)/cos(arctan(x*sqrt(d)/sqrt(2)))*EllipticPi(sin(arctan(x*sqrt(d)/sqrt(2))),1-2*b/(a*d),sqrt(1-2/3*f/d))*sqrt(3+f*x^2)/(a*sqrt(3)*sqrt(d)*sqrt(2+d*x^2)*sqrt((3+f*x^2)/(2+d*x^2)))],
[1/((a+b*x^2)*sqrt(2+d*x^2)*sqrt(3+f*x^2)),x,1,EllipticPi(x*sqrt(-d)/sqrt(2),2*b/(a*d),sqrt(2/3*f/d))/(a*sqrt(3)*sqrt(-d))],
[sqrt(1-x^2)/((-1+x^2)*sqrt(a+b*x^2)),x,3,-EllipticF(x,sqrt(-b/a))*sqrt(1+b*x^2/a)/sqrt(a+b*x^2)],

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

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

# q>0

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

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

# q>0
[sqrt(c-d*x^2)*sqrt(e+f*x^2)/(a+b*x^2)^2,x,11,1/2*x*sqrt(c-d*x^2)*sqrt(e+f*x^2)/(a*(a+b*x^2))+1/2*EllipticE(x*sqrt(d)/sqrt(c),sqrt(-c*f/(d*e)))*sqrt(c)*sqrt(d)*sqrt(1-d*x^2/c)*sqrt(e+f*x^2)/(a*b*sqrt(c-d*x^2)*sqrt(1+f*x^2/e))+1/2*(b^2*c*e+a^2*d*f)*EllipticPi(x*sqrt(d)/sqrt(c),-b*c/(a*d),sqrt(-c*f/(d*e)))*sqrt(c)*sqrt(1-d*x^2/c)*sqrt(1+f*x^2/e)/(a^2*b^2*sqrt(d)*sqrt(c-d*x^2)*sqrt(e+f*x^2))-1/2*(b*e+a*f)*EllipticF(x*sqrt(d)/sqrt(c),sqrt(-c*f/(d*e)))*sqrt(c)*sqrt(d)*sqrt(1-d*x^2/c)*sqrt(1+f*x^2/e)/(a*b^2*sqrt(c-d*x^2)*sqrt(e+f*x^2))],
[sqrt(c+d*x^2)*sqrt(e+f*x^2)/(a+b*x^2)^2,x,8,-1/2*f*x*sqrt(c+d*x^2)/(a*b*sqrt(e+f*x^2))+1/2*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(f)*sqrt(c+d*x^2)/(a*b*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/2*d*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(f)*sqrt(c+d*x^2)/(b^2*c*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/2*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/(a*(a+b*x^2))+1/2*(b^2*c*e-a^2*d*f)*EllipticPi(x*sqrt(d)/sqrt(-c),b*c/(a*d),sqrt(c*f/(d*e)))*sqrt(-c)*sqrt(1+d*x^2/c)*sqrt(1+f*x^2/e)/(a^2*b^2*sqrt(d)*sqrt(c+d*x^2)*sqrt(e+f*x^2))],

# q<0
[1/((a+b*x^2)^2*sqrt(c-d*x^2)*sqrt(e+f*x^2)),x,11,1/2*b^2*x*sqrt(c-d*x^2)*sqrt(e+f*x^2)/(a*(b*c+a*d)*(b*e-a*f)*(a+b*x^2))+1/2*b*EllipticE(x*sqrt(d)/sqrt(c),sqrt(-c*f/(d*e)))*sqrt(c)*sqrt(d)*sqrt(1-d*x^2/c)*sqrt(e+f*x^2)/(a*(b*c+a*d)*(b*e-a*f)*sqrt(c-d*x^2)*sqrt(1+f*x^2/e))+1/2*(b^2*c*e-3*a^2*d*f+a*b*(2*d*e-2*c*f))*EllipticPi(x*sqrt(d)/sqrt(c),-b*c/(a*d),sqrt(-c*f/(d*e)))*sqrt(c)*sqrt(1-d*x^2/c)*sqrt(1+f*x^2/e)/(a^2*(b*c+a*d)*(b*e-a*f)*sqrt(d)*sqrt(c-d*x^2)*sqrt(e+f*x^2))-1/2*EllipticF(x*sqrt(d)/sqrt(c),sqrt(-c*f/(d*e)))*sqrt(c)*sqrt(d)*sqrt(1-d*x^2/c)*sqrt(1+f*x^2/e)/(a*(b*c+a*d)*sqrt(c-d*x^2)*sqrt(e+f*x^2))],
[1/((a+b*x^2)^2*sqrt(c+d*x^2)*sqrt(e+f*x^2)),x,8,-1/2*b*f*x*sqrt(c+d*x^2)/(a*(b*c-a*d)*(b*e-a*f)*sqrt(e+f*x^2))+1/2*b*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticE(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(f)*sqrt(c+d*x^2)/(a*(b*c-a*d)*(b*e-a*f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))-1/2*d*sqrt(cos(arctan(x*sqrt(f)/sqrt(e)))^2)/cos(arctan(x*sqrt(f)/sqrt(e)))*EllipticF(sin(arctan(x*sqrt(f)/sqrt(e))),sqrt(1-d*e/(c*f)))*sqrt(e)*sqrt(f)*sqrt(c+d*x^2)/(c*(b*c-a*d)*(b*e-a*f)*sqrt(e*(c+d*x^2)/(c*(e+f*x^2)))*sqrt(e+f*x^2))+1/2*b^2*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/(a*(b*c-a*d)*(b*e-a*f)*(a+b*x^2))+1/2*(b^2*c*e+3*a^2*d*f-2*a*b*(d*e+c*f))*EllipticPi(x*sqrt(d)/sqrt(-c),b*c/(a*d),sqrt(c*f/(d*e)))*sqrt(-c)*sqrt(1+d*x^2/c)*sqrt(1+f*x^2/e)/(a^2*(b*c-a*d)*(b*e-a*f)*sqrt(d)*sqrt(c+d*x^2)*sqrt(e+f*x^2))],

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

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

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

# q>0

#  Not sure this is not integrable. 
[(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/sqrt(e+f*x^2),x,0,Unintegrable((a+b*x^2)^(3/2)*sqrt(c+d*x^2)/sqrt(e+f*x^2),x)],
[(a+b*x^2)^(1/2)*sqrt(c+d*x^2)/sqrt(e+f*x^2),x,7,1/2*d*x*sqrt(a+b*x^2)*sqrt(e+f*x^2)/(f*sqrt(c+d*x^2))+1/2*b*(d*e-c*f)*EllipticF(x*sqrt(b*e-a*f)/(sqrt(e)*sqrt(a+b*x^2)),sqrt((b*c-a*d)*e/(c*(b*e-a*f))))*sqrt(e)*sqrt(c+d*x^2)*sqrt(a*(e+f*x^2)/(e*(a+b*x^2)))/(d*f*sqrt(b*e-a*f)*sqrt(a*(c+d*x^2)/(c*(a+b*x^2)))*sqrt(e+f*x^2))-1/2*c*(b*d*e-b*c*f-a*d*f)*EllipticPi(x*sqrt(d*e-c*f)/(sqrt(e)*sqrt(c+d*x^2)),d*e/(d*e-c*f),sqrt(-(b*c-a*d)*e/(a*(d*e-c*f))))*sqrt(e)*sqrt(a+b*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2)))/(a*d*f*sqrt(d*e-c*f)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e+f*x^2))-1/2*EllipticE(x*sqrt(d*e-c*f)/(sqrt(e)*sqrt(c+d*x^2)),sqrt(-(b*c-a*d)*e/(a*(d*e-c*f))))*sqrt(e)*sqrt(d*e-c*f)*sqrt(a+b*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2)))/(f*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e+f*x^2))],
[sqrt(c+d*x^2)/((a+b*x^2)^(1/2)*sqrt(e+f*x^2)),x,2,c*EllipticPi(x*sqrt(d*e-c*f)/(sqrt(e)*sqrt(c+d*x^2)),d*e/(d*e-c*f),sqrt(-(b*c-a*d)*e/(a*(d*e-c*f))))*sqrt(e)*sqrt(a+b*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2)))/(a*sqrt(d*e-c*f)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e+f*x^2))],
[sqrt(c+d*x^2)/((a+b*x^2)^(3/2)*sqrt(e+f*x^2)),x,2,EllipticE(x*sqrt(b*e-a*f)/(sqrt(e)*sqrt(a+b*x^2)),sqrt((b*c-a*d)*e/(c*(b*e-a*f))))*sqrt(e)*sqrt(c+d*x^2)*sqrt(a*(e+f*x^2)/(e*(a+b*x^2)))/(a*sqrt(b*e-a*f)*sqrt(a*(c+d*x^2)/(c*(a+b*x^2)))*sqrt(e+f*x^2))],

#  Not sure this is not integrable. 
[(a+b*x^2)^(3/2)*sqrt(c+d*x^2)/(e+f*x^2)^(3/2),x,0,Unintegrable((a+b*x^2)^(3/2)*sqrt(c+d*x^2)/(e+f*x^2)^(3/2),x)],
[(a+b*x^2)^(1/2)*sqrt(c+d*x^2)/(e+f*x^2)^(3/2),x,8,-c^(3/2)*(b*e-a*f)*sqrt(cos(arctan(x*sqrt(d*e-c*f)/(sqrt(c)*sqrt(e+f*x^2))))^2)/cos(arctan(x*sqrt(d*e-c*f)/(sqrt(c)*sqrt(e+f*x^2))))*EllipticF(sin(arctan(x*sqrt(d*e-c*f)/(sqrt(c)*sqrt(e+f*x^2)))),sqrt(-(b*c-a*d)*e/(a*(d*e-c*f))))*sqrt(a+b*x^2)/(a*e*f*sqrt(d*e-c*f)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+sqrt(cos(arctan(x*sqrt(d*e-c*f)/(sqrt(c)*sqrt(e+f*x^2))))^2)/cos(arctan(x*sqrt(d*e-c*f)/(sqrt(c)*sqrt(e+f*x^2))))*EllipticE(sin(arctan(x*sqrt(d*e-c*f)/(sqrt(c)*sqrt(e+f*x^2)))),sqrt(-(b*c-a*d)*e/(a*(d*e-c*f))))*sqrt(c)*sqrt(d*e-c*f)*sqrt(a+b*x^2)/(e*f*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-(d*e-c*f)*x*sqrt(a+b*x^2)/(e*f*sqrt(c+d*x^2)*sqrt(e+f*x^2))+b*c*EllipticPi(x*sqrt(d*e-c*f)/(sqrt(e)*sqrt(c+d*x^2)),d*e/(d*e-c*f),sqrt(-(b*c-a*d)*e/(a*(d*e-c*f))))*sqrt(e)*sqrt(a+b*x^2)*sqrt(c*(e+f*x^2)/(e*(c+d*x^2)))/(a*f*sqrt(d*e-c*f)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(e+f*x^2))],
[sqrt(c+d*x^2)/((a+b*x^2)^(1/2)*(e+f*x^2)^(3/2)),x,5,c^(3/2)*sqrt(cos(arctan(x*sqrt(d*e-c*f)/(sqrt(c)*sqrt(e+f*x^2))))^2)/cos(arctan(x*sqrt(d*e-c*f)/(sqrt(c)*sqrt(e+f*x^2))))*EllipticF(sin(arctan(x*sqrt(d*e-c*f)/(sqrt(c)*sqrt(e+f*x^2)))),sqrt(-(b*c-a*d)*e/(a*(d*e-c*f))))*sqrt(a+b*x^2)/(a*e*sqrt(d*e-c*f)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))-sqrt(cos(arctan(x*sqrt(d*e-c*f)/(sqrt(c)*sqrt(e+f*x^2))))^2)/cos(arctan(x*sqrt(d*e-c*f)/(sqrt(c)*sqrt(e+f*x^2))))*EllipticE(sin(arctan(x*sqrt(d*e-c*f)/(sqrt(c)*sqrt(e+f*x^2)))),sqrt(-(b*c-a*d)*e/(a*(d*e-c*f))))*sqrt(c)*sqrt(d*e-c*f)*sqrt(a+b*x^2)/(e*(b*e-a*f)*sqrt(c*(a+b*x^2)/(a*(c+d*x^2)))*sqrt(c+d*x^2))+(d*e-c*f)*x*sqrt(a+b*x^2)/(e*(b*e-a*f)*sqrt(c+d*x^2)*sqrt(e+f*x^2))],
[sqrt(c+d*x^2)/((a+b*x^2)^(3/2)*(e+f*x^2)^(3/2)),x,0,Unintegrable(sqrt(c+d*x^2)/((a+b*x^2)^(3/2)*(e+f*x^2)^(3/2)),x)],
[sqrt(c+d*x^2)*sqrt(e+f*x^2)/sqrt(a+b*x^2),x,7,1/2*x*sqrt(c+d*x^2)*sqrt(e+f*x^2)/sqrt(a+b*x^2)-1/2*EllipticE(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)),sqrt(c*(b*e-a*f)/((b*c-a*d)*e)))*sqrt(c)*sqrt(b*c-a*d)*sqrt(a*(c+d*x^2)/(c*(a+b*x^2)))*sqrt(e+f*x^2)/(b*sqrt(c+d*x^2)*sqrt(a*(e+f*x^2)/(e*(a+b*x^2))))-1/2*a*(a*d*f-b*(d*e+c*f))*EllipticPi(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)),b*c/(b*c-a*d),sqrt(c*(b*e-a*f)/((b*c-a*d)*e)))*sqrt(c+d*x^2)*sqrt(a*(e+f*x^2)/(e*(a+b*x^2)))/(b^2*sqrt(c)*sqrt(b*c-a*d)*sqrt(a*(c+d*x^2)/(c*(a+b*x^2)))*sqrt(e+f*x^2))+1/2*(b*c-a*d)*(2*b*e-a*f)*EllipticF(x*sqrt(b*e-a*f)/(sqrt(e)*sqrt(a+b*x^2)),sqrt((b*c-a*d)*e/(c*(b*e-a*f))))*sqrt(e)*sqrt(c+d*x^2)*sqrt(a*(e+f*x^2)/(e*(a+b*x^2)))/(b^2*c*sqrt(b*e-a*f)*sqrt(a*(c+d*x^2)/(c*(a+b*x^2)))*sqrt(e+f*x^2))],

# q<0

#  Not sure this is not integrable. 
[(a+b*x^2)^(3/2)/(sqrt(c+d*x^2)*sqrt(e+f*x^2)),x,0,Unintegrable((a+b*x^2)^(3/2)/(sqrt(c+d*x^2)*sqrt(e+f*x^2)),x)],
[(a+b*x^2)^(1/2)/(sqrt(c+d*x^2)*sqrt(e+f*x^2)),x,2,a*EllipticPi(x*sqrt(b*c-a*d)/(sqrt(c)*sqrt(a+b*x^2)),b*c/(b*c-a*d),sqrt(c*(b*e-a*f)/((b*c-a*d)*e)))*sqrt(c+d*x^2)*sqrt(a*(e+f*x^2)/(e*(a+b*x^2)))/(sqrt(c)*sqrt(b*c-a*d)*sqrt(a*(c+d*x^2)/(c*(a+b*x^2)))*sqrt(e+f*x^2))],
[1/((a+b*x^2)^(1/2)*sqrt(c+d*x^2)*sqrt(e+f*x^2)),x,2,EllipticF(x*sqrt(b*e-a*f)/(sqrt(e)*sqrt(a+b*x^2)),sqrt((b*c-a*d)*e/(c*(b*e-a*f))))*sqrt(e)*sqrt(c+d*x^2)*sqrt(a*(e+f*x^2)/(e*(a+b*x^2)))/(c*sqrt(b*e-a*f)*sqrt(a*(c+d*x^2)/(c*(a+b*x^2)))*sqrt(e+f*x^2))],
[1/((a+b*x^2)^(3/2)*sqrt(c+d*x^2)*sqrt(e+f*x^2)),x,0,Unintegrable(1/((a+b*x^2)^(3/2)*sqrt(c+d*x^2)*sqrt(e+f*x^2)),x)]]:
