\[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b x^3+a \left (q+p x^3\right )^3\right )}{x^6} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {p^{2} x^{6}+2 p q \,x^{3}-2 p q \,x^{2}+q^{2}}\, \left (6 a \,p^{4} x^{12}+24 a \,p^{3} q \,x^{9}-4 a \,p^{3} q \,x^{8}+36 a \,p^{2} q^{2} x^{6}-8 a \,p^{2} q^{2} x^{5}-16 a \,p^{2} q^{2} x^{4}+24 a p \,q^{3} x^{3}+15 b p \,x^{6}-4 a p \,q^{3} x^{2}+6 a \,q^{4}+15 b q \,x^{3}\right )}{30 x^{5}}+b p q \ln \left (x \right )-b p q \ln \left (q +p \,x^{3}+\sqrt {p^{2} x^{6}+2 p q \,x^{3}-2 p q \,x^{2}+q^{2}}\right ) \]
command
Integrate[((-q + 2*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(b*x^3 + a*(q + p*x^3)^3))/x^6,x]
Mathematica 13.1 output
\[ \frac {\sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6} \left (15 b x^3 \left (q+p x^3\right )+2 a \left (3 q^4+3 p^4 x^{12}+2 p q^3 x^2 (-1+6 x)+2 p^3 q x^8 (-1+6 x)+2 p^2 q^2 x^4 \left (-4-2 x+9 x^2\right )\right )\right )}{30 x^5}-b p q \tanh ^{-1}\left (\frac {\sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6}}{q+p x^3}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b x^3+a \left (q+p x^3\right )^3\right )}{x^6} \, dx \]________________________________________________________________________________________