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