\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^6+a \left (q+p x^3\right )^3\right )}{x^{11}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {p^{2} x^{6}-2 p q \,x^{4}+2 p q \,x^{3}+q^{2}}\, \left (6 a \,p^{4} x^{12}-4 a \,p^{3} q \,x^{10}+24 a \,p^{3} q \,x^{9}-16 a \,p^{2} q^{2} x^{8}-8 a \,p^{2} q^{2} x^{7}+36 a \,p^{2} q^{2} x^{6}+15 b p \,x^{9}-4 a p \,q^{3} x^{4}+24 a p \,q^{3} x^{3}+15 b q \,x^{6}+6 a \,q^{4}\right )}{30 x^{10}}+2 b p q \ln \left (x \right )-b p q \ln \left (q +p \,x^{3}+\sqrt {p^{2} x^{6}-2 p q \,x^{4}+2 p q \,x^{3}+q^{2}}\right ) \]
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^6 + a*(q + p*x^3)^3))/x^11,x]
Mathematica 13.1 output
\[ \frac {\sqrt {q^2-2 p q (-1+x) x^3+p^2 x^6} \left (15 b x^6 \left (q+p x^3\right )+2 a \left (3 q^4-2 p q^3 (-6+x) x^3-2 p^3 q (-6+x) x^9+3 p^4 x^{12}-2 p^2 q^2 x^6 \left (-9+2 x+4 x^2\right )\right )\right )}{30 x^{10}}-b p q \tanh ^{-1}\left (\frac {\sqrt {q^2-2 p q (-1+x) x^3+p^2 x^6}}{q+p x^3}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^6+a \left (q+p x^3\right )^3\right )}{x^{11}} \, dx \]________________________________________________________________________________________