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