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