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