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