# Maple integration test file: "1 Algebraic functions\1.1 Binomial products\1.1.1 Linear\1.1.1.6 P(x) (a+b x)^m (c+d x)^n (e+f x)^p.txt"

lst:=[

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

# Integrands of the form (a+b x)^m (c+d x)^n (e+f x)^p (A+B x+C x^2) when b c+a d=0 and m=n

# Integrands of the form (a+b x)^(m/2) (c+d x)^(n/2) (e+f x)^p (A+B x+C x^2) when b c+a d=0, m=n, a>0 and c>0

# m>0
[(e+f*x)^3*(A+B*x+C*x^2)*sqrt(1-d*x)*sqrt(1+d*x),x,7,-1/70*(7*d^2*f*(B*e+2*A*f)-C*(3*d^2*e^2-8*f^2))*(e+f*x)^2*(1-d^2*x^2)^(3/2)/(d^4*f)+1/42*(3*C*e-7*B*f)*(e+f*x)^3*(1-d^2*x^2)^(3/2)/(d^2*f)-1/7*C*(e+f*x)^4*(1-d^2*x^2)^(3/2)/(d^2*f)+1/840*(8*(C*(3*d^4*e^4-30*d^2*e^2*f^2-8*f^4)-7*d^2*f*(2*A*f*(6*d^2*e^2+f^2)+B*(d^2*e^3+6*e*f^2)))+3*d^2*f*(6*C*d^2*e^3-14*B*d^2*e^2*f-41*C*e*f^2-98*A*d^2*e*f^2-35*B*f^3)*x)*(1-d^2*x^2)^(3/2)/(d^6*f)+1/16*(2*C*d^2*e^3+8*A*d^4*e^3+6*B*d^2*e^2*f+3*C*e*f^2+6*A*d^2*e*f^2+B*f^3)*arcsin(d*x)/d^5+1/16*(2*C*d^2*e^3+8*A*d^4*e^3+6*B*d^2*e^2*f+3*C*e*f^2+6*A*d^2*e*f^2+B*f^3)*x*sqrt(1-d^2*x^2)/d^4],
[(e+f*x)^2*(A+B*x+C*x^2)*sqrt(1-d*x)*sqrt(1+d*x),x,6,1/10*(C*e-2*B*f)*(e+f*x)^2*(1-d^2*x^2)^(3/2)/(d^2*f)-1/6*C*(e+f*x)^3*(1-d^2*x^2)^(3/2)/(d^2*f)+1/120*(8*(C*(d^2*e^3-4*e*f^2)-2*f*(5*A*d^2*e*f+B*(d^2*e^2+f^2)))-3*f*(5*(C+2*A*d^2)*f^2-2*d^2*e*(C*e-2*B*f))*x)*(1-d^2*x^2)^(3/2)/(d^4*f)+1/16*(C*(2*d^2*e^2+f^2)+2*d^2*(2*B*e*f+A*(4*d^2*e^2+f^2)))*arcsin(d*x)/d^5+1/16*(C*(2*d^2*e^2+f^2)+2*d^2*(2*B*e*f+A*(4*d^2*e^2+f^2)))*x*sqrt(1-d^2*x^2)/d^4],
[(e+f*x)*(A+B*x+C*x^2)*sqrt(1-d*x)*sqrt(1+d*x),x,5,-1/5*C*(e+f*x)^2*(1-d^2*x^2)^(3/2)/(d^2*f)-1/60*(4*(5*d^2*f*(B*e+A*f)-C*(3*d^2*e^2-2*f^2))-3*d^2*f*(3*C*e-5*B*f)*x)*(1-d^2*x^2)^(3/2)/(d^4*f)+1/8*(C*e+4*A*d^2*e+B*f)*arcsin(d*x)/d^3+1/8*(C*e+4*A*d^2*e+B*f)*x*sqrt(1-d^2*x^2)/d^2],
[(A+B*x+C*x^2)*sqrt(1-d*x)*sqrt(1+d*x),x,5,-1/3*B*(1-d^2*x^2)^(3/2)/d^2-1/4*C*x*(1-d^2*x^2)^(3/2)/d^2+1/8*(C+4*A*d^2)*arcsin(d*x)/d^3+1/8*(C+4*A*d^2)*x*sqrt(1-d^2*x^2)/d^2],
[(A+B*x+C*x^2)/((e+f*x)*sqrt(1-d*x)*sqrt(1+d*x)),x,6,-(C*e-B*f)*arcsin(d*x)/(d*f^2)+(C*e^2-B*e*f+A*f^2)*arctan((f+d^2*e*x)/(sqrt(d^2*e^2-f^2)*sqrt(1-d^2*x^2)))/(f^2*sqrt(d^2*e^2-f^2))-C*sqrt(1-d^2*x^2)/(d^2*f)],
[(A+B*x+C*x^2)/((e+f*x)^2*sqrt(1-d*x)*sqrt(1+d*x)),x,6,C*arcsin(d*x)/(d*f^2)-(C*d^2*e^3-2*C*e*f^2-A*d^2*e*f^2+B*f^3)*arctan((f+d^2*e*x)/(sqrt(d^2*e^2-f^2)*sqrt(1-d^2*x^2)))/(f^2*(d^2*e^2-f^2)^(3/2))+(C*e^2-B*e*f+A*f^2)*sqrt(1-d^2*x^2)/(f*(d^2*e^2-f^2)*(e+f*x))],
[(A+B*x+C*x^2)/((e+f*x)^3*sqrt(1-d*x)*sqrt(1+d*x)),x,5,1/2*(C*(d^2*e^2+2*f^2)-d^2*(3*B*e*f-A*(2*d^2*e^2+f^2)))*arctan((f+d^2*e*x)/(sqrt(d^2*e^2-f^2)*sqrt(1-d^2*x^2)))/(d^2*e^2-f^2)^(5/2)+1/2*(C*e^2-B*e*f+A*f^2)*sqrt(1-d^2*x^2)/(f*(d^2*e^2-f^2)*(e+f*x)^2)-1/2*(C*d^2*e^3+B*d^2*e^2*f-4*C*e*f^2-3*A*d^2*e*f^2+2*B*f^3)*sqrt(1-d^2*x^2)/(f*(d^2*e^2-f^2)^2*(e+f*x))],

# m<0
[(e+f*x)^3*(A+B*x+C*x^2)/(sqrt(1-d*x)*sqrt(1+d*x)),x,6,1/8*(4*C*d^2*e^3+8*A*d^4*e^3+12*B*d^2*e^2*f+9*C*e*f^2+12*A*d^2*e*f^2+3*B*f^3)*arcsin(d*x)/d^5-1/60*(4*(4*C+5*A*d^2)*f^2-3*d^2*e*(C*e-5*B*f))*(e+f*x)^2*sqrt(1-d^2*x^2)/(d^4*f)+1/20*(C*e-5*B*f)*(e+f*x)^3*sqrt(1-d^2*x^2)/(d^2*f)-1/5*C*(e+f*x)^4*sqrt(1-d^2*x^2)/(d^2*f)+1/120*(4*(C*(3*d^4*e^4-52*d^2*e^2*f^2-16*f^4)-5*d^2*f*(4*A*f*(4*d^2*e^2+f^2)+3*B*(d^2*e^3+4*e*f^2)))+d^2*f*(6*C*d^2*e^3-30*B*d^2*e^2*f-71*C*e*f^2-100*A*d^2*e*f^2-45*B*f^3)*x)*sqrt(1-d^2*x^2)/(d^6*f)],
[(e+f*x)^2*(A+B*x+C*x^2)/(sqrt(1-d*x)*sqrt(1+d*x)),x,5,1/8*(C*(4*d^2*e^2+3*f^2)+4*d^2*(2*B*e*f+A*(2*d^2*e^2+f^2)))*arcsin(d*x)/d^5+1/12*(C*e-4*B*f)*(e+f*x)^2*sqrt(1-d^2*x^2)/(d^2*f)-1/4*C*(e+f*x)^3*sqrt(1-d^2*x^2)/(d^2*f)+1/24*(4*(C*(d^2*e^3-8*e*f^2)-4*f*(3*A*d^2*e*f+B*(d^2*e^2+f^2)))-f*(3*(3*C+4*A*d^2)*f^2-2*d^2*e*(C*e-4*B*f))*x)*sqrt(1-d^2*x^2)/(d^4*f)],
[(e+f*x)*(A+B*x+C*x^2)/(sqrt(1-d*x)*sqrt(1+d*x)),x,4,1/2*(C*e+2*A*d^2*e+B*f)*arcsin(d*x)/d^3-1/3*C*(e+f*x)^2*sqrt(1-d^2*x^2)/(d^2*f)-1/6*(2*(3*d^2*f*(B*e+A*f)-C*(d^2*e^2-2*f^2))-d^2*f*(C*e-3*B*f)*x)*sqrt(1-d^2*x^2)/(d^4*f)],
[(A+B*x+C*x^2)/(sqrt(1-d*x)*sqrt(1+d*x)),x,4,1/2*(C+2*A*d^2)*arcsin(d*x)/d^3-B*sqrt(1-d^2*x^2)/d^2-1/2*C*x*sqrt(1-d^2*x^2)/d^2],
[(A+B*x+C*x^2)/((e+f*x)*sqrt(1-d*x)*sqrt(1+d*x)),x,6,-(C*e-B*f)*arcsin(d*x)/(d*f^2)+(C*e^2-B*e*f+A*f^2)*arctan((f+d^2*e*x)/(sqrt(d^2*e^2-f^2)*sqrt(1-d^2*x^2)))/(f^2*sqrt(d^2*e^2-f^2))-C*sqrt(1-d^2*x^2)/(d^2*f)],
[(A+B*x+C*x^2)/((e+f*x)^2*sqrt(1-d*x)*sqrt(1+d*x)),x,6,C*arcsin(d*x)/(d*f^2)-(C*d^2*e^3-2*C*e*f^2-A*d^2*e*f^2+B*f^3)*arctan((f+d^2*e*x)/(sqrt(d^2*e^2-f^2)*sqrt(1-d^2*x^2)))/(f^2*(d^2*e^2-f^2)^(3/2))+(C*e^2-B*e*f+A*f^2)*sqrt(1-d^2*x^2)/(f*(d^2*e^2-f^2)*(e+f*x))],
[(A+B*x+C*x^2)/((e+f*x)^3*sqrt(1-d*x)*sqrt(1+d*x)),x,5,1/2*(C*(d^2*e^2+2*f^2)-d^2*(3*B*e*f-A*(2*d^2*e^2+f^2)))*arctan((f+d^2*e*x)/(sqrt(d^2*e^2-f^2)*sqrt(1-d^2*x^2)))/(d^2*e^2-f^2)^(5/2)+1/2*(C*e^2-B*e*f+A*f^2)*sqrt(1-d^2*x^2)/(f*(d^2*e^2-f^2)*(e+f*x)^2)-1/2*(C*d^2*e^3+B*d^2*e^2*f-4*C*e*f^2-3*A*d^2*e*f^2+2*B*f^3)*sqrt(1-d^2*x^2)/(f*(d^2*e^2-f^2)^2*(e+f*x))],
[x*(a+b*x+c*x^2)/(sqrt(1-d*x)*sqrt(1+d*x)),x,4,1/2*b*arcsin(d*x)/d^3-1/3*c*x^2*sqrt(1-d^2*x^2)/d^2-1/6*(2*(2*c+3*a*d^2)+3*b*d^2*x)*sqrt(1-d^2*x^2)/d^4],
[(a+b*x+c*x^2)/(sqrt(1-d*x)*sqrt(1+d*x)),x,4,1/2*(c+2*a*d^2)*arcsin(d*x)/d^3-b*sqrt(1-d^2*x^2)/d^2-1/2*c*x*sqrt(1-d^2*x^2)/d^2],
[(a+b*x+c*x^2)/(x*sqrt(1-d*x)*sqrt(1+d*x)),x,7,b*arcsin(d*x)/d-a*arctanh(sqrt(1-d^2*x^2))-c*sqrt(1-d^2*x^2)/d^2],
[(a+b*x+c*x^2)/(x^2*sqrt(1-d*x)*sqrt(1+d*x)),x,7,c*arcsin(d*x)/d-b*arctanh(sqrt(1-d^2*x^2))-a*sqrt(1-d^2*x^2)/x],
[(a+b*x+c*x^2)/(x^3*sqrt(1-d*x)*sqrt(1+d*x)),x,6,-1/2*(2*c+a*d^2)*arctanh(sqrt(1-d^2*x^2))-1/2*a*sqrt(1-d^2*x^2)/x^2-b*sqrt(1-d^2*x^2)/x],

