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