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