\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^8+a \left (q+p x^3\right )^4\right )}{x^{13}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {p^{2} x^{6}-2 p q \,x^{4}+2 p q \,x^{3}+q^{2}}\, \left (2 a \,p^{5} x^{15}-a \,p^{4} q \,x^{13}+10 a \,p^{4} q \,x^{12}-3 a \,p^{3} q^{2} x^{11}-3 a \,p^{3} q^{2} x^{10}+20 a \,p^{3} q^{2} x^{9}-3 a \,p^{2} q^{3} x^{8}-3 a \,p^{2} q^{3} x^{7}+6 b p \,x^{11}+20 a \,p^{2} q^{3} x^{6}-a p \,q^{4} x^{4}+6 b q \,x^{8}+10 a p \,q^{4} x^{3}+2 a \,q^{5}\right )}{12 x^{12}}+\left (a \,p^{3} q^{3}+2 b p q \right ) \ln \left (x \right )+\frac {\left (-a \,p^{3} q^{3}-2 b p q \right ) \ln \left (q +p \,x^{3}+\sqrt {p^{2} x^{6}-2 p q \,x^{4}+2 p q \,x^{3}+q^{2}}\right )}{2} \]
command
Integrate[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(b*x^8 + a*(q + p*x^3)^4))/x^13,x]
Mathematica 13.1 output
\[ \frac {1}{12} \left (\frac {\left (q+p x^3\right ) \sqrt {q^2-2 p q (-1+x) x^3+p^2 x^6} \left (6 b x^8+a \left (2 q^4-p q^3 (-8+x) x^3-p^3 q (-8+x) x^9+2 p^4 x^{12}+p^2 q^2 x^6 \left (12-2 x-3 x^2\right )\right )\right )}{x^{12}}-6 p q \left (2 b+a p^2 q^2\right ) \tanh ^{-1}\left (\frac {\sqrt {q^2-2 p q (-1+x) x^3+p^2 x^6}}{q+p x^3}\right )\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^8+a \left (q+p x^3\right )^4\right )}{x^{13}} \, dx \]________________________________________________________________________________________