\[ \int \frac {1+x}{(-1+x) (1+2 x) \sqrt [3]{-1+3 x^2}} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\frac {\sqrt {3}\, \left (3 x^{2}-1\right )^{\frac {1}{3}}}{2 \,2^{\frac {1}{3}} x +\left (3 x^{2}-1\right )^{\frac {1}{3}}}\right ) 2^{\frac {2}{3}} \sqrt {3}}{6}+\frac {\ln \left (-2^{\frac {1}{3}} x +\left (3 x^{2}-1\right )^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{6}-\frac {\ln \left (2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} x \left (3 x^{2}-1\right )^{\frac {1}{3}}+\left (3 x^{2}-1\right )^{\frac {2}{3}}\right ) 2^{\frac {2}{3}}}{12} \]
command
Integrate[(1 + x)/((-1 + x)*(1 + 2*x)*(-1 + 3*x^2)^(1/3)),x]
Mathematica 13.1 output
\[ \frac {2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-1+3 x^2}}{2 \sqrt [3]{2} x+\sqrt [3]{-1+3 x^2}}\right )+2 \log \left (-\sqrt [3]{2} x+\sqrt [3]{-1+3 x^2}\right )-\log \left (2^{2/3} x^2+\left (-1+3 x^2\right )^{2/3}+x \sqrt [3]{-2+6 x^2}\right )}{6 \sqrt [3]{2}} \]
Mathematica 12.3 output
\[ \int \frac {1+x}{(-1+x) (1+2 x) \sqrt [3]{-1+3 x^2}} \, dx \]________________________________________________________________________________________