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

# m>0
[(e+f*x)^3*(A+B*x+C*x^2)*sqrt(a+b*x)*sqrt(a*c-b*c*x),x,8,1/16*(A*(8*b^4*e^3+6*a^2*b^2*e*f^2)+a^2*(a^2*f^2*(3*C*e+B*f)+2*b^2*e^2*(C*e+3*B*f)))*x*sqrt(a+b*x)*sqrt(a*c-b*c*x)/b^4-1/70*(8*a^2*C*f^2-b^2*(3*C*e^2-7*f*(B*e+2*A*f)))*(e+f*x)^2*(a^2-b^2*x^2)*sqrt(a+b*x)*sqrt(a*c-b*c*x)/(b^4*f)+1/42*(3*C*e-7*B*f)*(e+f*x)^3*(a^2-b^2*x^2)*sqrt(a+b*x)*sqrt(a*c-b*c*x)/(b^2*f)-1/7*C*(e+f*x)^4*(a^2-b^2*x^2)*sqrt(a+b*x)*sqrt(a*c-b*c*x)/(b^2*f)-1/840*(8*(8*a^4*C*f^4+2*a^2*b^2*f^2*(15*C*e^2+7*f*(3*B*e+A*f))-b^4*e^2*(3*C*e^2-7*f*(B*e+12*A*f)))+3*b^2*f*(a^2*f^2*(41*C*e+35*B*f)-2*b^2*e*(3*C*e^2-7*f*(B*e+7*A*f)))*x)*(a^2-b^2*x^2)*sqrt(a+b*x)*sqrt(a*c-b*c*x)/(b^6*f)+1/16*a^2*(A*(8*b^4*e^3+6*a^2*b^2*e*f^2)+a^2*(a^2*f^2*(3*C*e+B*f)+2*b^2*e^2*(C*e+3*B*f)))*arctan(b*x*sqrt(c)/sqrt(a^2*c-b^2*c*x^2))*sqrt(c)*sqrt(a+b*x)*sqrt(a*c-b*c*x)/(b^5*sqrt(a^2*c-b^2*c*x^2))],
[(e+f*x)^2*(A+B*x+C*x^2)*sqrt(a+b*x)*sqrt(a*c-b*c*x),x,7,1/16*(2*A*(4*b^4*e^2+a^2*b^2*f^2)+a^2*(a^2*C*f^2+2*b^2*e*(C*e+2*B*f)))*x*sqrt(a+b*x)*sqrt(a*c-b*c*x)/b^4+1/10*(C*e-2*B*f)*(e+f*x)^2*(a^2-b^2*x^2)*sqrt(a+b*x)*sqrt(a*c-b*c*x)/(b^2*f)-1/6*C*(e+f*x)^3*(a^2-b^2*x^2)*sqrt(a+b*x)*sqrt(a*c-b*c*x)/(b^2*f)-1/120*(8*(2*a^2*f^2*(2*C*e+B*f)-b^2*e*(C*e^2-2*f*(B*e+5*A*f)))+3*f*(5*a^2*C*f^2-b^2*(2*C*e^2-2*f*(2*B*e+5*A*f)))*x)*(a^2-b^2*x^2)*sqrt(a+b*x)*sqrt(a*c-b*c*x)/(b^4*f)+1/16*a^2*(2*A*(4*b^4*e^2+a^2*b^2*f^2)+a^2*(a^2*C*f^2+2*b^2*e*(C*e+2*B*f)))*arctan(b*x*sqrt(c)/sqrt(a^2*c-b^2*c*x^2))*sqrt(c)*sqrt(a+b*x)*sqrt(a*c-b*c*x)/(b^5*sqrt(a^2*c-b^2*c*x^2))],
[(e+f*x)*(A+B*x+C*x^2)*sqrt(a+b*x)*sqrt(a*c-b*c*x),x,6,1/8*(4*A*b^2*e+a^2*(C*e+B*f))*x*sqrt(a+b*x)*sqrt(a*c-b*c*x)/b^2-1/5*C*(e+f*x)^2*(a^2-b^2*x^2)*sqrt(a+b*x)*sqrt(a*c-b*c*x)/(b^2*f)-1/60*(4*(2*a^2*C*f^2-b^2*(3*C*e^2-5*f*(B*e+A*f)))-3*b^2*f*(3*C*e-5*B*f)*x)*(a^2-b^2*x^2)*sqrt(a+b*x)*sqrt(a*c-b*c*x)/(b^4*f)+1/8*a^2*(4*A*b^2*e+a^2*(C*e+B*f))*arctan(b*x*sqrt(c)/sqrt(a^2*c-b^2*c*x^2))*sqrt(c)*sqrt(a+b*x)*sqrt(a*c-b*c*x)/(b^3*sqrt(a^2*c-b^2*c*x^2))],
[(A+B*x+C*x^2)*sqrt(a+b*x)*sqrt(a*c-b*c*x),x,6,1/8*(4*A+a^2*C/b^2)*x*sqrt(a+b*x)*sqrt(a*c-b*c*x)-1/3*B*(a^2-b^2*x^2)*sqrt(a+b*x)*sqrt(a*c-b*c*x)/b^2-1/4*C*x*(a^2-b^2*x^2)*sqrt(a+b*x)*sqrt(a*c-b*c*x)/b^2+1/8*a^2*(4*A*b^2+a^2*C)*arctan(b*x*sqrt(c)/sqrt(a^2*c-b^2*c*x^2))*sqrt(c)*sqrt(a+b*x)*sqrt(a*c-b*c*x)/(b^3*sqrt(a^2*c-b^2*c*x^2))],
[(A+B*x+C*x^2)/((e+f*x)*sqrt(a+b*x)*sqrt(a*c-b*c*x)),x,7,-C*(a^2-b^2*x^2)/(b^2*f*sqrt(a+b*x)*sqrt(a*c-b*c*x))-(C*e-B*f)*arctan(b*x*sqrt(c)/sqrt(a^2*c-b^2*c*x^2))*sqrt(a^2*c-b^2*c*x^2)/(b*f^2*sqrt(c)*sqrt(a+b*x)*sqrt(a*c-b*c*x))+(C*e^2-B*e*f+A*f^2)*arctan((a^2*f+b^2*e*x)*sqrt(c)/(sqrt(b^2*e^2-a^2*f^2)*sqrt(a^2*c-b^2*c*x^2)))*sqrt(a^2*c-b^2*c*x^2)/(f^2*sqrt(c)*sqrt(b^2*e^2-a^2*f^2)*sqrt(a+b*x)*sqrt(a*c-b*c*x))],
[(A+B*x+C*x^2)/((e+f*x)^2*sqrt(a+b*x)*sqrt(a*c-b*c*x)),x,7,f*(A+e*(C*e-B*f)/f^2)*(a^2-b^2*x^2)/((b^2*e^2-a^2*f^2)*(e+f*x)*sqrt(a+b*x)*sqrt(a*c-b*c*x))+C*arctan(b*x*sqrt(c)/sqrt(a^2*c-b^2*c*x^2))*sqrt(a^2*c-b^2*c*x^2)/(b*f^2*sqrt(c)*sqrt(a+b*x)*sqrt(a*c-b*c*x))+(a^2*f^2*(2*C*e-B*f)-b^2*(C*e^3-A*e*f^2))*arctan((a^2*f+b^2*e*x)*sqrt(c)/(sqrt(b^2*e^2-a^2*f^2)*sqrt(a^2*c-b^2*c*x^2)))*sqrt(a^2*c-b^2*c*x^2)/(f^2*(b^2*e^2-a^2*f^2)^(3/2)*sqrt(c)*sqrt(a+b*x)*sqrt(a*c-b*c*x))],
[(A+B*x+C*x^2)/((e+f*x)^3*sqrt(a+b*x)*sqrt(a*c-b*c*x)),x,5,1/2*f*(A+e*(C*e-B*f)/f^2)*(a^2-b^2*x^2)/((b^2*e^2-a^2*f^2)*(e+f*x)^2*sqrt(a+b*x)*sqrt(a*c-b*c*x))+1/2*(2*a^2*f^2*(2*C*e-B*f)-b^2*e*(C*e^2+f*(B*e-3*A*f)))*(a^2-b^2*x^2)/(f*(b^2*e^2-a^2*f^2)^2*(e+f*x)*sqrt(a+b*x)*sqrt(a*c-b*c*x))+1/2*(A*(2*b^4*e^2+a^2*b^2*f^2)+a^2*(2*a^2*C*f^2+b^2*e*(C*e-3*B*f)))*arctan((a^2*f+b^2*e*x)*sqrt(c)/(sqrt(b^2*e^2-a^2*f^2)*sqrt(a^2*c-b^2*c*x^2)))*sqrt(a^2*c-b^2*c*x^2)/((b^2*e^2-a^2*f^2)^(5/2)*sqrt(c)*sqrt(a+b*x)*sqrt(a*c-b*c*x))],

# m<0
[(e+f*x)^3*(A+B*x+C*x^2)/(sqrt(a+b*x)*sqrt(a*c-b*c*x)),x,7,-1/60*(16*a^2*C*f^2-b^2*(3*C*e^2-5*f*(3*B*e+4*A*f)))*(e+f*x)^2*(a^2-b^2*x^2)/(b^4*f*sqrt(a+b*x)*sqrt(a*c-b*c*x))+1/20*(C*e-5*B*f)*(e+f*x)^3*(a^2-b^2*x^2)/(b^2*f*sqrt(a+b*x)*sqrt(a*c-b*c*x))-1/5*C*(e+f*x)^4*(a^2-b^2*x^2)/(b^2*f*sqrt(a+b*x)*sqrt(a*c-b*c*x))-1/120*(4*(16*a^4*C*f^4+4*a^2*b^2*f^2*(13*C*e^2+5*f*(3*B*e+A*f))-b^4*e^2*(3*C*e^2-5*f*(3*B*e+16*A*f)))+b^2*f*(a^2*f^2*(71*C*e+45*B*f)-2*b^2*e*(3*C*e^2-5*f*(3*B*e+10*A*f)))*x)*(a^2-b^2*x^2)/(b^6*f*sqrt(a+b*x)*sqrt(a*c-b*c*x))+1/8*(4*A*(2*b^4*e^3+3*a^2*b^2*e*f^2)+a^2*(3*a^2*f^2*(3*C*e+B*f)+4*b^2*e^2*(C*e+3*B*f)))*arctan(b*x*sqrt(c)/sqrt(a^2*c-b^2*c*x^2))*sqrt(a^2*c-b^2*c*x^2)/(b^5*sqrt(c)*sqrt(a+b*x)*sqrt(a*c-b*c*x))],
[(e+f*x)^2*(A+B*x+C*x^2)/(sqrt(a+b*x)*sqrt(a*c-b*c*x)),x,6,1/12*(C*e-4*B*f)*(e+f*x)^2*(a^2-b^2*x^2)/(b^2*f*sqrt(a+b*x)*sqrt(a*c-b*c*x))-1/4*C*(e+f*x)^3*(a^2-b^2*x^2)/(b^2*f*sqrt(a+b*x)*sqrt(a*c-b*c*x))-1/24*(4*(4*a^2*f^2*(2*C*e+B*f)-b^2*e*(C*e^2-4*f*(B*e+3*A*f)))+f*(9*a^2*C*f^2-b^2*(2*C*e^2-4*f*(2*B*e+3*A*f)))*x)*(a^2-b^2*x^2)/(b^4*f*sqrt(a+b*x)*sqrt(a*c-b*c*x))+1/8*(4*A*(2*b^4*e^2+a^2*b^2*f^2)+a^2*(3*a^2*C*f^2+4*b^2*e*(C*e+2*B*f)))*arctan(b*x*sqrt(c)/sqrt(a^2*c-b^2*c*x^2))*sqrt(a^2*c-b^2*c*x^2)/(b^5*sqrt(c)*sqrt(a+b*x)*sqrt(a*c-b*c*x))],
[(e+f*x)*(A+B*x+C*x^2)/(sqrt(a+b*x)*sqrt(a*c-b*c*x)),x,5,-1/3*C*(e+f*x)^2*(a^2-b^2*x^2)/(b^2*f*sqrt(a+b*x)*sqrt(a*c-b*c*x))-1/6*(2*(2*a^2*C*f^2-b^2*(C*e^2-3*f*(B*e+A*f)))-b^2*f*(C*e-3*B*f)*x)*(a^2-b^2*x^2)/(b^4*f*sqrt(a+b*x)*sqrt(a*c-b*c*x))+1/2*(2*A*b^2*e+a^2*(C*e+B*f))*arctan(b*x*sqrt(c)/sqrt(a^2*c-b^2*c*x^2))*sqrt(a^2*c-b^2*c*x^2)/(b^3*sqrt(c)*sqrt(a+b*x)*sqrt(a*c-b*c*x))],
[(A+B*x+C*x^2)/(sqrt(a+b*x)*sqrt(a*c-b*c*x)),x,5,-B*(a^2-b^2*x^2)/(b^2*sqrt(a+b*x)*sqrt(a*c-b*c*x))-1/2*C*x*(a^2-b^2*x^2)/(b^2*sqrt(a+b*x)*sqrt(a*c-b*c*x))+1/2*(2*A*b^2+a^2*C)*arctan(b*x*sqrt(c)/sqrt(a^2*c-b^2*c*x^2))*sqrt(a^2*c-b^2*c*x^2)/(b^3*sqrt(c)*sqrt(a+b*x)*sqrt(a*c-b*c*x))],
[(A+B*x+C*x^2)/((e+f*x)*sqrt(a+b*x)*sqrt(a*c-b*c*x)),x,7,-C*(a^2-b^2*x^2)/(b^2*f*sqrt(a+b*x)*sqrt(a*c-b*c*x))-(C*e-B*f)*arctan(b*x*sqrt(c)/sqrt(a^2*c-b^2*c*x^2))*sqrt(a^2*c-b^2*c*x^2)/(b*f^2*sqrt(c)*sqrt(a+b*x)*sqrt(a*c-b*c*x))+(C*e^2-B*e*f+A*f^2)*arctan((a^2*f+b^2*e*x)*sqrt(c)/(sqrt(b^2*e^2-a^2*f^2)*sqrt(a^2*c-b^2*c*x^2)))*sqrt(a^2*c-b^2*c*x^2)/(f^2*sqrt(c)*sqrt(b^2*e^2-a^2*f^2)*sqrt(a+b*x)*sqrt(a*c-b*c*x))],
[(A+B*x+C*x^2)/((e+f*x)^2*sqrt(a+b*x)*sqrt(a*c-b*c*x)),x,7,f*(A+e*(C*e-B*f)/f^2)*(a^2-b^2*x^2)/((b^2*e^2-a^2*f^2)*(e+f*x)*sqrt(a+b*x)*sqrt(a*c-b*c*x))+C*arctan(b*x*sqrt(c)/sqrt(a^2*c-b^2*c*x^2))*sqrt(a^2*c-b^2*c*x^2)/(b*f^2*sqrt(c)*sqrt(a+b*x)*sqrt(a*c-b*c*x))+(a^2*f^2*(2*C*e-B*f)-b^2*(C*e^3-A*e*f^2))*arctan((a^2*f+b^2*e*x)*sqrt(c)/(sqrt(b^2*e^2-a^2*f^2)*sqrt(a^2*c-b^2*c*x^2)))*sqrt(a^2*c-b^2*c*x^2)/(f^2*(b^2*e^2-a^2*f^2)^(3/2)*sqrt(c)*sqrt(a+b*x)*sqrt(a*c-b*c*x))],
[(A+B*x+C*x^2)/((e+f*x)^3*sqrt(a+b*x)*sqrt(a*c-b*c*x)),x,5,1/2*f*(A+e*(C*e-B*f)/f^2)*(a^2-b^2*x^2)/((b^2*e^2-a^2*f^2)*(e+f*x)^2*sqrt(a+b*x)*sqrt(a*c-b*c*x))+1/2*(2*a^2*f^2*(2*C*e-B*f)-b^2*e*(C*e^2+f*(B*e-3*A*f)))*(a^2-b^2*x^2)/(f*(b^2*e^2-a^2*f^2)^2*(e+f*x)*sqrt(a+b*x)*sqrt(a*c-b*c*x))+1/2*(A*(2*b^4*e^2+a^2*b^2*f^2)+a^2*(2*a^2*C*f^2+b^2*e*(C*e-3*B*f)))*arctan((a^2*f+b^2*e*x)*sqrt(c)/(sqrt(b^2*e^2-a^2*f^2)*sqrt(a^2*c-b^2*c*x^2)))*sqrt(a^2*c-b^2*c*x^2)/((b^2*e^2-a^2*f^2)^(5/2)*sqrt(c)*sqrt(a+b*x)*sqrt(a*c-b*c*x))],
[x*(a+b*x+c*x^2)/(sqrt(-1+d*x)*sqrt(1+d*x)),x,5,1/2*b*arccosh(d*x)/d^3+1/3*c*x^2*sqrt(-1+d*x)*sqrt(1+d*x)/d^2+1/6*(2*(2*c+3*a*d^2)+3*b*d^2*x)*sqrt(-1+d*x)*sqrt(1+d*x)/d^4,-1/3*c*x^2*(1-d^2*x^2)/(d^2*sqrt(-1+d*x)*sqrt(1+d*x))-1/6*(2*(2*c+3*a*d^2)+3*b*d^2*x)*(1-d^2*x^2)/(d^4*sqrt(-1+d*x)*sqrt(1+d*x))+1/2*b*arctanh(d*x/sqrt(-1+d^2*x^2))*sqrt(-1+d^2*x^2)/(d^3*sqrt(-1+d*x)*sqrt(1+d*x))],
[(a+b*x+c*x^2)/(sqrt(-1+d*x)*sqrt(1+d*x)),x,5,1/2*(c+2*a*d^2)*arccosh(d*x)/d^3+1/2*(2*b+c*x)*sqrt(-1+d*x)*sqrt(1+d*x)/d^2,-b*(1-d^2*x^2)/(d^2*sqrt(-1+d*x)*sqrt(1+d*x))-1/2*c*x*(1-d^2*x^2)/(d^2*sqrt(-1+d*x)*sqrt(1+d*x))+1/2*(c+2*a*d^2)*arctanh(d*x/sqrt(-1+d^2*x^2))*sqrt(-1+d^2*x^2)/(d^3*sqrt(-1+d*x)*sqrt(1+d*x))],
[(a+b*x+c*x^2)/(x*sqrt(-1+d*x)*sqrt(1+d*x)),x,8,b*arccosh(d*x)/d+a*arctan(sqrt(-1+d*x)*sqrt(1+d*x))+c*sqrt(-1+d*x)*sqrt(1+d*x)/d^2,-c*(1-d^2*x^2)/(d^2*sqrt(-1+d*x)*sqrt(1+d*x))+a*arctan(sqrt(-1+d^2*x^2))*sqrt(-1+d^2*x^2)/(sqrt(-1+d*x)*sqrt(1+d*x))+b*arctanh(d*x/sqrt(-1+d^2*x^2))*sqrt(-1+d^2*x^2)/(d*sqrt(-1+d*x)*sqrt(1+d*x))],
[(a+b*x+c*x^2)/(x^2*sqrt(-1+d*x)*sqrt(1+d*x)),x,8,c*arccosh(d*x)/d+b*arctan(sqrt(-1+d*x)*sqrt(1+d*x))+a*sqrt(-1+d*x)*sqrt(1+d*x)/x,-a*(1-d^2*x^2)/(x*sqrt(-1+d*x)*sqrt(1+d*x))+b*arctan(sqrt(-1+d^2*x^2))*sqrt(-1+d^2*x^2)/(sqrt(-1+d*x)*sqrt(1+d*x))+c*arctanh(d*x/sqrt(-1+d^2*x^2))*sqrt(-1+d^2*x^2)/(d*sqrt(-1+d*x)*sqrt(1+d*x))],
[(a+b*x+c*x^2)/(x^3*sqrt(-1+d*x)*sqrt(1+d*x)),x,6,1/2*(2*c+a*d^2)*arctan(sqrt(-1+d*x)*sqrt(1+d*x))+1/2*a*sqrt(-1+d*x)*sqrt(1+d*x)/x^2+b*sqrt(-1+d*x)*sqrt(1+d*x)/x,-1/2*a*(1-d^2*x^2)/(x^2*sqrt(-1+d*x)*sqrt(1+d*x))-b*(1-d^2*x^2)/(x*sqrt(-1+d*x)*sqrt(1+d*x))+1/2*(2*c+a*d^2)*arctan(sqrt(-1+d^2*x^2))*sqrt(-1+d^2*x^2)/(sqrt(-1+d*x)*sqrt(1+d*x))],
[(a+b*x+c*x^2)/(x^4*sqrt(-1+d*x)*sqrt(1+d*x)),x,7,1/2*b*d^2*arctan(sqrt(-1+d*x)*sqrt(1+d*x))+1/3*a*sqrt(-1+d*x)*sqrt(1+d*x)/x^3+1/2*b*sqrt(-1+d*x)*sqrt(1+d*x)/x^2+1/3*(3*c+2*a*d^2)*sqrt(-1+d*x)*sqrt(1+d*x)/x,-1/3*a*(1-d^2*x^2)/(x^3*sqrt(-1+d*x)*sqrt(1+d*x))-1/2*b*(1-d^2*x^2)/(x^2*sqrt(-1+d*x)*sqrt(1+d*x))-1/3*(3*c+2*a*d^2)*(1-d^2*x^2)/(x*sqrt(-1+d*x)*sqrt(1+d*x))+1/2*b*d^2*arctan(sqrt(-1+d^2*x^2))*sqrt(-1+d^2*x^2)/(sqrt(-1+d*x)*sqrt(1+d*x))],
[(a+b*x+c*x^2)/((d+e*x)^3*sqrt(-1+x)*sqrt(1+x)),x,5,((2*a+c)*d^2-3*b*d*e+(a+2*c)*e^2)*arctanh(sqrt(d+e)*sqrt(1+x)/(sqrt(d-e)*sqrt(-1+x)))/((d-e)^(5/2)*(d+e)^(5/2))-1/2*(c*d^2-b*d*e+a*e^2)*sqrt(-1+x)*sqrt(1+x)/(e*(d^2-e^2)*(d+e*x)^2)+1/2*(c*d^3+b*d^2*e-(3*a+4*c)*d*e^2+2*b*e^3)*sqrt(-1+x)*sqrt(1+x)/(e*(d^2-e^2)^2*(d+e*x)),1/2*(c*d^2-b*d*e+a*e^2)*(1-x^2)/(e*(d^2-e^2)*(d+e*x)^2*sqrt(-1+x)*sqrt(1+x))-1/2*(c*(d^3-4*d*e^2)-e*(3*a*d*e-b*(d^2+2*e^2)))*(1-x^2)/(e*(d^2-e^2)^2*(d+e*x)*sqrt(-1+x)*sqrt(1+x))-1/2*(3*b*d*e-a*(2*d^2+e^2)-c*(d^2+2*e^2))*arctanh((e+d*x)/(sqrt(d^2-e^2)*sqrt(-1+x^2)))*sqrt(-1+x^2)/((d^2-e^2)^(5/2)*sqrt(-1+x)*sqrt(1+x))],

