\[ \int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left (-5+6 x-6 x^2+x^3\right )} \, dx \]
Optimal antiderivative \[ \frac {\left (9 x^{3}-30 x^{2}+66 x -19\right )^{\frac {1}{3}}}{-3+2 x}+\frac {2^{\frac {1}{3}} \arctan \left (\frac {-3 \sqrt {3}+2 x \sqrt {3}}{-3+2 x +2^{\frac {2}{3}} \left (9 x^{3}-30 x^{2}+66 x -19\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}+\frac {2^{\frac {1}{3}} \ln \left (6-4 x +2^{\frac {2}{3}} \left (9 x^{3}-30 x^{2}+66 x -19\right )^{\frac {1}{3}}\right )}{3}-\frac {\ln \left (18-24 x +8 x^{2}+\left (-3 \,2^{\frac {2}{3}}+2 \,2^{\frac {2}{3}} x \right ) \left (9 x^{3}-30 x^{2}+66 x -19\right )^{\frac {1}{3}}+2^{\frac {1}{3}} \left (9 x^{3}-30 x^{2}+66 x -19\right )^{\frac {2}{3}}\right ) 2^{\frac {1}{3}}}{6} \]
command
Integrate[((2 + x)^2*(-19 + 66*x - 30*x^2 + 9*x^3)^(1/3))/((-3 + 2*x)^2*(-5 + 6*x - 6*x^2 + x^3)),x]
Mathematica 13.1 output
\[ \frac {\sqrt [3]{-19+66 x-30 x^2+9 x^3}}{-3+2 x}+\frac {\sqrt [3]{2} \text {ArcTan}\left (\frac {\sqrt {3} (-3+2 x)}{-3+2 x+2^{2/3} \sqrt [3]{-19+66 x-30 x^2+9 x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt [3]{2} \log \left (6-4 x+2^{2/3} \sqrt [3]{-19+66 x-30 x^2+9 x^3}\right )-\frac {\log \left (18-24 x+8 x^2+2^{2/3} (-3+2 x) \sqrt [3]{-19+66 x-30 x^2+9 x^3}+\sqrt [3]{2} \left (-19+66 x-30 x^2+9 x^3\right )^{2/3}\right )}{3\ 2^{2/3}} \]
Mathematica 12.3 output
\[ \int \frac {(2+x)^2 \sqrt [3]{-19+66 x-30 x^2+9 x^3}}{(-3+2 x)^2 \left (-5+6 x-6 x^2+x^3\right )} \, dx \]________________________________________________________________________________________