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