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