# Integrands of the form (a+b x)^m (c+d x)^n (e+f x)^p (A+B x+C x^2)

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

# n>0 & p>0
[(a+b*x)^2*(A+B*x+C*x^2)*sqrt(c+d*x)*sqrt(e+f*x),x,8,-1/20*(2*a*C*d*f-b*(4*B*d*f-3*C*(d*e+c*f)))*(a+b*x)^2*(c+d*x)^(3/2)*(e+f*x)^(3/2)/(b*d^2*f^2)+1/6*C*(a+b*x)^3*(c+d*x)^(3/2)*(e+f*x)^(3/2)/(b*d*f)-1/960*(c+d*x)^(3/2)*(e+f*x)^(3/2)*(64*a^3*C*d^3*f^3-8*a^2*b*d^2*f^2*(16*B*d*f-7*C*(d*e+c*f))-8*a*b^2*d*f*(C*(35*d^2*e^2+38*c*d*e*f+35*c^2*f^2)+10*d*f*(8*A*d*f-5*B*(d*e+c*f)))+b^3*(7*C*(15*d^3*e^3+17*c*d^2*e^2*f+17*c^2*d*e*f^2+15*c^3*f^3)+4*d*f*(50*A*d*f*(d*e+c*f)-B*(35*d^2*e^2+38*c*d*e*f+35*c^2*f^2)))+6*b*d*f*(10*b*d*f*(2*b*c*C*e+a*C*d*e+a*c*C*f-4*A*b*d*f)+(4*a*d*f-7*b*(d*e+c*f))*(2*a*C*d*f-b*(4*B*d*f-3*C*(d*e+c*f))))*x)/(b*d^4*f^4)-1/512*(d*e-c*f)^2*(8*a^2*d^2*f^2*(C*(5*d^2*e^2+6*c*d*e*f+5*c^2*f^2)+8*d*f*(2*A*d*f-B*(d*e+c*f)))-8*a*b*d*f*(C*(7*d^3*e^3+9*c*d^2*e^2*f+9*c^2*d*e*f^2+7*c^3*f^3)+2*d*f*(8*A*d*f*(d*e+c*f)-B*(5*d^2*e^2+6*c*d*e*f+5*c^2*f^2)))+b^2*(C*(21*d^4*e^4+28*c*d^3*e^3*f+30*c^2*d^2*e^2*f^2+28*c^3*d*e*f^3+21*c^4*f^4)+4*d*f*(2*A*d*f*(5*d^2*e^2+6*c*d*e*f+5*c^2*f^2)-B*(7*d^3*e^3+9*c*d^2*e^2*f+9*c^2*d*e*f^2+7*c^3*f^3))))*arctanh(sqrt(f)*sqrt(c+d*x)/(sqrt(d)*sqrt(e+f*x)))/(d^(11/2)*f^(11/2))+1/256*(8*a^2*d^2*f^2*(C*(5*d^2*e^2+6*c*d*e*f+5*c^2*f^2)+8*d*f*(2*A*d*f-B*(d*e+c*f)))-8*a*b*d*f*(C*(7*d^3*e^3+9*c*d^2*e^2*f+9*c^2*d*e*f^2+7*c^3*f^3)+2*d*f*(8*A*d*f*(d*e+c*f)-B*(5*d^2*e^2+6*c*d*e*f+5*c^2*f^2)))+b^2*(C*(21*d^4*e^4+28*c*d^3*e^3*f+30*c^2*d^2*e^2*f^2+28*c^3*d*e*f^3+21*c^4*f^4)+4*d*f*(2*A*d*f*(5*d^2*e^2+6*c*d*e*f+5*c^2*f^2)-B*(7*d^3*e^3+9*c*d^2*e^2*f+9*c^2*d*e*f^2+7*c^3*f^3))))*(c+d*x)^(3/2)*sqrt(e+f*x)/(d^5*f^4)+1/512*(d*e-c*f)*(8*a^2*d^2*f^2*(C*(5*d^2*e^2+6*c*d*e*f+5*c^2*f^2)+8*d*f*(2*A*d*f-B*(d*e+c*f)))-8*a*b*d*f*(C*(7*d^3*e^3+9*c*d^2*e^2*f+9*c^2*d*e*f^2+7*c^3*f^3)+2*d*f*(8*A*d*f*(d*e+c*f)-B*(5*d^2*e^2+6*c*d*e*f+5*c^2*f^2)))+b^2*(C*(21*d^4*e^4+28*c*d^3*e^3*f+30*c^2*d^2*e^2*f^2+28*c^3*d*e*f^3+21*c^4*f^4)+4*d*f*(2*A*d*f*(5*d^2*e^2+6*c*d*e*f+5*c^2*f^2)-B*(7*d^3*e^3+9*c*d^2*e^2*f+9*c^2*d*e*f^2+7*c^3*f^3))))*sqrt(c+d*x)*sqrt(e+f*x)/(d^5*f^5)],
[(a+b*x)*(A+B*x+C*x^2)*sqrt(c+d*x)*sqrt(e+f*x),x,7,1/5*C*(a+b*x)^2*(c+d*x)^(3/2)*(e+f*x)^(3/2)/(b*d*f)-1/240*(c+d*x)^(3/2)*(e+f*x)^(3/2)*(48*a^2*C*d^2*f^2-10*a*b*d*f*(8*B*d*f-5*C*(d*e+c*f))-b^2*(C*(35*d^2*e^2+38*c*d*e*f+35*c^2*f^2)+10*d*f*(8*A*d*f-5*B*(d*e+c*f)))+6*b*d*f*(6*a*C*d*f-b*(10*B*d*f-7*C*(d*e+c*f)))*x)/(b*d^3*f^3)-1/128*(d*e-c*f)^2*(2*a*d*f*(C*(5*d^2*e^2+6*c*d*e*f+5*c^2*f^2)+8*d*f*(2*A*d*f-B*(d*e+c*f)))-b*(C*(7*d^3*e^3+9*c*d^2*e^2*f+9*c^2*d*e*f^2+7*c^3*f^3)+2*d*f*(8*A*d*f*(d*e+c*f)-B*(5*d^2*e^2+6*c*d*e*f+5*c^2*f^2))))*arctanh(sqrt(f)*sqrt(c+d*x)/(sqrt(d)*sqrt(e+f*x)))/(d^(9/2)*f^(9/2))+1/64*(2*a*d*f*(C*(5*d^2*e^2+6*c*d*e*f+5*c^2*f^2)+8*d*f*(2*A*d*f-B*(d*e+c*f)))-b*(C*(7*d^3*e^3+9*c*d^2*e^2*f+9*c^2*d*e*f^2+7*c^3*f^3)+2*d*f*(8*A*d*f*(d*e+c*f)-B*(5*d^2*e^2+6*c*d*e*f+5*c^2*f^2))))*(c+d*x)^(3/2)*sqrt(e+f*x)/(d^4*f^3)+1/128*(d*e-c*f)*(2*a*d*f*(C*(5*d^2*e^2+6*c*d*e*f+5*c^2*f^2)+8*d*f*(2*A*d*f-B*(d*e+c*f)))-b*(C*(7*d^3*e^3+9*c*d^2*e^2*f+9*c^2*d*e*f^2+7*c^3*f^3)+2*d*f*(8*A*d*f*(d*e+c*f)-B*(5*d^2*e^2+6*c*d*e*f+5*c^2*f^2))))*sqrt(c+d*x)*sqrt(e+f*x)/(d^4*f^4)],
[(A+B*x+C*x^2)*sqrt(c+d*x)*sqrt(e+f*x),x,7,-1/24*(5*C*d*e+11*c*C*f-8*B*d*f)*(c+d*x)^(3/2)*(e+f*x)^(3/2)/(d^2*f^2)+1/4*C*(c+d*x)^(5/2)*(e+f*x)^(3/2)/(d^2*f)-1/64*(d*e-c*f)^2*(C*(5*d^2*e^2+6*c*d*e*f+5*c^2*f^2)+8*d*f*(2*A*d*f-B*(d*e+c*f)))*arctanh(sqrt(f)*sqrt(c+d*x)/(sqrt(d)*sqrt(e+f*x)))/(d^(7/2)*f^(7/2))+1/32*(C*(5*d^2*e^2+6*c*d*e*f+5*c^2*f^2)+8*d*f*(2*A*d*f-B*(d*e+c*f)))*(c+d*x)^(3/2)*sqrt(e+f*x)/(d^3*f^2)+1/64*(d*e-c*f)*(C*(5*d^2*e^2+6*c*d*e*f+5*c^2*f^2)+8*d*f*(2*A*d*f-B*(d*e+c*f)))*sqrt(c+d*x)*sqrt(e+f*x)/(d^3*f^3)],
[(A+B*x+C*x^2)*sqrt(c+d*x)*sqrt(e+f*x)/(a+b*x),x,9,1/3*C*(c+d*x)^(3/2)*(e+f*x)^(3/2)/(b*d*f)-1/8*(16*a^3*C*d^3*f^3-8*a^2*b*d^2*f^2*(C*d*e+c*C*f+2*B*d*f)-2*a*b^2*d*f*(C*(d*e-c*f)^2-4*d*f*(B*d*e+B*c*f+2*A*d*f))-b^3*(C*(d*e-c*f)^2*(d*e+c*f)-2*d*f*(B*(d*e-c*f)^2-4*A*d*f*(d*e+c*f))))*arctanh(sqrt(f)*sqrt(c+d*x)/(sqrt(d)*sqrt(e+f*x)))/(b^4*d^(5/2)*f^(5/2))-2*(A*b^2-a*(b*B-a*C))*arctanh(sqrt(b*e-a*f)*sqrt(c+d*x)/(sqrt(b*c-a*d)*sqrt(e+f*x)))*sqrt(b*c-a*d)*sqrt(b*e-a*f)/b^4-1/4*(2*a*C*d*f+b*(C*d*e+c*C*f-2*B*d*f))*(e+f*x)^(3/2)*sqrt(c+d*x)/(b^2*d*f^2)+1/8*(4*b*d*f*(2*A*b*d*f-a*C*(d*e+c*f))+(b*d*e-b*c*f+4*a*d*f)*(2*a*C*d*f+b*(C*d*e+c*C*f-2*B*d*f)))*sqrt(c+d*x)*sqrt(e+f*x)/(b^3*d^2*f^2)],
[(A+B*x+C*x^2)*sqrt(c+d*x)*sqrt(e+f*x)/(a+b*x)^2,x,9,-(A*b^2-a*(b*B-a*C))*(c+d*x)^(3/2)*(e+f*x)^(3/2)/(b*(b*c-a*d)*(b*e-a*f)*(a+b*x))+1/4*(24*a^2*C*d^2*f^2-8*a*b*d*f*(C*d*e+c*C*f+2*B*d*f)-b^2*(C*(d*e-c*f)^2-4*d*f*(B*d*e+B*c*f+2*A*d*f)))*arctanh(sqrt(f)*sqrt(c+d*x)/(sqrt(d)*sqrt(e+f*x)))/(b^4*d^(3/2)*f^(3/2))+(6*a^3*C*d*f-b^3*(2*B*c*e+A*d*e+A*c*f)+a*b^2*(4*c*C*e+3*B*d*e+3*B*c*f+2*A*d*f)-a^2*b*(4*B*d*f+5*C*(d*e+c*f)))*arctanh(sqrt(b*e-a*f)*sqrt(c+d*x)/(sqrt(b*c-a*d)*sqrt(e+f*x)))/(b^4*sqrt(b*c-a*d)*sqrt(b*e-a*f))+1/2*(3*a^2*C*d*f+b^2*(c*C*e+2*A*d*f)-a*b*(C*d*e+c*C*f+2*B*d*f))*(e+f*x)^(3/2)*sqrt(c+d*x)/(b^2*(b*c-a*d)*f*(b*e-a*f))+1/4*(12*a^2*C*d*f^2-a*b*f*(7*C*d*e+c*C*f+8*B*d*f)+b^2*(4*d*f*(B*e+A*f)-C*e*(d*e-c*f)))*sqrt(c+d*x)*sqrt(e+f*x)/(b^3*d*f*(b*e-a*f))],
[(A+B*x+C*x^2)*sqrt(c+d*x)*sqrt(e+f*x)/(a+b*x)^3,x,9,-1/2*(A*b^2-a*(b*B-a*C))*(c+d*x)^(3/2)*(e+f*x)^(3/2)/(b*(b*c-a*d)*(b*e-a*f)*(a+b*x)^2)-1/4*(24*a^4*C*d^2*f^2-3*a*b^3*(B*d^2*e^2+c^2*f*(8*C*e+B*f)+2*c*d*e*(4*C*e+3*B*f))-8*a^3*b*d*f*(B*d*f+5*C*(d*e+c*f))-b^4*(A*d^2*e^2-2*c*d*e*(2*B*e+A*f)-c^2*(8*C*e^2+4*B*e*f-A*f^2))+3*a^2*b^2*(4*B*d*f*(d*e+c*f)+C*(5*d^2*e^2+22*c*d*e*f+5*c^2*f^2)))*arctanh(sqrt(b*e-a*f)*sqrt(c+d*x)/(sqrt(b*c-a*d)*sqrt(e+f*x)))/(b^4*(b*c-a*d)^(3/2)*(b*e-a*f)^(3/2))-(6*a*C*d*f-b*(C*d*e+c*C*f+2*B*d*f))*arctanh(sqrt(f)*sqrt(c+d*x)/(sqrt(d)*sqrt(e+f*x)))/(b^4*sqrt(d)*sqrt(f))+1/4*(6*a^3*C*d*f-b^3*(4*B*c*e-A*d*e-A*c*f)+a*b^2*(8*c*C*e+3*B*d*e+3*B*c*f-2*A*d*f)-a^2*b*(2*B*d*f+7*C*(d*e+c*f)))*(e+f*x)^(3/2)*sqrt(c+d*x)/(b^2*(b*c-a*d)*(b*e-a*f)^2*(a+b*x))-1/4*(12*a^3*C*d*f^2-a^2*b*f*(17*C*d*e+11*c*C*f+4*B*d*f)+a*b^2*(B*f*(5*d*e+3*c*f)+4*C*e*(d*e+4*c*f))-b^3*(A*d*e*f+c*(4*C*e^2+4*B*e*f-A*f^2)))*sqrt(c+d*x)*sqrt(e+f*x)/(b^3*(b*c-a*d)*(b*e-a*f)^2)],

