\[ \int \frac {x \sqrt [3]{2-x^3+x^8} \left (-6+5 x^8\right )}{4+x^6+4 x^8+x^{16}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[(x*(2 - x^3 + x^8)^(1/3)*(-6 + 5*x^8))/(4 + x^6 + 4*x^8 + x^16),x]
Mathematica 13.1 output
\[ \frac {1}{2} \text {RootSum}\left [2+2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{2-x^3+x^8}-x \text {$\#$1}\right ) \text {$\#$1}}{1+\text {$\#$1}^3}\&\right ] \]
Mathematica 12.3 output
\[ \int \frac {x \sqrt [3]{2-x^3+x^8} \left (-6+5 x^8\right )}{4+x^6+4 x^8+x^{16}} \, dx \]________________________________________________________________________________________