\[ \int \frac {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx \]
Optimal antiderivative \[ \frac {3 \left (x^{3}+2 x -1\right )^{\frac {2}{3}}}{4 x^{2}}-\frac {3 \,3^{\frac {1}{6}} \arctan \left (\frac {3^{\frac {5}{6}} x}{3^{\frac {1}{3}} x +2 \,2^{\frac {1}{3}} \left (x^{3}+2 x -1\right )^{\frac {1}{3}}}\right ) 2^{\frac {1}{3}}}{4}+\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (-3 x +2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (x^{3}+2 x -1\right )^{\frac {1}{3}}\right )}{4}-\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (3 x^{2}+2^{\frac {1}{3}} 3^{\frac {2}{3}} x \left (x^{3}+2 x -1\right )^{\frac {1}{3}}+2^{\frac {2}{3}} 3^{\frac {1}{3}} \left (x^{3}+2 x -1\right )^{\frac {2}{3}}\right )}{8} \]
command
Integrate[((-3 + 4*x)*(-1 + 2*x + x^3)^(2/3))/(x^3*(2 - 4*x + x^3)),x]
Mathematica 13.1 output
\[ \frac {3 \left (-1+2 x+x^3\right )^{2/3}}{4 x^2}-\frac {3 \sqrt [6]{3} \text {ArcTan}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+2 x+x^3}}\right )}{2\ 2^{2/3}}+\frac {1}{2} \left (\frac {3}{2}\right )^{2/3} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+2 x+x^3}\right )-\frac {1}{4} \left (\frac {3}{2}\right )^{2/3} \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+2 x+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+2 x+x^3\right )^{2/3}\right ) \]
Mathematica 12.3 output
\[ \int \frac {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx \]________________________________________________________________________________________