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