# Maple integration test file: "1 Algebraic functions\1.1 Binomial products\1.1.1 Linear\1.1.1.7 P(x) (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q.txt"

lst:=[

# Integrands of the form P2[x] (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q

# Integrands of the form (a+b x)^m (c+d x)^(n/2) (A+B x) / (Sqrte+f x] Sqrt[g+h x])

# n>0

# n<0

#  {(a + b*x)^3*(A + B*x)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x, 10, (1/(105*d^3*f^3*h^3))*(2*b*(5*d*f*h*(7*a*(2*A*b + a*B)*d*f*h - b*B*(5*b*(d*e*g + c*f*g + c*e*h) + 2*a*(d*f*g + d*e*h + c*f*h))) + 2*(a*d*f*h - 2*b*(d*f*g + d*e*h + c*f*h))*(11*a*B*d*f*h + b*(7*A*d*f*h - 6*B*(d*f*g + d*e*h + c*f*h))))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]) + (2*b*(11*a*B*d*f*h + b*(7*A*d*f*h - 6*B*(d*f*g + d*e*h + c*f*h)))*(a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(35*d^2*f^2*h^2) + (2*b*B*(a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(7*d*f*h) - (1/(105*d^4*f^(7/2)*h^4*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)]))*(2*Sqrt[(-d)*e + c*f]*(2*b*(d*f*g + d*e*h + c*f*h)*(5*d*f*h*(7*a*(2*A*b + a*B)*d*f*h - b*B*(5*b*(d*e*g + c*f*g + c*e*h) + 2*a*(d*f*g + d*e*h + c*f*h))) + 2*(a*d*f*h - 2*b*(d*f*g + d*e*h + c*f*h))*(11*a*B*d*f*h + b*(7*A*d*f*h - 6*B*(d*f*g + d*e*h + c*f*h)))) + 3*d*f*h*(b*(3*b*(d*e*g + c*f*g + c*e*h) + 2*a*(d*f*g + d*e*h + c*f*h))*(11*a*B*d*f*h + b*(7*A*d*f*h - 6*B*(d*f*g + d*e*h + c*f*h))) + 5*d*f*h*(4*b^3*B*c*e*g - 7*a^3*B*d*f*h + 6*a*b^2*B*(d*e*g + c*f*g + c*e*h) - a^2*b*(21*A*d*f*h - 2*B*(d*f*g + d*e*h + c*f*h)))))*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[g + h*x]*EllipticE[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[(-d)*e + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))]) - (1/(105*d^4*f^(7/2)*h^4*Sqrt[e + f*x]*Sqrt[g + h*x]))*(2*Sqrt[(-d)*e + c*f]*(105*a^3*d^3*f^3*h^3*(B*g - A*h) + 105*a^2*b*d^2*f^2*h^2*(3*A*d*f*g*h - B*(c*h*(f*g - e*h) + d*g*(2*f*g + e*h))) - 21*a*b^2*d*f*h*(5*A*d*f*h*(c*h*(f*g - e*h) + d*g*(2*f*g + e*h)) - B*(4*c^2*f*h^2*(f*g - e*h) + c*d*h*(3*f^2*g^2 + e*f*g*h - 4*e^2*h^2) + d^2*g*(8*f^2*g^2 + 3*e*f*g*h + 4*e^2*h^2))) + b^3*(7*A*d*f*h*(4*c^2*f*h^2*(f*g - e*h) + c*d*h*(3*f^2*g^2 + e*f*g*h - 4*e^2*h^2) + d^2*g*(8*f^2*g^2 + 3*e*f*g*h + 4*e^2*h^2)) - B*(24*c^3*f^2*h^3*(f*g - e*h) + c^2*d*f*h^2*(17*f^2*g^2 + 6*e*f*g*h - 23*e^2*h^2) + 2*c*d^2*h*(8*f^3*g^3 + e*f^2*g^2*h + 3*e^2*f*g*h^2 - 12*e^3*h^3) + d^3*g*(48*f^3*g^3 + 16*e*f^2*g^2*h + 17*e^2*f*g*h^2 + 24*e^3*h^3))))*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*EllipticF[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[(-d)*e + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])} 
[(a+b*x)^2*(A+B*x)/(sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,9,2/15*b*(7*a*B*d*f*h+b*(5*A*d*f*h-4*B*(d*f*g+d*e*h+c*f*h)))*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/(d^2*f^2*h^2)+2/5*b*B*(a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/(d*f*h)+2/15*(15*a^2*B*d^2*f^2*h^2+10*a*b*d*f*h*(3*A*d*f*h-2*B*(d*f*g+d*e*h+c*f*h))-b^2*(10*A*d*f*h*(d*f*g+d*e*h+c*f*h)-B*(8*c^2*f^2*h^2+7*c*d*f*h*(f*g+e*h)+d^2*(8*f^2*g^2+7*e*f*g*h+8*e^2*h^2))))*EllipticE(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(g+h*x)/(d^3*f^(5/2)*h^3*sqrt(e+f*x)*sqrt(d*(g+h*x)/(d*g-c*h)))-2/15*(15*a^2*d^2*f^2*h^2*(B*g-A*h)+10*a*b*d*f*h*(3*A*d*f*g*h-B*(c*h*(f*g-e*h)+d*g*(2*f*g+e*h)))-b^2*(5*A*d*f*h*(c*h*(f*g-e*h)+d*g*(2*f*g+e*h))-B*(4*c^2*f*h^2*(f*g-e*h)+c*d*h*(3*f^2*g^2+e*f*g*h-4*e^2*h^2)+d^2*g*(8*f^2*g^2+3*e*f*g*h+4*e^2*h^2))))*EllipticF(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/(d^3*f^(5/2)*h^3*sqrt(e+f*x)*sqrt(g+h*x))],
[(a+b*x)*(A+B*x)/(sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,8,2/3*b*B*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/(d*f*h)+2/3*(3*a*B*d*f*h+b*(3*A*d*f*h-2*B*(d*f*g+d*e*h+c*f*h)))*EllipticE(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(g+h*x)/(d^2*f^(3/2)*h^2*sqrt(e+f*x)*sqrt(d*(g+h*x)/(d*g-c*h)))-2/3*(3*a*d*f*h*(B*g-A*h)+b*(3*A*d*f*g*h-B*(c*h*(f*g-e*h)+d*g*(2*f*g+e*h))))*EllipticF(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/(d^2*f^(3/2)*h^2*sqrt(e+f*x)*sqrt(g+h*x))],
[(A+B*x)/(sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,6,2*B*EllipticE(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(g+h*x)/(d*h*sqrt(f)*sqrt(e+f*x)*sqrt(d*(g+h*x)/(d*g-c*h)))-2*(B*g-A*h)*EllipticF(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/(d*h*sqrt(f)*sqrt(e+f*x)*sqrt(g+h*x))],
[(A+B*x)/((a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,9,2*B*EllipticF(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/(b*d*sqrt(f)*sqrt(e+f*x)*sqrt(g+h*x))-2*(A-a*B/b)*EllipticPi(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),-b*(d*e-c*f)/((b*c-a*d)*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/((b*c-a*d)*sqrt(f)*sqrt(e+f*x)*sqrt(g+h*x))],
[(A+B*x)/((a+b*x)^2*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,12,-b*(A*b-a*B)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)*(b*e-a*f)*(b*g-a*h)*(a+b*x))+(A*b-a*B)*EllipticE(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(f)*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(g+h*x)/((b*c-a*d)*(b*e-a*f)*(b*g-a*h)*sqrt(e+f*x)*sqrt(d*(g+h*x)/(d*g-c*h)))+(3*a^2*A*b*d*f*h-a^3*B*d*f*h-b^3*(2*B*c*e*g-A*(d*e*g+c*f*g+c*e*h))+a*b^2*(B*(d*e*g+c*f*g+c*e*h)-2*A*(d*f*g+d*e*h+c*f*h)))*EllipticPi(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),-b*(d*e-c*f)/((b*c-a*d)*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/(b*(b*c-a*d)^2*(b*e-a*f)*(b*g-a*h)*sqrt(f)*sqrt(e+f*x)*sqrt(g+h*x))-(A*b-a*B)*EllipticF(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(f)*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/(b*(b*c-a*d)*(b*e-a*f)*sqrt(e+f*x)*sqrt(g+h*x))],

