\[ \int \frac {\left (1+x^5\right ) \sqrt [3]{1+x^3+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((1 + x^5)*(1 + x^3 + x^5)^(1/3)*(-3 + 2*x^5))/(x^2*(2 - 2*x^3 + 4*x^5 - x^6 - 2*x^8 + 2*x^10)),x]
Mathematica 13.1 output
\[ \frac {3 \sqrt [3]{1+x^3+x^5}}{2 x}-\frac {1}{4} \text {RootSum}\left [3-6 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-3 \log (x)+3 \log \left (\sqrt [3]{1+x^3+x^5}-x \text {$\#$1}\right )+4 \log (x) \text {$\#$1}^3-4 \log \left (\sqrt [3]{1+x^3+x^5}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ] \]
Mathematica 12.3 output
\[ \int \frac {\left (1+x^5\right ) \sqrt [3]{1+x^3+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}\right )} \, dx \]________________________________________________________________________________________