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