# n>0 & p<0
[(a+b*x)^2*(A+B*x+C*x^2)*sqrt(c+d*x)/sqrt(e+f*x),x,7,1/128*(d*e-c*f)*(16*a^2*d^2*f^2*(2*d*f*(3*B*d*e+B*c*f-4*A*d*f)-C*(5*d^2*e^2+2*c*d*e*f+c^2*f^2))+4*a*b*d*f*(C*(35*d^3*e^3+15*c*d^2*e^2*f+9*c^2*d*e*f^2+5*c^3*f^3)+8*d*f*(2*A*d*f*(3*d*e+c*f)-B*(5*d^2*e^2+2*c*d*e*f+c^2*f^2)))-b^2*(C*(63*d^4*e^4+28*c*d^3*e^3*f+18*c^2*d^2*e^2*f^2+12*c^3*d*e*f^3+7*c^4*f^4)+2*d*f*(8*A*d*f*(5*d^2*e^2+2*c*d*e*f+c^2*f^2)-B*(35*d^3*e^3+15*c*d^2*e^2*f+9*c^2*d*e*f^2+5*c^3*f^3))))*arctanh(sqrt(f)*sqrt(c+d*x)/(sqrt(d)*sqrt(e+f*x)))/(d^(9/2)*f^(11/2))-1/40*(4*a*C*d*f+b*(9*C*d*e+7*c*C*f-10*B*d*f))*(a+b*x)^2*(c+d*x)^(3/2)*sqrt(e+f*x)/(b*d^2*f^2)+1/5*C*(a+b*x)^3*(c+d*x)^(3/2)*sqrt(e+f*x)/(b*d*f)-1/960*(c+d*x)^(3/2)*(96*a^3*C*d^3*f^3+8*a^2*b*d^2*f^2*(23*C*d*e+9*c*C*f-30*B*d*f)+20*a*b^2*d*f*(8*d*f*(5*B*d*e+3*B*c*f-6*A*d*f)-C*(35*d^2*e^2+22*c*d*e*f+15*c^2*f^2))+b^3*(C*(315*d^3*e^3+203*c*d^2*e^2*f+145*c^2*d*e*f^2+105*c^3*f^3)+10*d*f*(8*A*d*f*(5*d*e+3*c*f)-B*(35*d^2*e^2+22*c*d*e*f+15*c^2*f^2)))+4*b*d*f*(8*b*d*f*(6*b*c*C*e+3*a*C*d*e+a*c*C*f-10*A*b*d*f)-(7*b*d*e+5*b*c*f-4*a*d*f)*(4*a*C*d*f+b*(9*C*d*e+7*c*C*f-10*B*d*f)))*x)*sqrt(e+f*x)/(b*d^4*f^4)-1/128*(16*a^2*d^2*f^2*(2*d*f*(3*B*d*e+B*c*f-4*A*d*f)-C*(5*d^2*e^2+2*c*d*e*f+c^2*f^2))+4*a*b*d*f*(C*(35*d^3*e^3+15*c*d^2*e^2*f+9*c^2*d*e*f^2+5*c^3*f^3)+8*d*f*(2*A*d*f*(3*d*e+c*f)-B*(5*d^2*e^2+2*c*d*e*f+c^2*f^2)))-b^2*(C*(63*d^4*e^4+28*c*d^3*e^3*f+18*c^2*d^2*e^2*f^2+12*c^3*d*e*f^3+7*c^4*f^4)+2*d*f*(8*A*d*f*(5*d^2*e^2+2*c*d*e*f+c^2*f^2)-B*(35*d^3*e^3+15*c*d^2*e^2*f+9*c^2*d*e*f^2+5*c^3*f^3))))*sqrt(c+d*x)*sqrt(e+f*x)/(d^4*f^5)],
[(a+b*x)*(A+B*x+C*x^2)*sqrt(c+d*x)/sqrt(e+f*x),x,6,1/64*(d*e-c*f)*(8*a*d*f*(2*d*f*(3*B*d*e+B*c*f-4*A*d*f)-C*(5*d^2*e^2+2*c*d*e*f+c^2*f^2))+b*(C*(35*d^3*e^3+15*c*d^2*e^2*f+9*c^2*d*e*f^2+5*c^3*f^3)+8*d*f*(2*A*d*f*(3*d*e+c*f)-B*(5*d^2*e^2+2*c*d*e*f+c^2*f^2))))*arctanh(sqrt(f)*sqrt(c+d*x)/(sqrt(d)*sqrt(e+f*x)))/(d^(7/2)*f^(9/2))+1/4*C*(a+b*x)^2*(c+d*x)^(3/2)*sqrt(e+f*x)/(b*d*f)-1/96*(c+d*x)^(3/2)*(24*a^2*C*d^2*f^2+8*a*b*d*f*(5*C*d*e+3*c*C*f-6*B*d*f)+b^2*(8*d*f*(5*B*d*e+3*B*c*f-6*A*d*f)-C*(35*d^2*e^2+22*c*d*e*f+15*c^2*f^2))+4*b*d*f*(4*a*C*d*f+b*(7*C*d*e+5*c*C*f-8*B*d*f))*x)*sqrt(e+f*x)/(b*d^3*f^3)-1/64*(8*a*d*f*(2*d*f*(3*B*d*e+B*c*f-4*A*d*f)-C*(5*d^2*e^2+2*c*d*e*f+c^2*f^2))+b*(C*(35*d^3*e^3+15*c*d^2*e^2*f+9*c^2*d*e*f^2+5*c^3*f^3)+8*d*f*(2*A*d*f*(3*d*e+c*f)-B*(5*d^2*e^2+2*c*d*e*f+c^2*f^2))))*sqrt(c+d*x)*sqrt(e+f*x)/(d^3*f^4)],
[(A+B*x+C*x^2)*sqrt(c+d*x)/sqrt(e+f*x),x,6,-1/8*(d*e-c*f)*(C*(5*d^2*e^2+2*c*d*e*f+c^2*f^2)+2*d*f*(4*A*d*f-B*(3*d*e+c*f)))*arctanh(sqrt(f)*sqrt(c+d*x)/(sqrt(d)*sqrt(e+f*x)))/(d^(5/2)*f^(7/2))-1/12*(5*C*d*e+7*c*C*f-6*B*d*f)*(c+d*x)^(3/2)*sqrt(e+f*x)/(d^2*f^2)+1/3*C*(c+d*x)^(5/2)*sqrt(e+f*x)/(d^2*f)+1/8*(C*(5*d^2*e^2+2*c*d*e*f+c^2*f^2)+2*d*f*(4*A*d*f-B*(3*d*e+c*f)))*sqrt(c+d*x)*sqrt(e+f*x)/(d^2*f^3)],
[(A+B*x+C*x^2)*sqrt(c+d*x)/((a+b*x)*sqrt(e+f*x)),x,8,1/4*(2*b*d*f*(4*A*b*d*f-a*C*(3*d*e+c*f))+(b*d*e-b*c*f+2*a*d*f)*(4*a*C*d*f+b*(3*C*d*e+c*C*f-4*B*d*f)))*arctanh(sqrt(f)*sqrt(c+d*x)/(sqrt(d)*sqrt(e+f*x)))/(b^3*d^(3/2)*f^(5/2))-2*(A*b^2-a*(b*B-a*C))*arctanh(sqrt(b*e-a*f)*sqrt(c+d*x)/(sqrt(b*c-a*d)*sqrt(e+f*x)))*sqrt(b*c-a*d)/(b^3*sqrt(b*e-a*f))+1/2*C*(c+d*x)^(3/2)*sqrt(e+f*x)/(b*d*f)-1/4*(4*a*C*d*f+b*(3*C*d*e+c*C*f-4*B*d*f))*sqrt(c+d*x)*sqrt(e+f*x)/(b^2*d*f^2)],
[(A+B*x+C*x^2)*sqrt(c+d*x)/((a+b*x)^2*sqrt(e+f*x)),x,8,-(4*a*C*d*f+b*(C*d*e-c*C*f-2*B*d*f))*arctanh(sqrt(f)*sqrt(c+d*x)/(sqrt(d)*sqrt(e+f*x)))/(b^3*f^(3/2)*sqrt(d))+(4*a^3*C*d*f-b^3*(2*B*c*e+A*d*e-A*c*f)+a*b^2*(4*c*C*e+3*B*d*e+B*c*f)-a^2*b*(5*C*d*e+3*c*C*f+2*B*d*f))*arctanh(sqrt(b*e-a*f)*sqrt(c+d*x)/(sqrt(b*c-a*d)*sqrt(e+f*x)))/(b^3*(b*e-a*f)^(3/2)*sqrt(b*c-a*d))-(A*b^2-a*(b*B-a*C))*(c+d*x)^(3/2)*sqrt(e+f*x)/(b*(b*c-a*d)*(b*e-a*f)*(a+b*x))+(2*a^2*C*d*f+b^2*(c*C*e+A*d*f)-a*b*(C*d*e+c*C*f+B*d*f))*sqrt(c+d*x)*sqrt(e+f*x)/(b^2*(b*c-a*d)*f*(b*e-a*f))],
[(A+B*x+C*x^2)*sqrt(c+d*x)/((a+b*x)^3*sqrt(e+f*x)),x,8,-1/4*(8*a^4*C*d^2*f^2-4*a^3*b*C*d*f*(5*d*e+3*c*f)+3*a^2*b^2*C*(5*d^2*e^2+10*c*d*e*f+c^2*f^2)-a*b^3*(d^2*e*(3*B*e-4*A*f)+c^2*f*(8*C*e-B*f)+2*c*d*(12*C*e^2-B*e*f+2*A*f^2))-b^4*(A*d^2*e^2-2*c*d*e*(2*B*e-A*f)-c^2*(8*C*e^2-4*B*e*f+3*A*f^2)))*arctanh(sqrt(b*e-a*f)*sqrt(c+d*x)/(sqrt(b*c-a*d)*sqrt(e+f*x)))/(b^3*(b*c-a*d)^(3/2)*(b*e-a*f)^(5/2))+2*C*arctanh(sqrt(f)*sqrt(c+d*x)/(sqrt(d)*sqrt(e+f*x)))*sqrt(d)/(b^3*sqrt(f))-1/2*(A*b^2-a*(b*B-a*C))*(c+d*x)^(3/2)*sqrt(e+f*x)/(b*(b*c-a*d)*(b*e-a*f)*(a+b*x)^2)+1/4*(4*a^3*C*d*f-a^2*b*C*(7*d*e+5*c*f)-b^3*(4*B*c*e-A*d*e-3*A*c*f)+a*b^2*(8*c*C*e+3*B*d*e+B*c*f-4*A*d*f))*sqrt(c+d*x)*sqrt(e+f*x)/(b^2*(b*c-a*d)*(b*e-a*f)^2*(a+b*x))],
[(A+B*x+C*x^2)*sqrt(c+d*x)/((a+b*x)^4*sqrt(e+f*x)),x,6,-1/8*(d*e-c*f)*(b^2*(A*d^2*e^2-2*c*d*e*(B*e-A*f)+c^2*(8*C*e^2-6*B*e*f+5*A*f^2))+a*b*(d^2*e*(B*e-4*A*f)-c^2*f*(4*C*e-B*f)-2*c*d*(6*C*e^2-7*B*e*f+6*A*f^2))-a^2*(2*d*f*(3*B*d*e+B*c*f-4*A*d*f)-C*(5*d^2*e^2+2*c*d*e*f+c^2*f^2)))*arctanh(sqrt(b*e-a*f)*sqrt(c+d*x)/(sqrt(b*c-a*d)*sqrt(e+f*x)))/((b*c-a*d)^(5/2)*(b*e-a*f)^(7/2))-1/3*(A*b^2-a*(b*B-a*C))*(c+d*x)^(3/2)*sqrt(e+f*x)/(b*(b*c-a*d)*(b*e-a*f)*(a+b*x)^3)+1/12*(4*a^3*C*d*f-b^3*(6*B*c*e-3*A*d*e-5*A*c*f)+a*b^2*(12*c*C*e+3*B*d*e+B*c*f-8*A*d*f)-a^2*b*(9*C*d*e+7*c*C*f-2*B*d*f))*sqrt(c+d*x)*sqrt(e+f*x)/(b^2*(b*c-a*d)*(b*e-a*f)^2*(a+b*x)^2)-1/24*(8*a^4*C*d^2*f^2-2*a^3*b*d*f*(13*C*d*e+7*c*C*f-2*B*d*f)-b^4*(3*A*d^2*e^2-2*c*d*e*(3*B*e-2*A*f)-3*c^2*(8*C*e^2-6*B*e*f+5*A*f^2))-a*b^3*(d^2*e*(3*B*e-10*A*f)+3*c^2*f*(4*C*e-B*f)+2*c*d*(30*C*e^2-14*B*e*f+13*A*f^2))-a^2*b^2*(4*d*f*(4*B*d*e+B*c*f-2*A*d*f)-C*(33*d^2*e^2+44*c*d*e*f+3*c^2*f^2)))*sqrt(c+d*x)*sqrt(e+f*x)/(b^2*(b*c-a*d)^2*(b*e-a*f)^3*(a+b*x))],