#  {(A + B*x)/((a + b*x)^3*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x, 13, -((b*(A*b - a*B)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(2*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)*(a + b*x)^2)) - (b*(5*a^3*B*d*f*h + b^3*(4*B*c*e*g - 3*A*(d*e*g + c*f*g + c*e*h)) - a*b^2*(B*(d*e*g + c*f*g + c*e*h) - 6*A*(d*f*g + d*e*h + c*f*h)) - a^2*b*(9*A*d*f*h + 2*B*(d*f*g + d*e*h + c*f*h)))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(4*(b*c - a*d)^2*(b*e - a*f)^2*(b*g - a*h)^2*(a + b*x)) + (Sqrt[f]*Sqrt[(-d)*e + c*f]*(5*a^3*B*d*f*h + b^3*(4*B*c*e*g - 3*A*(d*e*g + c*f*g + c*e*h)) - a*b^2*(B*(d*e*g + c*f*g + c*e*h) - 6*A*(d*f*g + d*e*h + c*f*h)) - a^2*b*(9*A*d*f*h + 2*B*(d*f*g + d*e*h + c*f*h)))*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[g + h*x]*EllipticE[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[(-d)*e + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(4*(b*c - a*d)^2*(b*e - a*f)^2*(b*g - a*h)^2*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)]) - (Sqrt[f]*Sqrt[(-d)*e + c*f]*(3*a^3*B*d*f*h - a^2*b*d*f*(2*B*g + 7*A*h) + b^3*(4*B*c*e*g - A*(3*d*e*g + 3*c*f*g + c*e*h)) - a*b^2*(B*(d*e*g + c*f*g + 3*c*e*h) - 2*A*(3*d*f*g + 2*d*e*h + 2*c*f*h)))*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*EllipticF[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[(-d)*e + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(4*b*(b*c - a*d)^2*(b*e - a*f)^2*(b*g - a*h)*Sqrt[e + f*x]*Sqrt[g + h*x]) - (Sqrt[(-d)*e + c*f]*(15*a^4*A*b*d^2*f^2*h^2 - 3*a^5*B*d^2*f^2*h^2 + 10*a^3*b^2*d*f*h*(B*(d*e*g + c*f*g + c*e*h) - 2*A*(d*f*g + d*e*h + c*f*h)) + a*b^4*(B*(d^2*e^2*g^2 + 14*c*d*e*g*(f*g + e*h) + c^2*(f^2*g^2 + 14*e*f*g*h + e^2*h^2)) - 4*A*(2*d^2*e*g*(f*g + e*h) + 2*c^2*f*h*(f*g + e*h) + c*d*(2*f^2*g^2 + 3*e*f*g*h + 2*e^2*h^2))) - b^5*(4*B*c*e*g*(d*e*g + c*f*g + c*e*h) - A*(3*d^2*e^2*g^2 + 2*c*d*e*g*(f*g + e*h) + c^2*(3*f^2*g^2 + 2*e*f*g*h + 3*e^2*h^2))) - 2*a^2*b^3*(2*B*(d^2*e*g*(f*g + e*h) + c^2*f*h*(f*g + e*h) + c*d*(f^2*g^2 + 9*e*f*g*h + e^2*h^2)) - A*(4*c^2*f^2*h^2 + 11*c*d*f*h*(f*g + e*h) + d^2*(4*f^2*g^2 + 11*e*f*g*h + 4*e^2*h^2))))*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*EllipticPi[-((b*(d*e - c*f))/((b*c - a*d)*f)), ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[(-d)*e + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(4*b*(b*c - a*d)^3*Sqrt[f]*(b*e - a*f)^2*(b*g - a*h)^2*Sqrt[e + f*x]*Sqrt[g + h*x])} 

