\[ \int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx \]
Optimal antiderivative \[ 2 \arctan \left (\frac {\left (x^{3} a +b \right )^{\frac {1}{4}}}{x^{2}}\right )-2 \arctanh \left (\frac {x^{2}}{\left (x^{3} a +b \right )^{\frac {1}{4}}}\right ) \]
command
Integrate[(x*(8*b + 5*a*x^3))/((b + a*x^3)^(1/4)*(-b - a*x^3 + x^8)),x]
Mathematica 13.1 output
\[ 2 \text {ArcTan}\left (\frac {\sqrt [4]{b+a x^3}}{x^2}\right )-2 \tanh ^{-1}\left (\frac {x^2}{\sqrt [4]{b+a x^3}}\right ) \]
Mathematica 12.3 output
\[ \int \frac {x \left (8 b+5 a x^3\right )}{\sqrt [4]{b+a x^3} \left (-b-a x^3+x^8\right )} \, dx \]________________________________________________________________________________________