# n<0 & p<0
[(a+b*x)^2*(A+B*x+C*x^2)/(sqrt(c+d*x)*sqrt(e+f*x)),x,6,1/64*(16*a^2*d^2*f^2*(C*(3*d^2*e^2+2*c*d*e*f+3*c^2*f^2)+4*d*f*(2*A*d*f-B*(d*e+c*f)))-16*a*b*d*f*(C*(5*d^3*e^3+3*c*d^2*e^2*f+3*c^2*d*e*f^2+5*c^3*f^3)+2*d*f*(4*A*d*f*(d*e+c*f)-B*(3*d^2*e^2+2*c*d*e*f+3*c^2*f^2)))+b^2*(C*(35*d^4*e^4+20*c*d^3*e^3*f+18*c^2*d^2*e^2*f^2+20*c^3*d*e*f^3+35*c^4*f^4)+8*d*f*(2*A*d*f*(3*d^2*e^2+2*c*d*e*f+3*c^2*f^2)-B*(5*d^3*e^3+3*c*d^2*e^2*f+3*c^2*d*e*f^2+5*c^3*f^3))))*arctanh(sqrt(f)*sqrt(c+d*x)/(sqrt(d)*sqrt(e+f*x)))/(d^(9/2)*f^(9/2))-1/24*(2*a*C*d*f-b*(8*B*d*f-7*C*(d*e+c*f)))*(a+b*x)^2*sqrt(c+d*x)*sqrt(e+f*x)/(b*d^2*f^2)+1/4*C*(a+b*x)^3*sqrt(c+d*x)*sqrt(e+f*x)/(b*d*f)-1/192*(32*a^3*C*d^3*f^3-8*a^2*b*d^2*f^2*(16*B*d*f-11*C*(d*e+c*f))-16*a*b^2*d*f*(C*(15*d^2*e^2+14*c*d*e*f+15*c^2*f^2)+6*d*f*(4*A*d*f-3*B*(d*e+c*f)))+b^3*(5*C*(21*d^3*e^3+19*c*d^2*e^2*f+19*c^2*d*e*f^2+21*c^3*f^3)+8*d*f*(18*A*d*f*(d*e+c*f)-B*(15*d^2*e^2+14*c*d*e*f+15*c^2*f^2)))+2*b*d*f*(6*b*d*f*(6*b*c*C*e+a*C*d*e+a*c*C*f-8*A*b*d*f)+(4*a*d*f-5*b*(d*e+c*f))*(2*a*C*d*f-b*(8*B*d*f-7*C*(d*e+c*f))))*x)*sqrt(c+d*x)*sqrt(e+f*x)/(b*d^4*f^4)],
[(a+b*x)*(A+B*x+C*x^2)/(sqrt(c+d*x)*sqrt(e+f*x)),x,5,1/8*(2*a*d*f*(C*(3*d^2*e^2+2*c*d*e*f+3*c^2*f^2)+4*d*f*(2*A*d*f-B*(d*e+c*f)))-b*(C*(5*d^3*e^3+3*c*d^2*e^2*f+3*c^2*d*e*f^2+5*c^3*f^3)+2*d*f*(4*A*d*f*(d*e+c*f)-B*(3*d^2*e^2+2*c*d*e*f+3*c^2*f^2))))*arctanh(sqrt(f)*sqrt(c+d*x)/(sqrt(d)*sqrt(e+f*x)))/(d^(7/2)*f^(7/2))+1/3*C*(a+b*x)^2*sqrt(c+d*x)*sqrt(e+f*x)/(b*d*f)-1/24*(8*a^2*C*d^2*f^2-6*a*b*d*f*(4*B*d*f-3*C*(d*e+c*f))-b^2*(C*(15*d^2*e^2+14*c*d*e*f+15*c^2*f^2)+6*d*f*(4*A*d*f-3*B*(d*e+c*f)))+2*b*d*f*(2*a*C*d*f-b*(6*B*d*f-5*C*(d*e+c*f)))*x)*sqrt(c+d*x)*sqrt(e+f*x)/(b*d^3*f^3)],
[(A+B*x+C*x^2)/(sqrt(c+d*x)*sqrt(e+f*x)),x,5,1/4*(C*(3*d^2*e^2+2*c*d*e*f+3*c^2*f^2)+4*d*f*(2*A*d*f-B*(d*e+c*f)))*arctanh(sqrt(f)*sqrt(c+d*x)/(sqrt(d)*sqrt(e+f*x)))/(d^(5/2)*f^(5/2))+1/2*C*(c+d*x)^(3/2)*sqrt(e+f*x)/(d^2*f)-1/4*(3*C*d*e+5*c*C*f-4*B*d*f)*sqrt(c+d*x)*sqrt(e+f*x)/(d^2*f^2)],
[(A+B*x+C*x^2)/((a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)),x,7,-(2*a*C*d*f+b*(C*d*e+c*C*f-2*B*d*f))*arctanh(sqrt(f)*sqrt(c+d*x)/(sqrt(d)*sqrt(e+f*x)))/(b^2*d^(3/2)*f^(3/2))-2*(A*b^2-a*(b*B-a*C))*arctanh(sqrt(b*e-a*f)*sqrt(c+d*x)/(sqrt(b*c-a*d)*sqrt(e+f*x)))/(b^2*sqrt(b*c-a*d)*sqrt(b*e-a*f))+C*sqrt(c+d*x)*sqrt(e+f*x)/(b*d*f)],
[(A+B*x+C*x^2)/((a+b*x)^2*sqrt(c+d*x)*sqrt(e+f*x)),x,7,(2*a^3*C*d*f-3*a^2*b*C*(d*e+c*f)-b^3*(2*B*c*e-A*d*e-A*c*f)+a*b^2*(4*c*C*e+B*d*e+B*c*f-2*A*d*f))*arctanh(sqrt(b*e-a*f)*sqrt(c+d*x)/(sqrt(b*c-a*d)*sqrt(e+f*x)))/(b^2*(b*c-a*d)^(3/2)*(b*e-a*f)^(3/2))+2*C*arctanh(sqrt(f)*sqrt(c+d*x)/(sqrt(d)*sqrt(e+f*x)))/(b^2*sqrt(d)*sqrt(f))-(A*b^2-a*(b*B-a*C))*sqrt(c+d*x)*sqrt(e+f*x)/(b*(b*c-a*d)*(b*e-a*f)*(a+b*x))],
[(A+B*x+C*x^2)/((a+b*x)^3*sqrt(c+d*x)*sqrt(e+f*x)),x,5,-1/4*(b^2*(3*A*d^2*e^2-2*c*d*e*(2*B*e-A*f)+c^2*(8*C*e^2-4*B*e*f+3*A*f^2))+a*b*(d^2*e*(B*e-8*A*f)-c^2*f*(8*C*e-B*f)-2*c*d*(4*C*e^2-7*B*e*f+4*A*f^2))+a^2*(C*(3*d^2*e^2+2*c*d*e*f+3*c^2*f^2)+4*d*f*(2*A*d*f-B*(d*e+c*f))))*arctanh(sqrt(b*e-a*f)*sqrt(c+d*x)/(sqrt(b*c-a*d)*sqrt(e+f*x)))/((b*c-a*d)^(5/2)*(b*e-a*f)^(5/2))-1/2*(A*b^2-a*(b*B-a*C))*sqrt(c+d*x)*sqrt(e+f*x)/(b*(b*c-a*d)*(b*e-a*f)*(a+b*x)^2)+1/4*(2*a^3*C*d*f+a*b^2*(8*c*C*e+B*d*e+B*c*f-6*A*d*f)-b^3*(4*B*c*e-3*A*(d*e+c*f))+a^2*b*(2*B*d*f-5*C*(d*e+c*f)))*sqrt(c+d*x)*sqrt(e+f*x)/(b*(b*c-a*d)^2*(b*e-a*f)^2*(a+b*x))],
[(A+B*x+C*x^2)/((a+b*x)^4*sqrt(c+d*x)*sqrt(e+f*x)),x,6,1/8*(b^3*(5*A*d^3*e^3-3*c*d^2*e^2*(2*B*e-A*f)+c^2*d*e*(8*C*e^2-4*B*e*f+3*A*f^2)+c^3*f*(8*C*e^2-6*B*e*f+5*A*f^2))+a*b^2*(d^3*e^2*(B*e-18*A*f)-c^3*f^2*(4*C*e-B*f)-c*d^2*e*(4*C*e^2-23*B*e*f+12*A*f^2)-c^2*d*f*(40*C*e^2-23*B*e*f+18*A*f^2))-2*a^3*d*f*(C*(3*d^2*e^2+2*c*d*e*f+3*c^2*f^2)+4*d*f*(2*A*d*f-B*(d*e+c*f)))+a^2*b*(C*(d^3*e^3+23*c*d^2*e^2*f+23*c^2*d*e*f^2+c^3*f^3)+4*d*f*(6*A*d*f*(d*e+c*f)-B*(d^2*e^2+10*c*d*e*f+c^2*f^2))))*arctanh(sqrt(b*e-a*f)*sqrt(c+d*x)/(sqrt(b*c-a*d)*sqrt(e+f*x)))/((b*c-a*d)^(7/2)*(b*e-a*f)^(7/2))-1/3*(A*b^2-a*(b*B-a*C))*sqrt(c+d*x)*sqrt(e+f*x)/(b*(b*c-a*d)*(b*e-a*f)*(a+b*x)^3)+1/12*(2*a^3*C*d*f+a*b^2*(12*c*C*e+B*d*e+B*c*f-10*A*d*f)-b^3*(6*B*c*e-5*A*(d*e+c*f))+a^2*b*(4*B*d*f-7*C*(d*e+c*f)))*sqrt(c+d*x)*sqrt(e+f*x)/(b*(b*c-a*d)^2*(b*e-a*f)^2*(a+b*x)^2)+1/24*(4*a^4*C*d^2*f^2+8*a^3*b*d*f*(B*d*f-2*C*(d*e+c*f))-b^4*(15*A*d^2*e^2-2*c*d*e*(9*B*e-7*A*f)+3*c^2*(8*C*e^2-6*B*e*f+5*A*f^2))-a*b^3*(d^2*e*(3*B*e-44*A*f)-3*c^2*f*(4*C*e-B*f)-2*c*d*(6*C*e^2-29*B*e*f+22*A*f^2))-a^2*b^2*(C*(3*d^2*e^2-34*c*d*e*f+3*c^2*f^2)+2*d*f*(22*A*d*f-5*B*(d*e+c*f))))*sqrt(c+d*x)*sqrt(e+f*x)/(b*(b*c-a*d)^3*(b*e-a*f)^3*(a+b*x))],

