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