\[ \int \frac {\left (1+x^2\right ) \sqrt [4]{x^3+x^5}}{x^2 \left (-1+x^2\right )} \, dx \]
Optimal antiderivative \[ \frac {4 \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{x}+2^{\frac {1}{4}} \arctan \left (\frac {2^{\frac {1}{4}} x}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right )-\frac {\arctan \left (\frac {2^{\frac {3}{4}} x \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{x^{2} \sqrt {2}-\sqrt {x^{5}+x^{3}}}\right ) 2^{\frac {3}{4}}}{2}-2^{\frac {1}{4}} \arctanh \left (\frac {2^{\frac {1}{4}} x}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right )-\frac {\arctanh \left (\frac {\frac {x^{2} 2^{\frac {3}{4}}}{2}+\frac {\sqrt {x^{5}+x^{3}}\, 2^{\frac {1}{4}}}{2}}{x \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}}}{2} \]
command
Integrate[((1 + x^2)*(x^3 + x^5)^(1/4))/(x^2*(-1 + x^2)),x]
Mathematica 13.1 output
\[ -\frac {\sqrt [4]{x^3+x^5} \left (-8 \sqrt [4]{1+x^2}-2 \sqrt [4]{2} \sqrt [4]{x} \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+2^{3/4} \sqrt [4]{x} \text {ArcTan}\left (\frac {2^{3/4} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt {2} \sqrt {x}-\sqrt {1+x^2}}\right )+2 \sqrt [4]{2} \sqrt [4]{x} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+2^{3/4} \sqrt [4]{x} \tanh ^{-1}\left (\frac {2 \sqrt [4]{2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{2 \sqrt {x}+\sqrt {2} \sqrt {1+x^2}}\right )\right )}{2 x \sqrt [4]{1+x^2}} \]
Mathematica 12.3 output
\[ \int \frac {\left (1+x^2\right ) \sqrt [4]{x^3+x^5}}{x^2 \left (-1+x^2\right )} \, dx \]________________________________________________________________________________________