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

# n>0 & p>0
[(a+b*x)^(1/2)*(A+B*x+C*x^2)*sqrt(c+d*x)*sqrt(e+f*x),x,10,2/9*C*(a+b*x)^(3/2)*(c+d*x)^(3/2)*(e+f*x)^(3/2)/(b*d*f)-2/21*(2*a*C*d*f-b*(3*B*d*f-2*C*(d*e+c*f)))*(c+d*x)^(3/2)*(e+f*x)^(3/2)*sqrt(a+b*x)/(b*d^2*f^2)-2/105*(7*b*d*f*(b*c*C*e+a*C*d*e+a*c*C*f-3*A*b*d*f)+(a*d*f-4*b*(d*e+c*f))*(2*a*C*d*f-b*(3*B*d*f-2*C*(d*e+c*f))))*(e+f*x)^(3/2)*sqrt(a+b*x)*sqrt(c+d*x)/(b^2*d^2*f^3)+2/315*(8*a^3*C*d^3*f^3+3*a^2*b*d^2*f^2*(C*d*e-c*C*f-4*B*d*f)-3*a*b^2*d*f^2*((c^2*C-7*A*d^2)*f+B*d*(d*e-2*c*f))-b^3*(C*(16*d^3*e^3-3*c^2*d*e*f^2-8*c^3*f^3)+3*d*f*(7*A*d*f*(2*d*e-c*f)-B*(8*d^2*e^2-c*d*e*f-4*c^2*f^2))))*sqrt(a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)/(b^3*d^3*f^3)-2/315*(16*a^4*C*d^4*f^4-8*a^3*b*d^3*f^3*(C*d*e+c*C*f+3*B*d*f)+3*a^2*b^2*d^2*f^2*(d*f*(5*B*d*e+5*B*c*f+14*A*d*f)-2*C*(d^2*e^2-c*d*e*f+c^2*f^2))-a*b^3*d*f*(C*(8*d^3*e^3-6*c*d^2*e^2*f-6*c^2*d*e*f^2+8*c^3*f^3)+3*d*f*(14*A*d*f*(d*e+c*f)-B*(5*d^2*e^2-6*c*d*e*f+5*c^2*f^2)))+b^4*(2*C*(8*d^4*e^4-4*c*d^3*e^3*f-3*c^2*d^2*e^2*f^2-4*c^3*d*e*f^3+8*c^4*f^4)+3*d*f*(14*A*d*f*(d^2*e^2-c*d*e*f+c^2*f^2)-B*(8*d^3*e^3-5*c*d^2*e^2*f-5*c^2*d*e*f^2+8*c^3*f^3))))*EllipticE(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(-b*c+a*d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(e+f*x)/(b^4*d^(7/2)*f^4*sqrt(c+d*x)*sqrt(b*(e+f*x)/(b*e-a*f)))-2/315*(b*e-a*f)*(d*e-c*f)*(8*a^3*C*d^3*f^3+3*a^2*b*d^2*f^2*(C*d*e-c*C*f-4*B*d*f)-3*a*b^2*d*f^2*((c^2*C-7*A*d^2)*f+B*d*(d*e-2*c*f))-b^3*(C*(16*d^3*e^3-3*c^2*d*e*f^2-8*c^3*f^3)+3*d*f*(7*A*d*f*(2*d*e-c*f)-B*(8*d^2*e^2-c*d*e*f-4*c^2*f^2))))*EllipticF(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(-b*c+a*d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(b*(e+f*x)/(b*e-a*f))/(b^4*d^(7/2)*f^4*sqrt(c+d*x)*sqrt(e+f*x))],
[(A+B*x+C*x^2)*sqrt(c+d*x)*sqrt(e+f*x)/(a+b*x)^(1/2),x,9,2/7*C*(c+d*x)^(3/2)*(e+f*x)^(3/2)*sqrt(a+b*x)/(b*d*f)-2/35*(6*a*C*d*f-b*(7*B*d*f-4*C*(d*e+c*f)))*(e+f*x)^(3/2)*sqrt(a+b*x)*sqrt(c+d*x)/(b^2*d*f^2)-2/105*(5*b*d*f*(3*a*C*(d*e+c*f)+b*(c*C*e-7*A*d*f))-(2*b*d*e-b*c*f+4*a*d*f)*(6*a*C*d*f-b*(7*B*d*f-4*C*(d*e+c*f))))*sqrt(a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)/(b^3*d^2*f^2)-2/105*(3*b*d*f*(5*b*c*f*(3*a*C*(d*e+c*f)+b*(c*C*e-7*A*d*f))-(b*c*e+a*d*e+3*a*c*f)*(6*a*C*d*f-b*(7*B*d*f-4*C*(d*e+c*f))))+2*(1/2*b*d*e-(b*c+a*d)*f)*(5*b*d*f*(3*a*C*(d*e+c*f)+b*(c*C*e-7*A*d*f))-(2*b*d*e-b*c*f+4*a*d*f)*(6*a*C*d*f-b*(7*B*d*f-4*C*(d*e+c*f)))))*EllipticE(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(-b*c+a*d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(e+f*x)/(b^4*d^(5/2)*f^3*sqrt(c+d*x)*sqrt(b*(e+f*x)/(b*e-a*f)))-2/105*(b*e-a*f)*(d*e-c*f)*(24*a^2*C*d^2*f^2+a*b*d*f*(13*C*d*e-5*c*C*f-28*B*d*f)-b^2*(7*d*f*(2*B*d*e-B*c*f-5*A*d*f)-C*(8*d^2*e^2-c*d*e*f-4*c^2*f^2)))*EllipticF(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(-b*c+a*d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(b*(e+f*x)/(b*e-a*f))/(b^4*d^(5/2)*f^3*sqrt(c+d*x)*sqrt(e+f*x))],
[(A+B*x+C*x^2)*sqrt(c+d*x)*sqrt(e+f*x)/(a+b*x)^(3/2),x,9,-2*(A*b^2-a*(b*B-a*C))*(c+d*x)^(3/2)*(e+f*x)^(3/2)/(b*(b*c-a*d)*(b*e-a*f)*sqrt(a+b*x))+2/5*(6*a^2*C*d*f+b^2*(c*C*e+5*A*d*f)-a*b*(C*d*e+c*C*f+5*B*d*f))*(e+f*x)^(3/2)*sqrt(a+b*x)*sqrt(c+d*x)/(b^2*(b*c-a*d)*f*(b*e-a*f))+2/15*(24*a^2*C*d*f^2-a*b*f*(7*C*d*e+c*C*f+20*B*d*f)+b^2*(5*d*f*(B*e+3*A*f)-C*e*(2*d*e-c*f)))*sqrt(a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)/(b^3*d*f*(b*e-a*f))+2/15*(48*a^2*C*d^2*f^2-8*a*b*d*f*(C*d*e+c*C*f+5*B*d*f)+b^2*(5*d*f*(B*d*e+B*c*f+6*A*d*f)-2*C*(d^2*e^2-c*d*e*f+c^2*f^2)))*EllipticE(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(-b*c+a*d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(e+f*x)/(b^4*d^(3/2)*f^2*sqrt(c+d*x)*sqrt(b*(e+f*x)/(b*e-a*f)))-2/15*(d*e-c*f)*(24*a^2*C*d*f^2-a*b*f*(7*C*d*e+c*C*f+20*B*d*f)+b^2*(5*d*f*(B*e+3*A*f)-C*e*(2*d*e-c*f)))*EllipticF(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(-b*c+a*d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(b*(e+f*x)/(b*e-a*f))/(b^4*d^(3/2)*f^2*sqrt(c+d*x)*sqrt(e+f*x))],
[(A+B*x+C*x^2)*sqrt(c+d*x)*sqrt(e+f*x)/(a+b*x)^(5/2),x,9,-2/3*(A*b^2-a*(b*B-a*C))*(c+d*x)^(3/2)*(e+f*x)^(3/2)/(b*(b*c-a*d)*(b*e-a*f)*(a+b*x)^(3/2))-2*(b*B-2*a*C)*(e+f*x)^(3/2)*sqrt(c+d*x)/(b^2*(b*e-a*f)*sqrt(a+b*x))+2/3*(8*a^2*C*d*f+b^2*(c*C*e+3*B*c*f+A*d*f)-a*b*(C*d*e+7*c*C*f+4*B*d*f))*sqrt(a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)/(b^3*(b*c-a*d)*(b*e-a*f))+2/3*(16*a^3*C*d^2*f^2-8*a^2*b*d*f*(B*d*f+2*C*(d*e+c*f))-b^3*(c^2*C*e*f+A*d^2*e*f+c*d*(C*e^2+6*B*e*f+A*f^2))+a*b^2*(d*f*(7*B*d*e+7*B*c*f+2*A*d*f)+C*(d^2*e^2+16*c*d*e*f+c^2*f^2)))*EllipticE(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(e+f*x)/(b^4*f*(b*e-a*f)*sqrt(d)*sqrt(-b*c+a*d)*sqrt(c+d*x)*sqrt(b*(e+f*x)/(b*e-a*f)))+2/3*(d*e-c*f)*(8*a^2*C*d*f+b^2*(c*C*e+3*B*c*f+A*d*f)-a*b*(C*d*e+7*c*C*f+4*B*d*f))*EllipticF(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(b*(e+f*x)/(b*e-a*f))/(b^4*f*sqrt(d)*sqrt(-b*c+a*d)*sqrt(c+d*x)*sqrt(e+f*x))],
[(A+B*x+C*x^2)*sqrt(c+d*x)*sqrt(e+f*x)/(a+b*x)^(7/2),x,9,-2/5*(A*b^2-a*(b*B-a*C))*(c+d*x)^(3/2)*(e+f*x)^(3/2)/(b*(b*c-a*d)*(b*e-a*f)*(a+b*x)^(5/2))+2/15*(6*a^3*C*d*f+a*b^2*(10*c*C*e+3*B*d*e+3*B*c*f-4*A*d*f)-b^3*(5*B*c*e-2*A*(d*e+c*f))-a^2*b*(B*d*f+8*C*(d*e+c*f)))*(e+f*x)^(3/2)*sqrt(c+d*x)/(b^2*(b*c-a*d)*(b*e-a*f)^2*(a+b*x)^(3/2))+2/15*(24*a^3*C*d^2*f-a^2*b*d*(23*C*d*e+41*c*C*f+4*B*d*f)-b^3*(15*c^2*C*e-2*A*d^2*e+c*d*(5*B*e+A*f))+a*b^2*(15*c^2*C*f+d^2*(3*B*e-A*f)+c*(40*C*d*e+6*B*d*f)))*sqrt(c+d*x)*sqrt(e+f*x)/(b^3*(b*c-a*d)^2*(b*e-a*f)*sqrt(a+b*x))+2/15*(48*a^4*C*d^2*f^2-8*a^3*b*d*f*(B*d*f+11*C*(d*e+c*f))-b^4*(2*A*d^2*e^2-c*d*e*(5*B*e+2*A*f)-c^2*(30*C*e^2+5*B*e*f-2*A*f^2))-a*b^3*(d^2*e*(3*B*e-2*A*f)+c^2*f*(70*C*e+3*B*f)+2*c*d*(35*C*e^2+11*B*e*f-A*f^2))+a^2*b^2*(2*C*(19*d^2*e^2+81*c*d*e*f+19*c^2*f^2)-d*f*(2*A*d*f-13*B*(d*e+c*f))))*EllipticE(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(e+f*x)/(b^4*(-b*c+a*d)^(3/2)*(b*e-a*f)^2*sqrt(c+d*x)*sqrt(b*(e+f*x)/(b*e-a*f)))+2/15*(d*e-c*f)*(24*a^3*C*d^2*f-a^2*b*d*(23*C*d*e+41*c*C*f+4*B*d*f)-b^3*(15*c^2*C*e-2*A*d^2*e+c*d*(5*B*e+A*f))+a*b^2*(15*c^2*C*f+d^2*(3*B*e-A*f)+c*(40*C*d*e+6*B*d*f)))*EllipticF(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(b*(e+f*x)/(b*e-a*f))/(b^4*(-b*c+a*d)^(3/2)*(b*e-a*f)*sqrt(d)*sqrt(c+d*x)*sqrt(e+f*x))],
[(A+B*x+C*x^2)*sqrt(c+d*x)*sqrt(e+f*x)/(a+b*x)^(9/2),x,10,-2/7*(A*b^2-a*(b*B-a*C))*(c+d*x)^(3/2)*(e+f*x)^(3/2)/(b*(b*c-a*d)*(b*e-a*f)*(a+b*x)^(7/2))+2/35*(6*a^3*C*d*f+a*b^2*(14*c*C*e+3*B*d*e+3*B*c*f-8*A*d*f)-b^3*(7*B*c*e-4*A*(d*e+c*f))+a^2*b*(B*d*f-10*C*(d*e+c*f)))*(e+f*x)^(3/2)*sqrt(c+d*x)/(b^2*(b*c-a*d)*(b*e-a*f)^2*(a+b*x)^(5/2))-2/105*(24*a^4*C*d^2*f^2-a^3*b*d*f*(61*C*d*e+43*c*C*f-4*B*d*f)-3*a*b^3*(d^2*e*(B*e-3*A*f)+2*c^2*f*(7*C*e-B*f)+c*d*(28*C*e^2-5*B*e*f+5*A*f^2))-b^4*(4*A*d^2*e^2-c*d*e*(7*B*e-A*f)-c^2*(35*C*e^2-14*B*e*f+8*A*f^2))-3*a^2*b^2*(d*f*(3*B*d*e+2*B*c*f-A*d*f)-C*(15*d^2*e^2+37*c*d*e*f+5*c^2*f^2)))*sqrt(c+d*x)*sqrt(e+f*x)/(b^3*(b*c-a*d)^2*(b*e-a*f)^2*(a+b*x)^(3/2))+2/105*(48*a^5*C*d^3*f^3+8*a^4*b*d^2*f^2*(B*d*f-16*C*(d*e+c*f))-b^5*(8*A*d^3*e^3-c*d^2*e^2*(14*B*e+5*A*f)+c^2*d*e*(35*C*e^2+14*B*e*f-5*A*f^2)+c^3*f*(35*C*e^2-14*B*e*f+8*A*f^2))-a*b^4*(d^3*e^2*(6*B*e-19*A*f)-6*c^3*f^2*(7*C*e-B*f)-c^2*d*f*(238*C*e^2-19*f*(B*e-A*f))-c*d^2*e*(42*C*e^2-f*(19*B*e+20*A*f)))+a^3*b^2*d*f*(C*(103*d^2*e^2+344*c*d*e*f+103*c^2*f^2)+d*f*(6*A*d*f-19*B*(d*e+c*f)))-3*a^2*b^3*(C*(5*d^3*e^3+94*c*d^2*e^2*f+94*c^2*d*e*f^2+5*c^3*f^3)+d*f*(3*A*d*f*(d*e+c*f)-B*(3*d^2*e^2+16*c*d*e*f+3*c^2*f^2))))*sqrt(c+d*x)*sqrt(e+f*x)/(b^3*(b*c-a*d)^3*(b*e-a*f)^3*sqrt(a+b*x))+2/105*(48*a^5*C*d^3*f^3+8*a^4*b*d^2*f^2*(B*d*f-16*C*(d*e+c*f))-b^5*(8*A*d^3*e^3-c*d^2*e^2*(14*B*e+5*A*f)+c^2*d*e*(35*C*e^2+14*B*e*f-5*A*f^2)+c^3*f*(35*C*e^2-14*B*e*f+8*A*f^2))-a*b^4*(d^3*e^2*(6*B*e-19*A*f)-6*c^3*f^2*(7*C*e-B*f)-c^2*d*f*(238*C*e^2-19*f*(B*e-A*f))-c*d^2*e*(42*C*e^2-f*(19*B*e+20*A*f)))+a^3*b^2*d*f*(C*(103*d^2*e^2+344*c*d*e*f+103*c^2*f^2)+d*f*(6*A*d*f-19*B*(d*e+c*f)))-3*a^2*b^3*(C*(5*d^3*e^3+94*c*d^2*e^2*f+94*c^2*d*e*f^2+5*c^3*f^3)+d*f*(3*A*d*f*(d*e+c*f)-B*(3*d^2*e^2+16*c*d*e*f+3*c^2*f^2))))*EllipticE(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(e+f*x)/(b^4*(-b*c+a*d)^(5/2)*(b*e-a*f)^3*sqrt(c+d*x)*sqrt(b*(e+f*x)/(b*e-a*f)))+2/105*(d*e-c*f)*(24*a^4*C*d^2*f^2-a^3*b*d*f*(43*C*d*e+61*c*C*f-4*B*d*f)+b^4*(8*A*d^2*e^2-c*d*e*(14*B*e+A*f)+c^2*(35*C*e^2+7*B*e*f-4*A*f^2))+3*a*b^3*(d^2*e*(2*B*e-5*A*f)-c^2*f*(28*C*e+B*f)-c*d*(14*C*e^2-5*B*e*f-3*A*f^2))-3*a^2*b^2*(d*f*(2*B*d*e+3*B*c*f-A*d*f)-C*(5*d^2*e^2+37*c*d*e*f+15*c^2*f^2)))*EllipticF(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(b*(e+f*x)/(b*e-a*f))/(b^4*(-b*c+a*d)^(5/2)*(b*e-a*f)^2*sqrt(c+d*x)*sqrt(e+f*x))],