# Integrands of the form (a+b x)^(m/2) (c+d x)^(n/2) (A+B x) / (Sqrte+f x] Sqrt[g+h x])

# n>0

# n<0
[(a+b*x)^(3/2)*(A+B*x)/(sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,10,1/4*(4*d*f*h*(2*a*(2*A*b+a*B)*d*f*h-b*B*(b*(d*e*g+c*f*g+c*e*h)+a*(d*f*g+d*e*h+c*f*h)))-(a*d*f*h+b*(d*f*g+d*e*h+c*f*h))*(5*a*B*d*f*h+b*(4*A*d*f*h-3*B*(d*f*g+d*e*h+c*f*h))))*(a+b*x)*EllipticPi(sqrt(b*c-a*d)*sqrt(g+h*x)/(sqrt(-d*g+c*h)*sqrt(a+b*x)),-b*(d*g-c*h)/((b*c-a*d)*h),sqrt((b*e-a*f)*(d*g-c*h)/((b*c-a*d)*(f*g-e*h))))*sqrt(-d*g+c*h)*sqrt((b*g-a*h)*(c+d*x)/((d*g-c*h)*(a+b*x)))*sqrt((b*g-a*h)*(e+f*x)/((f*g-e*h)*(a+b*x)))/(b*d^2*f^2*h^3*sqrt(b*c-a*d)*sqrt(c+d*x)*sqrt(e+f*x))+1/4*(5*a*B*d*f*h+b*(4*A*d*f*h-3*B*(d*f*g+d*e*h+c*f*h)))*sqrt(a+b*x)*sqrt(e+f*x)*sqrt(g+h*x)/(d*f^2*h^2*sqrt(c+d*x))+1/2*b*B*sqrt(a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/(d*f*h)-1/4*(b*e-a*f)*(3*a*B*d*f*h+b*(4*A*d*f*h-B*(c*f*h+3*d*(f*g+e*h))))*EllipticF(sqrt(b*g-a*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(a+b*x)),sqrt(-(b*c-a*d)*(f*g-e*h)/((d*e-c*f)*(b*g-a*h))))*sqrt(b*g-a*h)*sqrt((b*e-a*f)*(c+d*x)/((d*e-c*f)*(a+b*x)))*sqrt(g+h*x)/(b*d*f^2*h^2*sqrt(f*g-e*h)*sqrt(c+d*x)*sqrt(-(b*e-a*f)*(g+h*x)/((f*g-e*h)*(a+b*x))))-1/4*(5*a*B*d*f*h+b*(4*A*d*f*h-3*B*(d*f*g+d*e*h+c*f*h)))*EllipticE(sqrt(d*g-c*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(c+d*x)),sqrt((b*c-a*d)*(f*g-e*h)/((b*e-a*f)*(d*g-c*h))))*sqrt(d*g-c*h)*sqrt(f*g-e*h)*sqrt(a+b*x)*sqrt(-(d*e-c*f)*(g+h*x)/((f*g-e*h)*(c+d*x)))/(d^2*f^2*h^2*sqrt((d*e-c*f)*(a+b*x)/((b*e-a*f)*(c+d*x)))*sqrt(g+h*x))],
[(a+b*x)^(1/2)*(A+B*x)/(sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,7,(2*A*b*d*f*h+B*(a*d*f*h-b*(d*f*g+d*e*h+c*f*h)))*(a+b*x)*EllipticPi(sqrt(b*c-a*d)*sqrt(g+h*x)/(sqrt(-d*g+c*h)*sqrt(a+b*x)),-b*(d*g-c*h)/((b*c-a*d)*h),sqrt((b*e-a*f)*(d*g-c*h)/((b*c-a*d)*(f*g-e*h))))*sqrt(-d*g+c*h)*sqrt((b*g-a*h)*(c+d*x)/((d*g-c*h)*(a+b*x)))*sqrt((b*g-a*h)*(e+f*x)/((f*g-e*h)*(a+b*x)))/(b*d*f*h^2*sqrt(b*c-a*d)*sqrt(c+d*x)*sqrt(e+f*x))+B*sqrt(a+b*x)*sqrt(e+f*x)*sqrt(g+h*x)/(f*h*sqrt(c+d*x))-B*(b*e-a*f)*EllipticF(sqrt(b*g-a*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(a+b*x)),sqrt(-(b*c-a*d)*(f*g-e*h)/((d*e-c*f)*(b*g-a*h))))*sqrt(b*g-a*h)*sqrt((b*e-a*f)*(c+d*x)/((d*e-c*f)*(a+b*x)))*sqrt(g+h*x)/(b*f*h*sqrt(f*g-e*h)*sqrt(c+d*x)*sqrt(-(b*e-a*f)*(g+h*x)/((f*g-e*h)*(a+b*x))))-B*EllipticE(sqrt(d*g-c*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(c+d*x)),sqrt((b*c-a*d)*(f*g-e*h)/((b*e-a*f)*(d*g-c*h))))*sqrt(d*g-c*h)*sqrt(f*g-e*h)*sqrt(a+b*x)*sqrt(-(d*e-c*f)*(g+h*x)/((f*g-e*h)*(c+d*x)))/(d*f*h*sqrt((d*e-c*f)*(a+b*x)/((b*e-a*f)*(c+d*x)))*sqrt(g+h*x))],
[(A+B*x)/((a+b*x)^(1/2)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,5,2*B*(a+b*x)*EllipticPi(sqrt(b*c-a*d)*sqrt(g+h*x)/(sqrt(-d*g+c*h)*sqrt(a+b*x)),-b*(d*g-c*h)/((b*c-a*d)*h),sqrt((b*e-a*f)*(d*g-c*h)/((b*c-a*d)*(f*g-e*h))))*sqrt(-d*g+c*h)*sqrt((b*g-a*h)*(c+d*x)/((d*g-c*h)*(a+b*x)))*sqrt((b*g-a*h)*(e+f*x)/((f*g-e*h)*(a+b*x)))/(b*h*sqrt(b*c-a*d)*sqrt(c+d*x)*sqrt(e+f*x))+2*(A*b-a*B)*EllipticF(sqrt(b*g-a*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(a+b*x)),sqrt(-(b*c-a*d)*(f*g-e*h)/((d*e-c*f)*(b*g-a*h))))*sqrt((b*e-a*f)*(c+d*x)/((d*e-c*f)*(a+b*x)))*sqrt(g+h*x)/(b*sqrt(b*g-a*h)*sqrt(f*g-e*h)*sqrt(c+d*x)*sqrt(-(b*e-a*f)*(g+h*x)/((f*g-e*h)*(a+b*x))))],
[(A+B*x)/((a+b*x)^(3/2)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,7,2*(A*b-a*B)*d*sqrt(a+b*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)*(b*e-a*f)*(b*g-a*h)*sqrt(c+d*x))-2*b*(A*b-a*B)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)*(b*e-a*f)*(b*g-a*h)*sqrt(a+b*x))+2*(B*c-A*d)*EllipticF(sqrt(b*g-a*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(a+b*x)),sqrt(-(b*c-a*d)*(f*g-e*h)/((d*e-c*f)*(b*g-a*h))))*sqrt((b*e-a*f)*(c+d*x)/((d*e-c*f)*(a+b*x)))*sqrt(g+h*x)/((b*c-a*d)*sqrt(b*g-a*h)*sqrt(f*g-e*h)*sqrt(c+d*x)*sqrt(-(b*e-a*f)*(g+h*x)/((f*g-e*h)*(a+b*x))))-2*(A*b-a*B)*EllipticE(sqrt(d*g-c*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(c+d*x)),sqrt((b*c-a*d)*(f*g-e*h)/((b*e-a*f)*(d*g-c*h))))*sqrt(d*g-c*h)*sqrt(f*g-e*h)*sqrt(a+b*x)*sqrt(-(d*e-c*f)*(g+h*x)/((f*g-e*h)*(c+d*x)))/((b*c-a*d)*(b*e-a*f)*(b*g-a*h)*sqrt((d*e-c*f)*(a+b*x)/((b*e-a*f)*(c+d*x)))*sqrt(g+h*x))],
[(A+B*x)/((a+b*x)^(5/2)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,8,2/3*d*(3*a^3*B*d*f*h+b^3*(3*B*c*e*g-2*A*(d*e*g+c*f*g+c*e*h))-a*b^2*(B*(d*e*g+c*f*g+c*e*h)-4*A*(d*f*g+d*e*h+c*f*h))-a^2*b*(6*A*d*f*h+B*(d*f*g+d*e*h+c*f*h)))*sqrt(a+b*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)^2*(b*e-a*f)^2*(b*g-a*h)^2*sqrt(c+d*x))-2/3*b*(A*b-a*B)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)*(b*e-a*f)*(b*g-a*h)*(a+b*x)^(3/2))-2/3*b*(3*a^3*B*d*f*h+b^3*(3*B*c*e*g-2*A*(d*e*g+c*f*g+c*e*h))-a*b^2*(B*(d*e*g+c*f*g+c*e*h)-4*A*(d*f*g+d*e*h+c*f*h))-a^2*b*(6*A*d*f*h+B*(d*f*g+d*e*h+c*f*h)))*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)^2*(b*e-a*f)^2*(b*g-a*h)^2*sqrt(a+b*x))-2/3*(3*a^2*d*(B*c-A*d)*f*h+b^2*(3*B*c*d*e*g-A*(2*d^2*e*g-c^2*f*h+c*d*(f*g+e*h)))+a*b*(3*A*d^2*(f*g+e*h)-B*(d^2*e*g+c^2*f*h+2*c*d*(f*g+e*h))))*EllipticF(sqrt(b*g-a*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(a+b*x)),sqrt(-(b*c-a*d)*(f*g-e*h)/((d*e-c*f)*(b*g-a*h))))*sqrt((b*e-a*f)*(c+d*x)/((d*e-c*f)*(a+b*x)))*sqrt(g+h*x)/((b*c-a*d)^2*(b*e-a*f)*(b*g-a*h)^(3/2)*sqrt(f*g-e*h)*sqrt(c+d*x)*sqrt(-(b*e-a*f)*(g+h*x)/((f*g-e*h)*(a+b*x))))-2/3*(3*a^3*B*d*f*h+b^3*(3*B*c*e*g-2*A*(d*e*g+c*f*g+c*e*h))-a*b^2*(B*(d*e*g+c*f*g+c*e*h)-4*A*(d*f*g+d*e*h+c*f*h))-a^2*b*(6*A*d*f*h+B*(d*f*g+d*e*h+c*f*h)))*EllipticE(sqrt(d*g-c*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(c+d*x)),sqrt((b*c-a*d)*(f*g-e*h)/((b*e-a*f)*(d*g-c*h))))*sqrt(d*g-c*h)*sqrt(f*g-e*h)*sqrt(a+b*x)*sqrt(-(d*e-c*f)*(g+h*x)/((f*g-e*h)*(c+d*x)))/((b*c-a*d)^2*(b*e-a*f)^2*(b*g-a*h)^2*sqrt((d*e-c*f)*(a+b*x)/((b*e-a*f)*(c+d*x)))*sqrt(g+h*x))],
