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