# n>0 & p<0
[(a+b*x)^(3/2)*(A+B*x+C*x^2)*sqrt(c+d*x)/sqrt(e+f*x),x,10,-2/63*(4*a*C*d*f+b*(8*C*d*e+6*c*C*f-9*B*d*f))*(a+b*x)^(3/2)*(c+d*x)^(3/2)*sqrt(e+f*x)/(b*d^2*f^2)+2/9*C*(a+b*x)^(5/2)*(c+d*x)^(3/2)*sqrt(e+f*x)/(b*d*f)-2/315*(7*b*d*f*(5*b*c*C*e+3*a*C*d*e+a*c*C*f-9*A*b*d*f)-(6*b*d*e+4*b*c*f-3*a*d*f)*(4*a*C*d*f+b*(8*C*d*e+6*c*C*f-9*B*d*f)))*(c+d*x)^(3/2)*sqrt(a+b*x)*sqrt(e+f*x)/(b*d^3*f^3)-2/945*(5*b*d*f*(7*a*d*f*(5*b*c*C*e+3*a*C*d*e+a*c*C*f-9*A*b*d*f)-(3*b*c*e+3*a*d*e+a*c*f)*(4*a*C*d*f+b*(8*C*d*e+6*c*C*f-9*B*d*f)))+2*(1/2*a*d*f-b*(2*d*e+c*f))*(7*b*d*f*(5*b*c*C*e+3*a*C*d*e+a*c*C*f-9*A*b*d*f)-(6*b*d*e+4*b*c*f-3*a*d*f)*(4*a*C*d*f+b*(8*C*d*e+6*c*C*f-9*B*d*f))))*sqrt(a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)/(b^2*d^3*f^4)+2/315*(8*a^4*C*d^4*f^4+a^3*b*d^3*f^3*(11*C*d*e-7*c*C*f-18*B*d*f)-3*a^2*b^2*d^2*f^2*(3*d*f*(4*B*d*e-3*B*c*f-7*A*d*f)-C*(9*d^2*e^2-5*c*d*e*f-3*c^2*f^2))-a*b^3*d*f*(2*C*(92*d^3*e^3-33*c*d^2*e^2*f-18*c^2*d*e*f^2-16*c^3*f^3)+3*d*f*(7*A*d*f*(13*d*e-7*c*f)-B*(72*d^2*e^2-29*c*d*e*f-19*c^2*f^2)))+b^4*(C*(128*d^4*e^4-40*c*d^3*e^3*f-21*c^2*d^2*e^2*f^2-16*c^3*d*e*f^3-16*c^4*f^4)+3*d*f*(7*A*d*f*(8*d^2*e^2-3*c*d*e*f-2*c^2*f^2)-B*(48*d^3*e^3-16*c*d^2*e^2*f-9*c^2*d*e*f^2-8*c^3*f^3))))*EllipticE(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(-b*c+a*d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(e+f*x)/(b^3*d^(7/2)*f^5*sqrt(c+d*x)*sqrt(b*(e+f*x)/(b*e-a*f)))+2/315*(b*e-a*f)*(d*e-c*f)*(4*a^3*C*d^3*f^3+3*a^2*b*d^2*f^2*(3*C*d*e-c*C*f-3*B*d*f)-3*a*b^2*d*f*(3*d*f*(16*B*d*e+3*B*c*f-21*A*d*f)-5*C*(8*d^2*e^2+2*c*d*e*f+c^2*f^2))-b^3*(C*(128*d^3*e^3+24*c*d^2*e^2*f+15*c^2*d*e*f^2+8*c^3*f^3)+3*d*f*(7*A*d*f*(8*d*e+c*f)-4*B*(12*d^2*e^2+2*c*d*e*f+c^2*f^2))))*EllipticF(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(-b*c+a*d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(b*(e+f*x)/(b*e-a*f))/(b^3*d^(7/2)*f^5*sqrt(c+d*x)*sqrt(e+f*x))],
[(a+b*x)^(1/2)*(A+B*x+C*x^2)*sqrt(c+d*x)/sqrt(e+f*x),x,9,2/7*C*(a+b*x)^(3/2)*(c+d*x)^(3/2)*sqrt(e+f*x)/(b*d*f)-2/35*(4*a*C*d*f+b*(6*C*d*e+4*c*C*f-7*B*d*f))*(c+d*x)^(3/2)*sqrt(a+b*x)*sqrt(e+f*x)/(b*d^2*f^2)-2/105*(5*b*d*f*(3*b*c*C*e+3*a*C*d*e+a*c*C*f-7*A*b*d*f)+(a*d*f-2*b*(2*d*e+c*f))*(4*a*C*d*f+b*(6*C*d*e+4*c*C*f-7*B*d*f)))*sqrt(a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)/(b^2*d^2*f^3)-2/105*(3*b*d*f*(5*a*d*f*(3*b*c*C*e+3*a*C*d*e+a*c*C*f-7*A*b*d*f)-(b*c*e+3*a*d*e+a*c*f)*(4*a*C*d*f+b*(6*C*d*e+4*c*C*f-7*B*d*f)))+2*(1/2*b*c*f-d*(b*e+a*f))*(5*b*d*f*(3*b*c*C*e+3*a*C*d*e+a*c*C*f-7*A*b*d*f)+(a*d*f-2*b*(2*d*e+c*f))*(4*a*C*d*f+b*(6*C*d*e+4*c*C*f-7*B*d*f))))*EllipticE(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(-b*c+a*d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(e+f*x)/(b^3*d^(5/2)*f^4*sqrt(c+d*x)*sqrt(b*(e+f*x)/(b*e-a*f)))+2/105*(b*e-a*f)*(d*e-c*f)*(4*a^2*C*d^2*f^2+a*b*d*f*(8*C*d*e-2*c*C*f-7*B*d*f)-b^2*(7*d*f*(8*B*d*e+B*c*f-10*A*d*f)-4*C*(12*d^2*e^2+2*c*d*e*f+c^2*f^2)))*EllipticF(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(-b*c+a*d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(b*(e+f*x)/(b*e-a*f))/(b^3*d^(5/2)*f^4*sqrt(c+d*x)*sqrt(e+f*x))],
[(A+B*x+C*x^2)*sqrt(c+d*x)/((a+b*x)^(1/2)*sqrt(e+f*x)),x,8,2/5*C*(c+d*x)^(3/2)*sqrt(a+b*x)*sqrt(e+f*x)/(b*d*f)-2/15*(4*a*C*d*f+b*(4*C*d*e+2*c*C*f-5*B*d*f))*sqrt(a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)/(b^2*d*f^2)-2/15*(3*b*d*f*(b*c*C*e+3*a*C*d*e+a*c*C*f-5*A*b*d*f)-(2*b*d*e-b*c*f+2*a*d*f)*(4*a*C*d*f+b*(4*C*d*e+2*c*C*f-5*B*d*f)))*EllipticE(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(-b*c+a*d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(e+f*x)/(b^3*d^(3/2)*f^3*sqrt(c+d*x)*sqrt(b*(e+f*x)/(b*e-a*f)))-2/15*(d*e-c*f)*(4*a^2*C*d*f^2+a*b*f*(3*C*d*e-c*C*f-5*B*d*f)-b^2*(5*d*f*(2*B*e-3*A*f)-C*e*(8*d*e+c*f)))*EllipticF(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(-b*c+a*d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(b*(e+f*x)/(b*e-a*f))/(b^3*d^(3/2)*f^3*sqrt(c+d*x)*sqrt(e+f*x))],
[(A+B*x+C*x^2)*sqrt(c+d*x)/((a+b*x)^(3/2)*sqrt(e+f*x)),x,8,-2*(A*b^2-a*(b*B-a*C))*(c+d*x)^(3/2)*sqrt(e+f*x)/(b*(b*c-a*d)*(b*e-a*f)*sqrt(a+b*x))+2/3*(4*a^2*C*d*f+b^2*(c*C*e+3*A*d*f)-a*b*(C*d*e+c*C*f+3*B*d*f))*sqrt(a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)/(b^2*(b*c-a*d)*f*(b*e-a*f))+2/3*(8*a^2*C*d*f^2-a*b*f*(3*C*d*e+c*C*f+6*B*d*f)+b^2*(3*d*f*(B*e+A*f)-C*e*(2*d*e-c*f)))*EllipticE(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(-b*c+a*d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(e+f*x)/(b^3*f^2*(b*e-a*f)*sqrt(d)*sqrt(c+d*x)*sqrt(b*(e+f*x)/(b*e-a*f)))+2/3*(d*e-c*f)*(2*b*C*e-3*b*B*f+4*a*C*f)*EllipticF(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(-b*c+a*d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(b*(e+f*x)/(b*e-a*f))/(b^3*f^2*sqrt(d)*sqrt(c+d*x)*sqrt(e+f*x))],
[(A+B*x+C*x^2)*sqrt(c+d*x)/((a+b*x)^(5/2)*sqrt(e+f*x)),x,8,-2/3*(A*b^2-a*(b*B-a*C))*(c+d*x)^(3/2)*sqrt(e+f*x)/(b*(b*c-a*d)*(b*e-a*f)*(a+b*x)^(3/2))-2/3*(4*a^2*C*f+b^2*(3*B*e-2*A*f)-a*b*(6*C*e+B*f))*sqrt(c+d*x)*sqrt(e+f*x)/(b^2*(b*e-a*f)^2*sqrt(a+b*x))+2/3*(8*a^3*C*d*f^2-a^2*b*f*(13*C*d*e+7*c*C*f+2*B*d*f)+a*b^2*(3*C*e*(d*e+4*c*f)+f*(4*B*d*e+B*c*f-A*d*f))-b^3*(A*d*e*f+c*(3*C*e^2+3*B*e*f-2*A*f^2)))*EllipticE(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(e+f*x)/(b^3*f*(b*e-a*f)^2*sqrt(-b*c+a*d)*sqrt(c+d*x)*sqrt(b*(e+f*x)/(b*e-a*f)))+2/3*(d*e-c*f)*(4*a^2*C*d*f+b^2*(3*c*C*e+A*d*f)-a*b*(B*d*f+3*C*(d*e+c*f)))*EllipticF(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(b*(e+f*x)/(b*e-a*f))/(b^3*f*(b*e-a*f)*sqrt(d)*sqrt(-b*c+a*d)*sqrt(c+d*x)*sqrt(e+f*x))],
[(A+B*x+C*x^2)*sqrt(c+d*x)/((a+b*x)^(7/2)*sqrt(e+f*x)),x,9,-2/5*(A*b^2-a*(b*B-a*C))*(c+d*x)^(3/2)*sqrt(e+f*x)/(b*(b*c-a*d)*(b*e-a*f)*(a+b*x)^(5/2))+2/15*(4*a^3*C*d*f-b^3*(5*B*c*e-2*A*d*e-4*A*c*f)+a*b^2*(10*c*C*e+3*B*d*e+B*c*f-6*A*d*f)-a^2*b*(8*C*d*e+6*c*C*f-B*d*f))*sqrt(c+d*x)*sqrt(e+f*x)/(b^2*(b*c-a*d)*(b*e-a*f)^2*(a+b*x)^(3/2))-2/15*(8*a^4*C*d^2*f^2-a^3*b*d*f*(23*C*d*e+13*c*C*f-2*B*d*f)-b^4*(2*A*d^2*e^2-c*d*e*(5*B*e-3*A*f)-c^2*(15*C*e^2-10*B*e*f+8*A*f^2))-a^2*b^2*(d*f*(7*B*d*e+2*B*c*f-3*A*d*f)-C*(23*d^2*e^2+37*c*d*e*f+3*c^2*f^2))-a*b^3*(d^2*e*(3*B*e-7*A*f)+2*c^2*f*(5*C*e-B*f)+c*d*(40*C*e^2-13*f*(B*e-A*f))))*sqrt(c+d*x)*sqrt(e+f*x)/(b^2*(b*c-a*d)^2*(b*e-a*f)^3*sqrt(a+b*x))+2/15*(8*a^4*C*d^2*f^2-a^3*b*d*f*(23*C*d*e+13*c*C*f-2*B*d*f)-b^4*(2*A*d^2*e^2-c*d*e*(5*B*e-3*A*f)-c^2*(15*C*e^2-10*B*e*f+8*A*f^2))-a^2*b^2*(d*f*(7*B*d*e+2*B*c*f-3*A*d*f)-C*(23*d^2*e^2+37*c*d*e*f+3*c^2*f^2))-a*b^3*(d^2*e*(3*B*e-7*A*f)+2*c^2*f*(5*C*e-B*f)+c*d*(40*C*e^2-13*f*(B*e-A*f))))*EllipticE(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(e+f*x)/(b^3*(-b*c+a*d)^(3/2)*(b*e-a*f)^3*sqrt(c+d*x)*sqrt(b*(e+f*x)/(b*e-a*f)))+2/15*(d*e-c*f)*(4*a^3*C*d*f-b^3*(5*B*c*e-2*A*d*e-4*A*c*f)+a*b^2*(10*c*C*e+3*B*d*e+B*c*f-6*A*d*f)-a^2*b*(8*C*d*e+6*c*C*f-B*d*f))*EllipticF(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(b*(e+f*x)/(b*e-a*f))/(b^3*(-b*c+a*d)^(3/2)*(b*e-a*f)^2*sqrt(c+d*x)*sqrt(e+f*x))],

