\[ \int \frac {(-2+x) \left (1-x+x^2\right )}{x^3 \left (-1+x+x^2\right ) \sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}}} \, dx \]
Optimal antiderivative \[ \frac {\left (-3 x^{2}+x -1\right ) \left (\frac {2 x^{2}-x +1}{3 x^{2}-x +1}\right )^{\frac {2}{3}}}{x^{2}}-2 \,2^{\frac {2}{3}} 3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}}{3}+\frac {2 \,2^{\frac {2}{3}} \left (\frac {2 x^{2}-x +1}{3 x^{2}-x +1}\right )^{\frac {1}{3}} 3^{\frac {1}{6}}}{3}\right )+\frac {7 \arctan \left (\frac {\sqrt {3}}{3}+\frac {2 \left (\frac {2 x^{2}-x +1}{3 x^{2}-x +1}\right )^{\frac {1}{3}} \sqrt {3}}{3}\right ) \sqrt {3}}{3}+\frac {7 \ln \left (-1+\left (\frac {2 x^{2}-x +1}{3 x^{2}-x +1}\right )^{\frac {1}{3}}\right )}{3}-\frac {2 \,2^{\frac {2}{3}} \ln \left (-3+6^{\frac {2}{3}} \left (\frac {2 x^{2}-x +1}{3 x^{2}-x +1}\right )^{\frac {1}{3}}\right ) 3^{\frac {2}{3}}}{3}-\frac {7 \ln \left (1+\left (\frac {2 x^{2}-x +1}{3 x^{2}-x +1}\right )^{\frac {1}{3}}+\left (\frac {2 x^{2}-x +1}{3 x^{2}-x +1}\right )^{\frac {2}{3}}\right )}{6}+\frac {2^{\frac {2}{3}} \ln \left (3+6^{\frac {2}{3}} \left (\frac {2 x^{2}-x +1}{3 x^{2}-x +1}\right )^{\frac {1}{3}}+2 \,6^{\frac {1}{3}} \left (\frac {2 x^{2}-x +1}{3 x^{2}-x +1}\right )^{\frac {2}{3}}\right ) 3^{\frac {2}{3}}}{3} \]
command
Integrate[((-2 + x)*(1 - x + x^2))/(x^3*(-1 + x + x^2)*((1 - x + 2*x^2)/(1 - x + 3*x^2))^(1/3)),x]
Mathematica 13.1 output
\[ -\frac {6 \sqrt [3]{1-x+3 x^2}-6 x \sqrt [3]{1-x+3 x^2}+12 x^2 \sqrt [3]{1-x+3 x^2}+14 \sqrt {3} x^2 \sqrt [3]{1-x+2 x^2} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{1-x+3 x^2}}{\sqrt [3]{1-x+2 x^2}}}{\sqrt {3}}\right )-12\ 2^{2/3} \sqrt [6]{3} x^2 \sqrt [3]{1-x+2 x^2} \text {ArcTan}\left (\frac {1+\frac {\sqrt [3]{6-6 x+18 x^2}}{\sqrt [3]{1-x+2 x^2}}}{\sqrt {3}}\right )-14 x^2 \sqrt [3]{1-x+2 x^2} \log \left (-1+\frac {\sqrt [3]{1-x+3 x^2}}{\sqrt [3]{1-x+2 x^2}}\right )+7 x^2 \sqrt [3]{1-x+2 x^2} \log \left (1+\frac {\sqrt [3]{1-x+3 x^2}}{\sqrt [3]{1-x+2 x^2}}+\frac {\left (1-x+3 x^2\right )^{2/3}}{\left (1-x+2 x^2\right )^{2/3}}\right )+4\ 6^{2/3} x^2 \sqrt [3]{1-x+2 x^2} \log \left (-2+\frac {\sqrt [3]{6-6 x+18 x^2}}{\sqrt [3]{1-x+2 x^2}}\right )-2\ 6^{2/3} x^2 \sqrt [3]{1-x+2 x^2} \log \left (4+\frac {2 \sqrt [3]{6-6 x+18 x^2}}{\sqrt [3]{1-x+2 x^2}}+\frac {\left (6-6 x+18 x^2\right )^{2/3}}{\left (1-x+2 x^2\right )^{2/3}}\right )}{6 x^2 \sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}} \sqrt [3]{1-x+3 x^2}} \]
Mathematica 12.3 output
\[ \int \frac {(-2+x) \left (1-x+x^2\right )}{x^3 \left (-1+x+x^2\right ) \sqrt [3]{\frac {1-x+2 x^2}{1-x+3 x^2}}} \, dx \]________________________________________________________________________________________