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