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