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