\[ \int \frac {1-x^2+x^3}{\left (-1-x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx \]
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
\[ \frac {x^{2/3} \sqrt [3]{1+x} \left (6 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{1+x}}\right )-6 \log \left (-\sqrt [3]{x}+\sqrt [3]{1+x}\right )+3 \log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )-4 \text {RootSum}\left [-3+4 \text {$\#$1}^3-3 \text {$\#$1}^6+\text {$\#$1}^9\&,\frac {\log (x)-3 \log \left (\sqrt [3]{1+x}-\sqrt [3]{x} \text {$\#$1}\right )-2 \log (x) \text {$\#$1}^3+6 \log \left (\sqrt [3]{1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^3+\log (x) \text {$\#$1}^6-3 \log \left (\sqrt [3]{1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^6}{4 \text {$\#$1}-6 \text {$\#$1}^4+3 \text {$\#$1}^7}\&\right ]\right )}{6 \sqrt [3]{x^2 (1+x)}} \]
Mathematica 12.3 output
\[ \int \frac {1-x^2+x^3}{\left (-1-x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx \]________________________________________________________________________________________