# n<0 & p<0
[(a+b*x)^(3/2)*(A+B*x+C*x^2)/(sqrt(c+d*x)*sqrt(e+f*x)),x,9,-2/35*(2*a*C*d*f-b*(7*B*d*f-6*C*(d*e+c*f)))*(a+b*x)^(3/2)*sqrt(c+d*x)*sqrt(e+f*x)/(b*d^2*f^2)+2/7*C*(a+b*x)^(5/2)*sqrt(c+d*x)*sqrt(e+f*x)/(b*d*f)-2/105*(5*b*d*f*(5*b*c*C*e+a*C*d*e+a*c*C*f-7*A*b*d*f)+(3*a*d*f-4*b*(d*e+c*f))*(2*a*C*d*f-b*(7*B*d*f-6*C*(d*e+c*f))))*sqrt(a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)/(b*d^3*f^3)-2/105*(3*b*d*f*(5*a*d*f*(5*b*c*C*e+a*C*d*e+a*c*C*f-7*A*b*d*f)-(3*b*c*e+a*d*e+a*c*f)*(2*a*C*d*f-b*(7*B*d*f-6*C*(d*e+c*f))))+2*(1/2*a*d*f-b*(d*e+c*f))*(5*b*d*f*(5*b*c*C*e+a*C*d*e+a*c*C*f-7*A*b*d*f)+(3*a*d*f-4*b*(d*e+c*f))*(2*a*C*d*f-b*(7*B*d*f-6*C*(d*e+c*f)))))*EllipticE(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(-b*c+a*d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(e+f*x)/(b^2*d^(7/2)*f^4*sqrt(c+d*x)*sqrt(b*(e+f*x)/(b*e-a*f)))-2/105*(b*e-a*f)*(3*a^2*C*d^2*f^2*(d*e-c*f)-3*a*b*d*f*(7*d*f*(3*B*d*e+2*B*c*f-5*A*d*f)-C*(16*d^2*e^2+8*c*d*e*f+11*c^2*f^2))-b^2*(C*(48*d^3*e^3+16*c*d^2*e^2*f+17*c^2*d*e*f^2+24*c^3*f^3)+7*d*f*(5*A*d*f*(2*d*e+c*f)-B*(8*d^2*e^2+3*c*d*e*f+4*c^2*f^2))))*EllipticF(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(-b*c+a*d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(b*(e+f*x)/(b*e-a*f))/(b^2*d^(7/2)*f^4*sqrt(c+d*x)*sqrt(e+f*x))],
[(a+b*x)^(1/2)*(A+B*x+C*x^2)/(sqrt(c+d*x)*sqrt(e+f*x)),x,8,2/5*C*(a+b*x)^(3/2)*sqrt(c+d*x)*sqrt(e+f*x)/(b*d*f)-2/15*(2*a*C*d*f-b*(5*B*d*f-4*C*(d*e+c*f)))*sqrt(a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)/(b*d^2*f^2)-2/15*(3*b*d*f*(3*b*c*C*e+a*C*d*e+a*c*C*f-5*A*b*d*f)+(a*d*f-2*b*(d*e+c*f))*(2*a*C*d*f-b*(5*B*d*f-4*C*(d*e+c*f))))*EllipticE(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(-b*c+a*d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(e+f*x)/(b^2*d^(5/2)*f^3*sqrt(c+d*x)*sqrt(b*(e+f*x)/(b*e-a*f)))-2/15*(b*e-a*f)*(a*C*d*f*(d*e-c*f)-b*(5*d*f*(2*B*d*e+B*c*f-3*A*d*f)-C*(8*d^2*e^2+3*c*d*e*f+4*c^2*f^2)))*EllipticF(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(-b*c+a*d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(b*(e+f*x)/(b*e-a*f))/(b^2*d^(5/2)*f^3*sqrt(c+d*x)*sqrt(e+f*x))],
[(A+B*x+C*x^2)/((a+b*x)^(1/2)*sqrt(c+d*x)*sqrt(e+f*x)),x,7,2/3*C*sqrt(a+b*x)*sqrt(c+d*x)*sqrt(e+f*x)/(b*d*f)-2/3*(2*a*C*d*f-b*(3*B*d*f-2*C*(d*e+c*f)))*EllipticE(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(-b*c+a*d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(e+f*x)/(b^2*d^(3/2)*f^2*sqrt(c+d*x)*sqrt(b*(e+f*x)/(b*e-a*f)))+2/3*(a*C*f*(d*e-c*f)-b*(3*d*f*(B*e-A*f)-C*e*(2*d*e+c*f)))*EllipticF(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(-b*c+a*d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(b*(e+f*x)/(b*e-a*f))/(b^2*d^(3/2)*f^2*sqrt(c+d*x)*sqrt(e+f*x))],
[(A+B*x+C*x^2)/((a+b*x)^(3/2)*sqrt(c+d*x)*sqrt(e+f*x)),x,7,-2*(A*b^2-a*(b*B-a*C))*sqrt(c+d*x)*sqrt(e+f*x)/(b*(b*c-a*d)*(b*e-a*f)*sqrt(a+b*x))-2*(2*a^2*C*d*f+b^2*(c*C*e+A*d*f)-a*b*(C*d*e+c*C*f+B*d*f))*EllipticE(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(e+f*x)/(b^2*f*(b*e-a*f)*sqrt(d)*sqrt(-b*c+a*d)*sqrt(c+d*x)*sqrt(b*(e+f*x)/(b*e-a*f)))-2*(a*C*(d*e-c*f)-b*(c*C*e-B*c*f+A*d*f))*EllipticF(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(b*(e+f*x)/(b*e-a*f))/(b^2*f*sqrt(d)*sqrt(-b*c+a*d)*sqrt(c+d*x)*sqrt(e+f*x))],
[(A+B*x+C*x^2)/((a+b*x)^(5/2)*sqrt(c+d*x)*sqrt(e+f*x)),x,8,-2/3*(A*b^2-a*(b*B-a*C))*sqrt(c+d*x)*sqrt(e+f*x)/(b*(b*c-a*d)*(b*e-a*f)*(a+b*x)^(3/2))+2/3*(2*a^3*C*d*f+a*b^2*(6*c*C*e+B*d*e+B*c*f-4*A*d*f)-b^3*(3*B*c*e-2*A*(d*e+c*f))+a^2*b*(B*d*f-4*C*(d*e+c*f)))*sqrt(c+d*x)*sqrt(e+f*x)/(b*(b*c-a*d)^2*(b*e-a*f)^2*sqrt(a+b*x))-2/3*(2*a^3*C*d*f+a*b^2*(6*c*C*e+B*d*e+B*c*f-4*A*d*f)-b^3*(3*B*c*e-2*A*(d*e+c*f))+a^2*b*(B*d*f-4*C*(d*e+c*f)))*EllipticE(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(e+f*x)/(b^2*(-b*c+a*d)^(3/2)*(b*e-a*f)^2*sqrt(c+d*x)*sqrt(b*(e+f*x)/(b*e-a*f)))-2/3*(a^2*C*d*(d*e-c*f)-b^2*(3*c^2*C*e-3*B*c*d*e+2*A*d^2*e+A*c*d*f)+a*b*(3*(c^2*C+A*d^2)*f-B*d*(d*e+2*c*f)))*EllipticF(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(b*(e+f*x)/(b*e-a*f))/(b^2*(-b*c+a*d)^(3/2)*(b*e-a*f)*sqrt(d)*sqrt(c+d*x)*sqrt(e+f*x))],
[(A+B*x+C*x^2)/((a+b*x)^(7/2)*sqrt(c+d*x)*sqrt(e+f*x)),x,9,-2/5*(A*b^2-a*(b*B-a*C))*sqrt(c+d*x)*sqrt(e+f*x)/(b*(b*c-a*d)*(b*e-a*f)*(a+b*x)^(5/2))+2/15*(2*a^3*C*d*f+a*b^2*(10*c*C*e+B*d*e+B*c*f-8*A*d*f)-b^3*(5*B*c*e-4*A*(d*e+c*f))+3*a^2*b*(B*d*f-2*C*(d*e+c*f)))*sqrt(c+d*x)*sqrt(e+f*x)/(b*(b*c-a*d)^2*(b*e-a*f)^2*(a+b*x)^(3/2))+2/15*(2*a^4*C*d^2*f^2+a^3*b*d*f*(3*B*d*f-7*C*(d*e+c*f))-b^4*(8*A*d^2*e^2-c*d*e*(10*B*e-7*A*f)+c^2*(15*C*e^2-10*B*e*f+8*A*f^2))-a*b^3*(d^2*e*(2*B*e-23*A*f)-2*c^2*f*(5*C*e-B*f)-c*d*(10*C*e^2-33*B*e*f+23*A*f^2))-a^2*b^2*(C*(3*d^2*e^2-13*c*d*e*f+3*c^2*f^2)+d*f*(23*A*d*f-7*B*(d*e+c*f))))*sqrt(c+d*x)*sqrt(e+f*x)/(b*(b*c-a*d)^3*(b*e-a*f)^3*sqrt(a+b*x))+2/15*(2*a^4*C*d^2*f^2+a^3*b*d*f*(3*B*d*f-7*C*(d*e+c*f))-b^4*(8*A*d^2*e^2-c*d*e*(10*B*e-7*A*f)+c^2*(15*C*e^2-10*B*e*f+8*A*f^2))-a*b^3*(d^2*e*(2*B*e-23*A*f)-2*c^2*f*(5*C*e-B*f)-c*d*(10*C*e^2-33*B*e*f+23*A*f^2))-a^2*b^2*(C*(3*d^2*e^2-13*c*d*e*f+3*c^2*f^2)+d*f*(23*A*d*f-7*B*(d*e+c*f))))*EllipticE(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(e+f*x)/(b^2*(-b*c+a*d)^(5/2)*(b*e-a*f)^3*sqrt(c+d*x)*sqrt(b*(e+f*x)/(b*e-a*f)))+2/15*(a^3*C*d*f*(d*e-c*f)+b^3*(8*A*d^2*e^2-c*d*e*(10*B*e-3*A*f)+c^2*(15*C*e^2-5*B*e*f+4*A*f^2))+a*b^2*(d^2*e*(2*B*e-19*A*f)-c^2*f*(20*C*e-B*f)-c*d*(10*C*e^2-27*B*e*f+11*A*f^2))-3*a^2*b*(d*f*(2*B*d*e+3*B*c*f-5*A*d*f)-C*(d^2*e^2+c*d*e*f+3*c^2*f^2)))*EllipticF(sqrt(d)*sqrt(a+b*x)/sqrt(-b*c+a*d),sqrt((b*c-a*d)*f/(d*(b*e-a*f))))*sqrt(d)*sqrt(b*(c+d*x)/(b*c-a*d))*sqrt(b*(e+f*x)/(b*e-a*f))/(b^2*(-b*c+a*d)^(5/2)*(b*e-a*f)^2*sqrt(c+d*x)*sqrt(e+f*x))]]:
