\[ \int \frac {3+x}{\sqrt [3]{-1+x^2} \left (5-x+2 x^2\right )} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\frac {3^{\frac {5}{6}} \left (x^{2}-1\right )^{\frac {1}{3}}}{2 \,2^{\frac {1}{3}}-2 \,2^{\frac {1}{3}} x +3^{\frac {1}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}}\right ) 2^{\frac {2}{3}} 3^{\frac {5}{6}}}{6}+\frac {\ln \left (-2^{\frac {1}{3}} 3^{\frac {2}{3}}+2^{\frac {1}{3}} 3^{\frac {2}{3}} x +3 \left (x^{2}-1\right )^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}}}{6}-\frac {\ln \left (2^{\frac {2}{3}} 3^{\frac {1}{3}}-2 \,2^{\frac {2}{3}} 3^{\frac {1}{3}} x +2^{\frac {2}{3}} 3^{\frac {1}{3}} x^{2}+\left (2^{\frac {1}{3}} 3^{\frac {2}{3}}-2^{\frac {1}{3}} 3^{\frac {2}{3}} x \right ) \left (x^{2}-1\right )^{\frac {1}{3}}+3 \left (x^{2}-1\right )^{\frac {2}{3}}\right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}}}{12} \]
command
Integrate[(3 + x)/((-1 + x^2)^(1/3)*(5 - x + 2*x^2)),x]
Mathematica 13.1 output
\[ \frac {2 \sqrt {3} \text {ArcTan}\left (\frac {3^{5/6} \sqrt [3]{-1+x^2}}{2 \sqrt [3]{2}-2 \sqrt [3]{2} x+\sqrt [3]{3} \sqrt [3]{-1+x^2}}\right )+2 \log \left (-\sqrt [3]{2} 3^{2/3}+\sqrt [3]{2} 3^{2/3} x+3 \sqrt [3]{-1+x^2}\right )-\log \left (2^{2/3} \sqrt [3]{3}-2\ 2^{2/3} \sqrt [3]{3} x+2^{2/3} \sqrt [3]{3} x^2-\sqrt [3]{2} 3^{2/3} (-1+x) \sqrt [3]{-1+x^2}+3 \left (-1+x^2\right )^{2/3}\right )}{2 \sqrt [3]{2} 3^{2/3}} \]
Mathematica 12.3 output
\[ \int \frac {3+x}{\sqrt [3]{-1+x^2} \left (5-x+2 x^2\right )} \, dx \]________________________________________________________________________________________