[(a+b*x)^(3/2)*(d*e+c*f+2*d*f*x)/(sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,10,-1/2*(6*a*b*d^2*f^2*g*h-3*a^2*d^2*f^2*h^2+b^2*(2*c*d*e*f*h^2-c^2*f^2*h^2-d^2*(3*f^2*g^2+e^2*h^2)))*(a+b*x)*EllipticPi(sqrt(b*c-a*d)*sqrt(g+h*x)/(sqrt(-d*g+c*h)*sqrt(a+b*x)),-b*(d*g-c*h)/((b*c-a*d)*h),sqrt((b*e-a*f)*(d*g-c*h)/((b*c-a*d)*(f*g-e*h))))*sqrt(-d*g+c*h)*sqrt((b*g-a*h)*(c+d*x)/((d*g-c*h)*(a+b*x)))*sqrt((b*g-a*h)*(e+f*x)/((f*g-e*h)*(a+b*x)))/(b*d*f*h^3*sqrt(b*c-a*d)*sqrt(c+d*x)*sqrt(e+f*x))+1/2*(5*a*d*f*h-b*(3*d*f*g+d*e*h+c*f*h))*sqrt(a+b*x)*sqrt(e+f*x)*sqrt(g+h*x)/(f*h^2*sqrt(c+d*x))+b*sqrt(a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/h-1/2*(b*e-a*f)*(3*a*d*f*h+b*(c*f*h-d*(3*f*g+e*h)))*EllipticF(sqrt(b*g-a*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(a+b*x)),sqrt(-(b*c-a*d)*(f*g-e*h)/((d*e-c*f)*(b*g-a*h))))*sqrt(b*g-a*h)*sqrt((b*e-a*f)*(c+d*x)/((d*e-c*f)*(a+b*x)))*sqrt(g+h*x)/(b*f*h^2*sqrt(f*g-e*h)*sqrt(c+d*x)*sqrt(-(b*e-a*f)*(g+h*x)/((f*g-e*h)*(a+b*x))))-1/2*(5*a*d*f*h-b*(3*d*f*g+d*e*h+c*f*h))*EllipticE(sqrt(d*g-c*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(c+d*x)),sqrt((b*c-a*d)*(f*g-e*h)/((b*e-a*f)*(d*g-c*h))))*sqrt(d*g-c*h)*sqrt(f*g-e*h)*sqrt(a+b*x)*sqrt(-(d*e-c*f)*(g+h*x)/((f*g-e*h)*(c+d*x)))/(d*f*h^2*sqrt((d*e-c*f)*(a+b*x)/((b*e-a*f)*(c+d*x)))*sqrt(g+h*x))],
[(a+b*x)^(1/2)*(d*e+c*f+2*d*f*x)/(sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,5,-2*d*(b*g-a*h)^(3/2)*(e+f*x)*EllipticPi(sqrt(b*e-a*f)*sqrt(g+h*x)/(sqrt(b*g-a*h)*sqrt(e+f*x)),f*(b*g-a*h)/((b*e-a*f)*h),sqrt((d*e-c*f)*(b*g-a*h)/((b*e-a*f)*(d*g-c*h))))*sqrt((f*g-e*h)*(a+b*x)/((b*g-a*h)*(e+f*x)))*sqrt((f*g-e*h)*(c+d*x)/((d*g-c*h)*(e+f*x)))/(h^2*sqrt(b*e-a*f)*sqrt(a+b*x)*sqrt(c+d*x))+2*b*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/(h*sqrt(a+b*x))-2*EllipticE(sqrt(b*g-a*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(a+b*x)),sqrt(-(b*c-a*d)*(f*g-e*h)/((d*e-c*f)*(b*g-a*h))))*sqrt(b*g-a*h)*sqrt(f*g-e*h)*sqrt(c+d*x)*sqrt(-(b*e-a*f)*(g+h*x)/((f*g-e*h)*(a+b*x)))/(h*sqrt((b*e-a*f)*(c+d*x)/((d*e-c*f)*(a+b*x)))*sqrt(g+h*x))],
[(d*e+c*f+2*d*f*x)/((a+b*x)^(1/2)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,5,4*d*f*(a+b*x)*EllipticPi(sqrt(b*c-a*d)*sqrt(g+h*x)/(sqrt(-d*g+c*h)*sqrt(a+b*x)),-b*(d*g-c*h)/((b*c-a*d)*h),sqrt((b*e-a*f)*(d*g-c*h)/((b*c-a*d)*(f*g-e*h))))*sqrt(-d*g+c*h)*sqrt((b*g-a*h)*(c+d*x)/((d*g-c*h)*(a+b*x)))*sqrt((b*g-a*h)*(e+f*x)/((f*g-e*h)*(a+b*x)))/(b*h*sqrt(b*c-a*d)*sqrt(c+d*x)*sqrt(e+f*x))+2*(b*d*e+b*c*f-2*a*d*f)*EllipticF(sqrt(b*g-a*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(a+b*x)),sqrt(-(b*c-a*d)*(f*g-e*h)/((d*e-c*f)*(b*g-a*h))))*sqrt((b*e-a*f)*(c+d*x)/((d*e-c*f)*(a+b*x)))*sqrt(g+h*x)/(b*sqrt(b*g-a*h)*sqrt(f*g-e*h)*sqrt(c+d*x)*sqrt(-(b*e-a*f)*(g+h*x)/((f*g-e*h)*(a+b*x))))],
[(d*e+c*f+2*d*f*x)/((a+b*x)^(3/2)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,7,2*d*(b*d*e+b*c*f-2*a*d*f)*sqrt(a+b*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)*(b*e-a*f)*(b*g-a*h)*sqrt(c+d*x))-2*b*(b*d*e+b*c*f-2*a*d*f)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)*(b*e-a*f)*(b*g-a*h)*sqrt(a+b*x))-2*d*(d*e-c*f)*EllipticF(sqrt(b*g-a*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(a+b*x)),sqrt(-(b*c-a*d)*(f*g-e*h)/((d*e-c*f)*(b*g-a*h))))*sqrt((b*e-a*f)*(c+d*x)/((d*e-c*f)*(a+b*x)))*sqrt(g+h*x)/((b*c-a*d)*sqrt(b*g-a*h)*sqrt(f*g-e*h)*sqrt(c+d*x)*sqrt(-(b*e-a*f)*(g+h*x)/((f*g-e*h)*(a+b*x))))-2*(b*d*e+b*c*f-2*a*d*f)*EllipticE(sqrt(d*g-c*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(c+d*x)),sqrt((b*c-a*d)*(f*g-e*h)/((b*e-a*f)*(d*g-c*h))))*sqrt(d*g-c*h)*sqrt(f*g-e*h)*sqrt(a+b*x)*sqrt(-(d*e-c*f)*(g+h*x)/((f*g-e*h)*(c+d*x)))/((b*c-a*d)*(b*e-a*f)*(b*g-a*h)*sqrt((d*e-c*f)*(a+b*x)/((b*e-a*f)*(c+d*x)))*sqrt(g+h*x))],
[(d*e+c*f+2*d*f*x)/((a+b*x)^(5/2)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,8,4/3*d*(3*a^3*d^2*f^2*h-a^2*b*d*f*(d*f*g+4*d*e*h+4*c*f*h)-b^3*(d^2*e^2*g-c*d*e*(f*g-e*h)+c^2*f*(f*g+e*h))+a*b^2*(2*c^2*f^2*h+d^2*e*(f*g+2*e*h)+c*d*f*(f*g+3*e*h)))*sqrt(a+b*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)^2*(b*e-a*f)^2*(b*g-a*h)^2*sqrt(c+d*x))-2/3*b*(b*d*e+b*c*f-2*a*d*f)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)*(b*e-a*f)*(b*g-a*h)*(a+b*x)^(3/2))-4/3*b*(3*a^3*d^2*f^2*h-a^2*b*d*f*(d*f*g+4*d*e*h+4*c*f*h)-b^3*(d^2*e^2*g-c*d*e*(f*g-e*h)+c^2*f*(f*g+e*h))+a*b^2*(2*c^2*f^2*h+d^2*e*(f*g+2*e*h)+c*d*f*(f*g+3*e*h)))*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)^2*(b*e-a*f)^2*(b*g-a*h)^2*sqrt(a+b*x))+2/3*(d*e-c*f)*(3*a^2*d^2*f*h-a*b*d*(d*f*g+3*d*e*h+2*c*f*h)+b^2*(2*d^2*e*g-c*d*f*g+c*d*e*h+c^2*f*h))*EllipticF(sqrt(b*g-a*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(a+b*x)),sqrt(-(b*c-a*d)*(f*g-e*h)/((d*e-c*f)*(b*g-a*h))))*sqrt((b*e-a*f)*(c+d*x)/((d*e-c*f)*(a+b*x)))*sqrt(g+h*x)/((b*c-a*d)^2*(b*e-a*f)*(b*g-a*h)^(3/2)*sqrt(f*g-e*h)*sqrt(c+d*x)*sqrt(-(b*e-a*f)*(g+h*x)/((f*g-e*h)*(a+b*x))))-4/3*(3*a^3*d^2*f^2*h-a^2*b*d*f*(d*f*g+4*d*e*h+4*c*f*h)-b^3*(d^2*e^2*g-c*d*e*(f*g-e*h)+c^2*f*(f*g+e*h))+a*b^2*(2*c^2*f^2*h+d^2*e*(f*g+2*e*h)+c*d*f*(f*g+3*e*h)))*EllipticE(sqrt(d*g-c*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(c+d*x)),sqrt((b*c-a*d)*(f*g-e*h)/((b*e-a*f)*(d*g-c*h))))*sqrt(d*g-c*h)*sqrt(f*g-e*h)*sqrt(a+b*x)*sqrt(-(d*e-c*f)*(g+h*x)/((f*g-e*h)*(c+d*x)))/((b*c-a*d)^2*(b*e-a*f)^2*(b*g-a*h)^2*sqrt((d*e-c*f)*(a+b*x)/((b*e-a*f)*(c+d*x)))*sqrt(g+h*x))],

