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