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