# Integrands of the form P3[x] (a+b x)^m (c+d x)^n (e+f x)^p (g+h x)^q

# Integrands of the form (a+b x)^m (c+d x)^(n/2) (e+f x)^(p/2) (g+h x)^(q/2) (A+B x+C x^2) with A b^2-a b B+a^2 C=0

# Integrands of the form (a+b x)^m (c+d x)^(n/2) (A+B x+C x^2) / (Sqrte+f x] Sqrt[g+h x])

# n>0

# n<0
[(a+b*x)*(a*b*B-a^2*C+b^2*B*x+b^2*C*x^2)/(sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,8,2/15*b^2*(5*b*B*d*f*h+2*C*(a*d*f*h-2*b*(d*f*g+d*e*h+c*f*h)))*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/(d^2*f^2*h^2)+2/5*b^2*C*(a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/(d*f*h)-2/15*b*(15*a^2*C*d^2*f^2*h^2-10*a*b*d*f*h*(3*B*d*f*h-C*(d*f*g+d*e*h+c*f*h))+b^2*(10*B*d*f*h*(d*f*g+d*e*h+c*f*h)-C*(8*c^2*f^2*h^2+7*c*d*f*h*(f*g+e*h)+d^2*(8*f^2*g^2+7*e*f*g*h+8*e^2*h^2))))*EllipticE(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(g+h*x)/(d^3*f^(5/2)*h^3*sqrt(e+f*x)*sqrt(d*(g+h*x)/(d*g-c*h)))-2/15*(15*a^3*C*d^2*f^2*h^3-15*a^2*b*d^2*f^2*h^2*(C*g+B*h)+5*a*b^2*d*f*h*(6*B*d*f*g*h-C*(c*h*(f*g-e*h)+d*g*(2*f*g+e*h)))-b^3*(5*B*d*f*h*(c*h*(f*g-e*h)+d*g*(2*f*g+e*h))-C*(4*c^2*f*h^2*(f*g-e*h)+c*d*h*(3*f^2*g^2+e*f*g*h-4*e^2*h^2)+d^2*g*(8*f^2*g^2+3*e*f*g*h+4*e^2*h^2))))*EllipticF(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/(d^3*f^(5/2)*h^3*sqrt(e+f*x)*sqrt(g+h*x))],
[(a*b*B-a^2*C+b^2*B*x+b^2*C*x^2)/(sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,7,2/3*b^2*C*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/(d*f*h)+2/3*b^2*(3*B*d*f*h-2*C*(d*f*g+d*e*h+c*f*h))*EllipticE(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(g+h*x)/(d^2*f^(3/2)*h^2*sqrt(e+f*x)*sqrt(d*(g+h*x)/(d*g-c*h)))+2/3*(3*a*b*B*d*f*h^2-3*a^2*C*d*f*h^2-b^2*(3*B*d*f*g*h-C*(c*h*(f*g-e*h)+d*g*(2*f*g+e*h))))*EllipticF(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/(d^2*f^(3/2)*h^2*sqrt(e+f*x)*sqrt(g+h*x))],
[(a*b*B-a^2*C+b^2*B*x+b^2*C*x^2)/((a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,7,2*b*C*EllipticE(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(g+h*x)/(d*h*sqrt(f)*sqrt(e+f*x)*sqrt(d*(g+h*x)/(d*g-c*h)))-2*(b*C*g-b*B*h+a*C*h)*EllipticF(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/(d*h*sqrt(f)*sqrt(e+f*x)*sqrt(g+h*x))],
[(a*b*B-a^2*C+b^2*B*x+b^2*C*x^2)/((a+b*x)^2*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,10,2*C*EllipticF(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/(d*sqrt(f)*sqrt(e+f*x)*sqrt(g+h*x))-2*(b*B-2*a*C)*EllipticPi(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),-b*(d*e-c*f)/((b*c-a*d)*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/((b*c-a*d)*sqrt(f)*sqrt(e+f*x)*sqrt(g+h*x))],
[(a*b*B-a^2*C+b^2*B*x+b^2*C*x^2)/((a+b*x)^3*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,13,-b^2*(b*B-2*a*C)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)*(b*e-a*f)*(b*g-a*h)*(a+b*x))+b*(b*B-2*a*C)*EllipticE(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(f)*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(g+h*x)/((b*c-a*d)*(b*e-a*f)*(b*g-a*h)*sqrt(e+f*x)*sqrt(d*(g+h*x)/(d*g-c*h)))-(4*a^3*C*d*f*h+2*a*b^2*B*(d*f*g+d*e*h+c*f*h)-b^3*(B*d*e*g-c*(2*C*e*g-B*f*g-B*e*h))-a^2*b*(3*B*d*f*h+2*C*(d*f*g+d*e*h+c*f*h)))*EllipticPi(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),-b*(d*e-c*f)/((b*c-a*d)*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/((b*c-a*d)^2*(b*e-a*f)*(b*g-a*h)*sqrt(f)*sqrt(e+f*x)*sqrt(g+h*x))-(b*B-2*a*C)*EllipticF(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(f)*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/((b*c-a*d)*(b*e-a*f)*sqrt(e+f*x)*sqrt(g+h*x))],

