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