\[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x+2 x^3}}{x^2 \left (1+x^4\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((1 + x^2)*(-x + 2*x^3)^(1/3))/(x^2*(1 + x^4)),x]
Mathematica 13.1 output
\[ \frac {18-36 x^2+x^{2/3} \left (-1+2 x^2\right )^{2/3} \text {RootSum}\left [5-4 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-10 \log (x)+15 \log \left (\sqrt [3]{-1+2 x^2}-x^{2/3} \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^3-3 \log \left (\sqrt [3]{-1+2 x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}^3}{-2 \text {$\#$1}^2+\text {$\#$1}^5}\&\right ]}{12 \left (x \left (-1+2 x^2\right )\right )^{2/3}} \]
Mathematica 12.3 output
\[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x+2 x^3}}{x^2 \left (1+x^4\right )} \, dx \]________________________________________________________________________________________