# Integrands of the form (a+b x)^(m/2) (c+d x)^(n/2) (A+B x+C x^2) / (Sqrte+f x] Sqrt[g+h x])

# n>0

# n<0
[(a+b*x)^(1/2)*(a*b*B-a^2*C+b^2*B*x+b^2*C*x^2)/(sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,9,-1/4*((a*d*f*h+b*(d*f*g+d*e*h+c*f*h))*(4*b*B*d*f*h+C*(a*d*f*h-3*b*(d*f*g+d*e*h+c*f*h)))+4*d*f*h*(2*a^2*C*d*f*h+b^2*C*(d*e*g+c*f*g+c*e*h)-a*b*(4*B*d*f*h-C*(d*f*g+d*e*h+c*f*h))))*(a+b*x)*EllipticPi(sqrt(b*c-a*d)*sqrt(g+h*x)/(sqrt(-d*g+c*h)*sqrt(a+b*x)),-b*(d*g-c*h)/((b*c-a*d)*h),sqrt((b*e-a*f)*(d*g-c*h)/((b*c-a*d)*(f*g-e*h))))*sqrt(-d*g+c*h)*sqrt((b*g-a*h)*(c+d*x)/((d*g-c*h)*(a+b*x)))*sqrt((b*g-a*h)*(e+f*x)/((f*g-e*h)*(a+b*x)))/(d^2*f^2*h^3*sqrt(b*c-a*d)*sqrt(c+d*x)*sqrt(e+f*x))+1/4*b*(4*b*B*d*f*h+C*(a*d*f*h-3*b*(d*f*g+d*e*h+c*f*h)))*sqrt(a+b*x)*sqrt(e+f*x)*sqrt(g+h*x)/(d*f^2*h^2*sqrt(c+d*x))+1/2*b^2*C*sqrt(a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/(d*f*h)+1/4*(b*e-a*f)*(a*C*d*f*h-b*(4*B*d*f*h-C*(3*d*f*g+3*d*e*h+c*f*h)))*EllipticF(sqrt(b*g-a*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(a+b*x)),sqrt(-(b*c-a*d)*(f*g-e*h)/((d*e-c*f)*(b*g-a*h))))*sqrt(b*g-a*h)*sqrt((b*e-a*f)*(c+d*x)/((d*e-c*f)*(a+b*x)))*sqrt(g+h*x)/(d*f^2*h^2*sqrt(f*g-e*h)*sqrt(c+d*x)*sqrt(-(b*e-a*f)*(g+h*x)/((f*g-e*h)*(a+b*x))))-1/4*b*(4*b*B*d*f*h+C*(a*d*f*h-3*b*(d*f*g+d*e*h+c*f*h)))*EllipticE(sqrt(d*g-c*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(c+d*x)),sqrt((b*c-a*d)*(f*g-e*h)/((b*e-a*f)*(d*g-c*h))))*sqrt(d*g-c*h)*sqrt(f*g-e*h)*sqrt(a+b*x)*sqrt(-(d*e-c*f)*(g+h*x)/((f*g-e*h)*(c+d*x)))/(d^2*f^2*h^2*sqrt((d*e-c*f)*(a+b*x)/((b*e-a*f)*(c+d*x)))*sqrt(g+h*x))],
[(a*b*B-a^2*C+b^2*B*x+b^2*C*x^2)/((a+b*x)^(1/2)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,8,-(a*C*d*f*h-b*(2*B*d*f*h-C*(d*f*g+d*e*h+c*f*h)))*(a+b*x)*EllipticPi(sqrt(b*c-a*d)*sqrt(g+h*x)/(sqrt(-d*g+c*h)*sqrt(a+b*x)),-b*(d*g-c*h)/((b*c-a*d)*h),sqrt((b*e-a*f)*(d*g-c*h)/((b*c-a*d)*(f*g-e*h))))*sqrt(-d*g+c*h)*sqrt((b*g-a*h)*(c+d*x)/((d*g-c*h)*(a+b*x)))*sqrt((b*g-a*h)*(e+f*x)/((f*g-e*h)*(a+b*x)))/(d*f*h^2*sqrt(b*c-a*d)*sqrt(c+d*x)*sqrt(e+f*x))+b*C*sqrt(a+b*x)*sqrt(e+f*x)*sqrt(g+h*x)/(f*h*sqrt(c+d*x))-C*(b*e-a*f)*EllipticF(sqrt(b*g-a*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(a+b*x)),sqrt(-(b*c-a*d)*(f*g-e*h)/((d*e-c*f)*(b*g-a*h))))*sqrt(b*g-a*h)*sqrt((b*e-a*f)*(c+d*x)/((d*e-c*f)*(a+b*x)))*sqrt(g+h*x)/(f*h*sqrt(f*g-e*h)*sqrt(c+d*x)*sqrt(-(b*e-a*f)*(g+h*x)/((f*g-e*h)*(a+b*x))))-b*C*EllipticE(sqrt(d*g-c*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(c+d*x)),sqrt((b*c-a*d)*(f*g-e*h)/((b*e-a*f)*(d*g-c*h))))*sqrt(d*g-c*h)*sqrt(f*g-e*h)*sqrt(a+b*x)*sqrt(-(d*e-c*f)*(g+h*x)/((f*g-e*h)*(c+d*x)))/(d*f*h*sqrt((d*e-c*f)*(a+b*x)/((b*e-a*f)*(c+d*x)))*sqrt(g+h*x))],
[(a*b*B-a^2*C+b^2*B*x+b^2*C*x^2)/((a+b*x)^(3/2)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,6,2*C*(a+b*x)*EllipticPi(sqrt(b*c-a*d)*sqrt(g+h*x)/(sqrt(-d*g+c*h)*sqrt(a+b*x)),-b*(d*g-c*h)/((b*c-a*d)*h),sqrt((b*e-a*f)*(d*g-c*h)/((b*c-a*d)*(f*g-e*h))))*sqrt(-d*g+c*h)*sqrt((b*g-a*h)*(c+d*x)/((d*g-c*h)*(a+b*x)))*sqrt((b*g-a*h)*(e+f*x)/((f*g-e*h)*(a+b*x)))/(h*sqrt(b*c-a*d)*sqrt(c+d*x)*sqrt(e+f*x))+2*(b*B-2*a*C)*EllipticF(sqrt(b*g-a*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(a+b*x)),sqrt(-(b*c-a*d)*(f*g-e*h)/((d*e-c*f)*(b*g-a*h))))*sqrt((b*e-a*f)*(c+d*x)/((d*e-c*f)*(a+b*x)))*sqrt(g+h*x)/(sqrt(b*g-a*h)*sqrt(f*g-e*h)*sqrt(c+d*x)*sqrt(-(b*e-a*f)*(g+h*x)/((f*g-e*h)*(a+b*x))))],
[(a*b*B-a^2*C+b^2*B*x+b^2*C*x^2)/((a+b*x)^(5/2)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,8,2*b*(b*B-2*a*C)*d*sqrt(a+b*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)*(b*e-a*f)*(b*g-a*h)*sqrt(c+d*x))-2*b^2*(b*B-2*a*C)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)*(b*e-a*f)*(b*g-a*h)*sqrt(a+b*x))+2*(b*c*C-b*B*d+a*C*d)*EllipticF(sqrt(b*g-a*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(a+b*x)),sqrt(-(b*c-a*d)*(f*g-e*h)/((d*e-c*f)*(b*g-a*h))))*sqrt((b*e-a*f)*(c+d*x)/((d*e-c*f)*(a+b*x)))*sqrt(g+h*x)/((b*c-a*d)*sqrt(b*g-a*h)*sqrt(f*g-e*h)*sqrt(c+d*x)*sqrt(-(b*e-a*f)*(g+h*x)/((f*g-e*h)*(a+b*x))))-2*b*(b*B-2*a*C)*EllipticE(sqrt(d*g-c*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(c+d*x)),sqrt((b*c-a*d)*(f*g-e*h)/((b*e-a*f)*(d*g-c*h))))*sqrt(d*g-c*h)*sqrt(f*g-e*h)*sqrt(a+b*x)*sqrt(-(d*e-c*f)*(g+h*x)/((f*g-e*h)*(c+d*x)))/((b*c-a*d)*(b*e-a*f)*(b*g-a*h)*sqrt((d*e-c*f)*(a+b*x)/((b*e-a*f)*(c+d*x)))*sqrt(g+h*x))],
[(a*b*B-a^2*C+b^2*B*x+b^2*C*x^2)/((a+b*x)^(7/2)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,9,2/3*b*d*(9*a^3*C*d*f*h-b^3*(2*B*d*e*g-c*(3*C*e*g-2*B*f*g-2*B*e*h))+a*b^2*(C*(d*e*g+c*f*g+c*e*h)+4*B*(d*f*g+d*e*h+c*f*h))-a^2*b*(6*B*d*f*h+5*C*(d*f*g+d*e*h+c*f*h)))*sqrt(a+b*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)^2*(b*e-a*f)^2*(b*g-a*h)^2*sqrt(c+d*x))-2/3*b^2*(b*B-2*a*C)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)*(b*e-a*f)*(b*g-a*h)*(a+b*x)^(3/2))-2/3*b^2*(9*a^3*C*d*f*h-b^3*(2*B*d*e*g-c*(3*C*e*g-2*B*f*g-2*B*e*h))+a*b^2*(C*(d*e*g+c*f*g+c*e*h)+4*B*(d*f*g+d*e*h+c*f*h))-a^2*b*(6*B*d*f*h+5*C*(d*f*g+d*e*h+c*f*h)))*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)^2*(b*e-a*f)^2*(b*g-a*h)^2*sqrt(a+b*x))-2/3*(3*a^3*C*d^2*f*h-b^3*(2*B*d^2*e*g-B*c^2*f*h-c*d*(3*C*e*g-B*f*g-B*e*h))-3*a^2*b*d*(B*d*f*h+C*(d*f*g+d*e*h-c*f*h))+a*b^2*(3*B*d^2*(f*g+e*h)+C*(d^2*e*g-c*d*f*g-c*d*e*h-2*c^2*f*h)))*EllipticF(sqrt(b*g-a*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(a+b*x)),sqrt(-(b*c-a*d)*(f*g-e*h)/((d*e-c*f)*(b*g-a*h))))*sqrt((b*e-a*f)*(c+d*x)/((d*e-c*f)*(a+b*x)))*sqrt(g+h*x)/((b*c-a*d)^2*(b*e-a*f)*(b*g-a*h)^(3/2)*sqrt(f*g-e*h)*sqrt(c+d*x)*sqrt(-(b*e-a*f)*(g+h*x)/((f*g-e*h)*(a+b*x))))-2/3*b*(9*a^3*C*d*f*h-b^3*(2*B*d*e*g-c*(3*C*e*g-2*B*f*g-2*B*e*h))+a*b^2*(C*(d*e*g+c*f*g+c*e*h)+4*B*(d*f*g+d*e*h+c*f*h))-a^2*b*(6*B*d*f*h+5*C*(d*f*g+d*e*h+c*f*h)))*EllipticE(sqrt(d*g-c*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(c+d*x)),sqrt((b*c-a*d)*(f*g-e*h)/((b*e-a*f)*(d*g-c*h))))*sqrt(d*g-c*h)*sqrt(f*g-e*h)*sqrt(a+b*x)*sqrt(-(d*e-c*f)*(g+h*x)/((f*g-e*h)*(c+d*x)))/((b*c-a*d)^2*(b*e-a*f)^2*(b*g-a*h)^2*sqrt((d*e-c*f)*(a+b*x)/((b*e-a*f)*(c+d*x)))*sqrt(g+h*x))],

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

# Integrands of the form (a+b x)^m (c+d x)^(n/2) (A+B x+C x^2) / (Sqrte+f x] Sqrt[g+h x])

# n>0

# n<0
[(a+b*x)^2*(A+C*x^2)/(sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,9,2/105*(4*C*(2*a*d*f*h-3*b*(d*f*g+d*e*h+c*f*h))*(a*d*f*h-2*b*(d*f*g+d*e*h+c*f*h))+5*b*d*f*h*(7*A*b*d*f*h-C*(5*b*(d*e*g+c*f*g+c*e*h)+2*a*(d*f*g+d*e*h+c*f*h))))*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/(d^3*f^3*h^3)+4/35*C*(2*a*d*f*h-3*b*(d*f*g+d*e*h+c*f*h))*(a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/(d^2*f^2*h^2)+2/7*C*(a+b*x)^2*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/(d*f*h)-4/105*(35*a^2*C*d^2*f^2*h^2*(d*f*g+d*e*h+c*f*h)-7*a*b*d*f*h*(15*A*d^2*f^2*h^2+C*(8*c^2*f^2*h^2+7*c*d*f*h*(f*g+e*h)+d^2*(8*f^2*g^2+7*e*f*g*h+8*e^2*h^2)))+b^2*(35*A*d^2*f^2*h^2*(d*f*g+d*e*h+c*f*h)+2*C*(12*c^3*f^3*h^3+10*c^2*d*f^2*h^2*(f*g+e*h)+c*d^2*f*h*(10*f^2*g^2+9*e*f*g*h+10*e^2*h^2)+2*d^3*(6*f^3*g^3+5*e*f^2*g^2*h+5*e^2*f*g*h^2+6*e^3*h^3))))*EllipticE(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(g+h*x)/(d^4*f^(7/2)*h^4*sqrt(e+f*x)*sqrt(d*(g+h*x)/(d*g-c*h)))+2/105*(35*a^2*d^2*f^2*h^2*(3*A*d*f*h^2+C*(c*h*(f*g-e*h)+d*g*(2*f*g+e*h)))-14*a*b*d*f*h*(15*A*d^2*f^2*g*h^2+C*(4*c^2*f*h^2*(f*g-e*h)+c*d*h*(3*f^2*g^2+e*f*g*h-4*e^2*h^2)+d^2*g*(8*f^2*g^2+3*e*f*g*h+4*e^2*h^2)))+b^2*(35*A*d^2*f^2*h^2*(c*h*(f*g-e*h)+d*g*(2*f*g+e*h))+C*(24*c^3*f^2*h^3*(f*g-e*h)+c^2*d*f*h^2*(17*f^2*g^2+6*e*f*g*h-23*e^2*h^2)+2*c*d^2*h*(8*f^3*g^3+e*f^2*g^2*h+3*e^2*f*g*h^2-12*e^3*h^3)+d^3*g*(48*f^3*g^3+16*e*f^2*g^2*h+17*e^2*f*g*h^2+24*e^3*h^3))))*EllipticF(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/(d^4*f^(7/2)*h^4*sqrt(e+f*x)*sqrt(g+h*x))],
[(a+b*x)*(A+C*x^2)/(sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,8,4/15*C*(a*d*f*h-2*b*(d*f*g+d*e*h+c*f*h))*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/(d^2*f^2*h^2)+2/5*C*(a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/(d*f*h)-2/15*(10*a*C*d*f*h*(d*f*g+d*e*h+c*f*h)-b*(15*A*d^2*f^2*h^2+C*(8*c^2*f^2*h^2+7*c*d*f*h*(f*g+e*h)+d^2*(8*f^2*g^2+7*e*f*g*h+8*e^2*h^2))))*EllipticE(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(g+h*x)/(d^3*f^(5/2)*h^3*sqrt(e+f*x)*sqrt(d*(g+h*x)/(d*g-c*h)))+2/15*(5*a*d*f*h*(3*A*d*f*h^2+C*(c*h*(f*g-e*h)+d*g*(2*f*g+e*h)))-b*(15*A*d^2*f^2*g*h^2+C*(4*c^2*f*h^2*(f*g-e*h)+c*d*h*(3*f^2*g^2+e*f*g*h-4*e^2*h^2)+d^2*g*(8*f^2*g^2+3*e*f*g*h+4*e^2*h^2))))*EllipticF(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/(d^3*f^(5/2)*h^3*sqrt(e+f*x)*sqrt(g+h*x))],
[(A+C*x^2)/(sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,7,2/3*C*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/(d*f*h)-4/3*C*(d*f*g+d*e*h+c*f*h)*EllipticE(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(g+h*x)/(d^2*f^(3/2)*h^2*sqrt(e+f*x)*sqrt(d*(g+h*x)/(d*g-c*h)))+2/3*(3*A*d*f*h^2+C*(c*h*(f*g-e*h)+d*g*(2*f*g+e*h)))*EllipticF(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/(d^2*f^(3/2)*h^2*sqrt(e+f*x)*sqrt(g+h*x))],
[(A+C*x^2)/((a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,11,2*C*EllipticE(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(g+h*x)/(b*d*h*sqrt(f)*sqrt(e+f*x)*sqrt(d*(g+h*x)/(d*g-c*h)))-2*C*(b*g+a*h)*EllipticF(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/(b^2*d*h*sqrt(f)*sqrt(e+f*x)*sqrt(g+h*x))-2*(A+a^2*C/b^2)*EllipticPi(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),-b*(d*e-c*f)/((b*c-a*d)*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/((b*c-a*d)*sqrt(f)*sqrt(e+f*x)*sqrt(g+h*x))],
[(A+C*x^2)/((a+b*x)^2*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,12,-(A*b^2+a^2*C)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)*(b*e-a*f)*(b*g-a*h)*(a+b*x))+(A*b+a^2*C/b)*EllipticE(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(f)*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(g+h*x)/((b*c-a*d)*(b*e-a*f)*(b*g-a*h)*sqrt(e+f*x)*sqrt(d*(g+h*x)/(d*g-c*h)))+(a^2*C*d*f-2*a*b*C*(d*e+c*f)+b^2*(2*c*C*e-A*d*f))*EllipticF(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/(b^2*d*(b*c-a*d)*(b*e-a*f)*sqrt(f)*sqrt(e+f*x)*sqrt(g+h*x))-(a^4*C*d*f*h-A*b^4*(d*e*g+c*f*g+c*e*h)-2*a^3*b*C*(d*f*g+d*e*h+c*f*h)-2*a*b^3*(2*c*C*e*g-A*d*f*g-A*d*e*h-A*c*f*h)-3*a^2*b^2*(A*d*f*h-C*(d*e*g+c*f*g+c*e*h)))*EllipticPi(sqrt(f)*sqrt(c+d*x)/sqrt(-d*e+c*f),-b*(d*e-c*f)/((b*c-a*d)*f),sqrt((d*e-c*f)*h/(f*(d*g-c*h))))*sqrt(-d*e+c*f)*sqrt(d*(e+f*x)/(d*e-c*f))*sqrt(d*(g+h*x)/(d*g-c*h))/(b^2*(b*c-a*d)^2*(b*e-a*f)*(b*g-a*h)*sqrt(f)*sqrt(e+f*x)*sqrt(g+h*x))],

# Integrands of the form (a+b x)^(m/2) (c+d x)^(n/2) (A+B x+C x^2) / (Sqrte+f x] Sqrt[g+h x])

# n>0

# n<0
[(a+b*x)^(3/2)*(A+C*x^2)/(sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,10,-1/24*(4*b*d*f*h*(C*(b*(d*e*g+c*f*g+c*e*h)+a*(d*f*g+d*e*h+c*f*h))*(3*a*d*f*h-5*b*(d*f*g+d*e*h+c*f*h))+2*d*f*h*(3*b^2*c*C*e*g+2*a^2*C*(d*f*g+d*e*h+c*f*h)-a*b*(12*A*d*f*h-5*C*(d*e*g+c*f*g+c*e*h))))+(a*d*f*h+b*(d*f*g+d*e*h+c*f*h))*(C*(3*a*d*f*h-5*b*(d*f*g+d*e*h+c*f*h))*(a*d*f*h-3*b*(d*f*g+d*e*h+c*f*h))+8*b*d*f*h*(3*A*b*d*f*h-C*(2*b*(d*e*g+c*f*g+c*e*h)+a*(d*f*g+d*e*h+c*f*h)))))*(a+b*x)*EllipticPi(sqrt(b*c-a*d)*sqrt(g+h*x)/(sqrt(-d*g+c*h)*sqrt(a+b*x)),-b*(d*g-c*h)/((b*c-a*d)*h),sqrt((b*e-a*f)*(d*g-c*h)/((b*c-a*d)*(f*g-e*h))))*sqrt(-d*g+c*h)*sqrt((b*g-a*h)*(c+d*x)/((d*g-c*h)*(a+b*x)))*sqrt((b*g-a*h)*(e+f*x)/((f*g-e*h)*(a+b*x)))/(b^2*d^3*f^3*h^4*sqrt(b*c-a*d)*sqrt(c+d*x)*sqrt(e+f*x))+1/24*(C*(3*a*d*f*h-5*b*(d*f*g+d*e*h+c*f*h))*(a*d*f*h-3*b*(d*f*g+d*e*h+c*f*h))+8*b*d*f*h*(3*A*b*d*f*h-C*(2*b*(d*e*g+c*f*g+c*e*h)+a*(d*f*g+d*e*h+c*f*h))))*sqrt(a+b*x)*sqrt(e+f*x)*sqrt(g+h*x)/(b*d^2*f^3*h^3*sqrt(c+d*x))+1/3*C*(a+b*x)^(3/2)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/(d*f*h)+1/12*C*(3*a*d*f*h-5*b*(d*f*g+d*e*h+c*f*h))*sqrt(a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/(d^2*f^2*h^2)+1/24*(b*e-a*f)*(3*a^2*C*d^2*f^2*h^2+6*a*b*C*d*f*h*(c*f*h+2*d*(f*g+e*h))-b^2*(24*A*d^2*f^2*h^2+C*(5*c^2*f^2*h^2+4*c*d*f*h*(f*g+e*h)+d^2*(15*f^2*g^2+14*e*f*g*h+15*e^2*h^2))))*EllipticF(sqrt(b*g-a*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(a+b*x)),sqrt(-(b*c-a*d)*(f*g-e*h)/((d*e-c*f)*(b*g-a*h))))*sqrt(b*g-a*h)*sqrt((b*e-a*f)*(c+d*x)/((d*e-c*f)*(a+b*x)))*sqrt(g+h*x)/(b^2*d^2*f^3*h^3*sqrt(f*g-e*h)*sqrt(c+d*x)*sqrt(-(b*e-a*f)*(g+h*x)/((f*g-e*h)*(a+b*x))))-1/24*(C*(3*a*d*f*h-5*b*(d*f*g+d*e*h+c*f*h))*(a*d*f*h-3*b*(d*f*g+d*e*h+c*f*h))+8*b*d*f*h*(3*A*b*d*f*h-C*(2*b*(d*e*g+c*f*g+c*e*h)+a*(d*f*g+d*e*h+c*f*h))))*EllipticE(sqrt(d*g-c*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(c+d*x)),sqrt((b*c-a*d)*(f*g-e*h)/((b*e-a*f)*(d*g-c*h))))*sqrt(d*g-c*h)*sqrt(f*g-e*h)*sqrt(a+b*x)*sqrt(-(d*e-c*f)*(g+h*x)/((f*g-e*h)*(c+d*x)))/(b*d^3*f^3*h^3*sqrt((d*e-c*f)*(a+b*x)/((b*e-a*f)*(c+d*x)))*sqrt(g+h*x))],
[(a+b*x)^(1/2)*(A+C*x^2)/(sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,9,-1/4*(C*(a*d*f*h-3*b*(d*f*g+d*e*h+c*f*h))*(a*d*f*h+b*(d*f*g+d*e*h+c*f*h))-4*b*d*f*h*(2*A*b*d*f*h-C*(b*(d*e*g+c*f*g+c*e*h)+a*(d*f*g+d*e*h+c*f*h))))*(a+b*x)*EllipticPi(sqrt(b*c-a*d)*sqrt(g+h*x)/(sqrt(-d*g+c*h)*sqrt(a+b*x)),-b*(d*g-c*h)/((b*c-a*d)*h),sqrt((b*e-a*f)*(d*g-c*h)/((b*c-a*d)*(f*g-e*h))))*sqrt(-d*g+c*h)*sqrt((b*g-a*h)*(c+d*x)/((d*g-c*h)*(a+b*x)))*sqrt((b*g-a*h)*(e+f*x)/((f*g-e*h)*(a+b*x)))/(b^2*d^2*f^2*h^3*sqrt(b*c-a*d)*sqrt(c+d*x)*sqrt(e+f*x))+1/4*C*(a*d*f*h-3*b*(d*f*g+d*e*h+c*f*h))*sqrt(a+b*x)*sqrt(e+f*x)*sqrt(g+h*x)/(b*d*f^2*h^2*sqrt(c+d*x))+1/2*C*sqrt(a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/(d*f*h)+1/4*C*(b*e-a*f)*(a*d*f*h+b*(c*f*h+3*d*(f*g+e*h)))*EllipticF(sqrt(b*g-a*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(a+b*x)),sqrt(-(b*c-a*d)*(f*g-e*h)/((d*e-c*f)*(b*g-a*h))))*sqrt(b*g-a*h)*sqrt((b*e-a*f)*(c+d*x)/((d*e-c*f)*(a+b*x)))*sqrt(g+h*x)/(b^2*d*f^2*h^2*sqrt(f*g-e*h)*sqrt(c+d*x)*sqrt(-(b*e-a*f)*(g+h*x)/((f*g-e*h)*(a+b*x))))-1/4*C*(a*d*f*h-3*b*(d*f*g+d*e*h+c*f*h))*EllipticE(sqrt(d*g-c*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(c+d*x)),sqrt((b*c-a*d)*(f*g-e*h)/((b*e-a*f)*(d*g-c*h))))*sqrt(d*g-c*h)*sqrt(f*g-e*h)*sqrt(a+b*x)*sqrt(-(d*e-c*f)*(g+h*x)/((f*g-e*h)*(c+d*x)))/(b*d^2*f^2*h^2*sqrt((d*e-c*f)*(a+b*x)/((b*e-a*f)*(c+d*x)))*sqrt(g+h*x))],
[(A+C*x^2)/((a+b*x)^(1/2)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,8,-C*(a*d*f*h+b*(d*f*g+d*e*h+c*f*h))*(a+b*x)*EllipticPi(sqrt(b*c-a*d)*sqrt(g+h*x)/(sqrt(-d*g+c*h)*sqrt(a+b*x)),-b*(d*g-c*h)/((b*c-a*d)*h),sqrt((b*e-a*f)*(d*g-c*h)/((b*c-a*d)*(f*g-e*h))))*sqrt(-d*g+c*h)*sqrt((b*g-a*h)*(c+d*x)/((d*g-c*h)*(a+b*x)))*sqrt((b*g-a*h)*(e+f*x)/((f*g-e*h)*(a+b*x)))/(b^2*d*f*h^2*sqrt(b*c-a*d)*sqrt(c+d*x)*sqrt(e+f*x))+C*sqrt(a+b*x)*sqrt(e+f*x)*sqrt(g+h*x)/(b*f*h*sqrt(c+d*x))+(a^2*C*f*h+a*b*C*(f*g+e*h)-b^2*(C*e*g-2*A*f*h))*EllipticF(sqrt(b*g-a*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(a+b*x)),sqrt(-(b*c-a*d)*(f*g-e*h)/((d*e-c*f)*(b*g-a*h))))*sqrt((b*e-a*f)*(c+d*x)/((d*e-c*f)*(a+b*x)))*sqrt(g+h*x)/(b^2*f*h*sqrt(b*g-a*h)*sqrt(f*g-e*h)*sqrt(c+d*x)*sqrt(-(b*e-a*f)*(g+h*x)/((f*g-e*h)*(a+b*x))))-C*EllipticE(sqrt(d*g-c*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(c+d*x)),sqrt((b*c-a*d)*(f*g-e*h)/((b*e-a*f)*(d*g-c*h))))*sqrt(d*g-c*h)*sqrt(f*g-e*h)*sqrt(a+b*x)*sqrt(-(d*e-c*f)*(g+h*x)/((f*g-e*h)*(c+d*x)))/(b*d*f*h*sqrt((d*e-c*f)*(a+b*x)/((b*e-a*f)*(c+d*x)))*sqrt(g+h*x))],
[(A+C*x^2)/((a+b*x)^(3/2)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,9,2*C*(a+b*x)*EllipticPi(sqrt(b*c-a*d)*sqrt(g+h*x)/(sqrt(-d*g+c*h)*sqrt(a+b*x)),-b*(d*g-c*h)/((b*c-a*d)*h),sqrt((b*e-a*f)*(d*g-c*h)/((b*c-a*d)*(f*g-e*h))))*sqrt(-d*g+c*h)*sqrt((b*g-a*h)*(c+d*x)/((d*g-c*h)*(a+b*x)))*sqrt((b*g-a*h)*(e+f*x)/((f*g-e*h)*(a+b*x)))/(b^2*h*sqrt(b*c-a*d)*sqrt(c+d*x)*sqrt(e+f*x))+2*(A*b^2+a^2*C)*d*sqrt(a+b*x)*sqrt(e+f*x)*sqrt(g+h*x)/(b*(b*c-a*d)*(b*e-a*f)*(b*g-a*h)*sqrt(c+d*x))-2*(A*b^2+a^2*C)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)*(b*e-a*f)*(b*g-a*h)*sqrt(a+b*x))-2*(2*a*b*c*C+A*b^2*d-a^2*C*d)*EllipticF(sqrt(b*g-a*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(a+b*x)),sqrt(-(b*c-a*d)*(f*g-e*h)/((d*e-c*f)*(b*g-a*h))))*sqrt((b*e-a*f)*(c+d*x)/((d*e-c*f)*(a+b*x)))*sqrt(g+h*x)/(b^2*(b*c-a*d)*sqrt(b*g-a*h)*sqrt(f*g-e*h)*sqrt(c+d*x)*sqrt(-(b*e-a*f)*(g+h*x)/((f*g-e*h)*(a+b*x))))-2*(A*b^2+a^2*C)*EllipticE(sqrt(d*g-c*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(c+d*x)),sqrt((b*c-a*d)*(f*g-e*h)/((b*e-a*f)*(d*g-c*h))))*sqrt(d*g-c*h)*sqrt(f*g-e*h)*sqrt(a+b*x)*sqrt(-(d*e-c*f)*(g+h*x)/((f*g-e*h)*(c+d*x)))/(b*(b*c-a*d)*(b*e-a*f)*(b*g-a*h)*sqrt((d*e-c*f)*(a+b*x)/((b*e-a*f)*(c+d*x)))*sqrt(g+h*x))],
[(A+C*x^2)/((a+b*x)^(5/2)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)),x,8,-4/3*d*(A*b^3*(d*e*g+c*f*g+c*e*h)+a^3*C*(d*f*g+d*e*h+c*f*h)+a^2*b*(3*A*d*f*h-2*C*(d*e*g+c*f*g+c*e*h))-a*b^2*(2*A*d*(f*g+e*h)-c*(3*C*e*g-2*A*f*h)))*sqrt(a+b*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)^2*(b*e-a*f)^2*(b*g-a*h)^2*sqrt(c+d*x))-2/3*(A*b^2+a^2*C)*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)*(b*e-a*f)*(b*g-a*h)*(a+b*x)^(3/2))+4/3*b*(A*b^3*(d*e*g+c*f*g+c*e*h)+a^3*C*(d*f*g+d*e*h+c*f*h)+a^2*b*(3*A*d*f*h-2*C*(d*e*g+c*f*g+c*e*h))-a*b^2*(2*A*d*(f*g+e*h)-c*(3*C*e*g-2*A*f*h)))*sqrt(c+d*x)*sqrt(e+f*x)*sqrt(g+h*x)/((b*c-a*d)^2*(b*e-a*f)^2*(b*g-a*h)^2*sqrt(a+b*x))-2/3*(3*a*b*(c^2*C+A*d^2)*(f*g+e*h)-b^2*(2*A*d^2*e*g+A*c*d*(f*g+e*h)+c^2*(3*C*e*g-A*f*h))-a^2*(3*A*d^2*f*h-C*(d^2*e*g-c*d*f*g-c*d*e*h-2*c^2*f*h)))*EllipticF(sqrt(b*g-a*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(a+b*x)),sqrt(-(b*c-a*d)*(f*g-e*h)/((d*e-c*f)*(b*g-a*h))))*sqrt((b*e-a*f)*(c+d*x)/((d*e-c*f)*(a+b*x)))*sqrt(g+h*x)/((b*c-a*d)^2*(b*e-a*f)*(b*g-a*h)^(3/2)*sqrt(f*g-e*h)*sqrt(c+d*x)*sqrt(-(b*e-a*f)*(g+h*x)/((f*g-e*h)*(a+b*x))))+4/3*(A*b^3*(d*e*g+c*f*g+c*e*h)+a^3*C*(d*f*g+d*e*h+c*f*h)+a^2*b*(3*A*d*f*h-2*C*(d*e*g+c*f*g+c*e*h))-a*b^2*(2*A*d*(f*g+e*h)-c*(3*C*e*g-2*A*f*h)))*EllipticE(sqrt(d*g-c*h)*sqrt(e+f*x)/(sqrt(f*g-e*h)*sqrt(c+d*x)),sqrt((b*c-a*d)*(f*g-e*h)/((b*e-a*f)*(d*g-c*h))))*sqrt(d*g-c*h)*sqrt(f*g-e*h)*sqrt(a+b*x)*sqrt(-(d*e-c*f)*(g+h*x)/((f*g-e*h)*(c+d*x)))/((b*c-a*d)^2*(b*e-a*f)^2*(b*g-a*h)^2*sqrt((d*e-c*f)*(a+b*x)/((b*e-a*f)*(c+d*x)))*sqrt(g+h*x))]]:
