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