\[ \int \frac {6+2 x+x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx \]
Optimal antiderivative \[ -\sqrt {3}\, \arctan \left (\frac {\frac {2 \sqrt {3}}{3}-\frac {2 x \sqrt {3}}{3}+\frac {\left (x^{2}+2\right )^{\frac {1}{3}} \sqrt {3}}{3}}{\left (x^{2}+2\right )^{\frac {1}{3}}}\right )+\ln \left (-1+x +\left (x^{2}+2\right )^{\frac {1}{3}}\right )-\frac {\ln \left (1-2 x +x^{2}+\left (1-x \right ) \left (x^{2}+2\right )^{\frac {1}{3}}+\left (x^{2}+2\right )^{\frac {2}{3}}\right )}{2} \]
command
Integrate[(6 + 2*x + x^2)/((2 + x^2)^(1/3)*(1 + 3*x - 2*x^2 + x^3)),x]
Mathematica 13.1 output
\[ -\sqrt {3} \text {ArcTan}\left (\frac {2-2 x+\sqrt [3]{2+x^2}}{\sqrt {3} \sqrt [3]{2+x^2}}\right )+\log \left (-1+x+\sqrt [3]{2+x^2}\right )-\frac {1}{2} \log \left (1-2 x+x^2-(-1+x) \sqrt [3]{2+x^2}+\left (2+x^2\right )^{2/3}\right ) \]
Mathematica 12.3 output
\[ \int \frac {6+2 x+x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx \]________________________________________________________________________________________