\[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \,2^{\frac {1}{4}} \arctan \left (\frac {2^{\frac {1}{4}} x}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right )}{3}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{-x^{2}+\sqrt {x^{5}+x^{3}}}\right )}{3}-\frac {2 \,2^{\frac {1}{4}} \arctanh \left (\frac {2^{\frac {1}{4}} x}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right )}{3}+\frac {\sqrt {2}\, \arctanh \left (\frac {\frac {x^{2} \sqrt {2}}{2}+\frac {\sqrt {x^{5}+x^{3}}\, \sqrt {2}}{2}}{x \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right )}{3} \]
command
Integrate[((1 + x)*(x^3 + x^5)^(1/4))/(x*(-1 + x^3)),x]
Mathematica 13.1 output
\[ \frac {\sqrt [4]{2} \sqrt [4]{x^3+x^5} \left (2 \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+\sqrt [4]{2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt {x}-\sqrt {1+x^2}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+\sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt {x}+\sqrt {1+x^2}}\right )\right )}{3 x^{3/4} \sqrt [4]{1+x^2}} \]
Mathematica 12.3 output
\[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx \]________________________________________________________________________________________