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