\[ \int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx \]
Optimal antiderivative \[ -\frac {\sqrt {3}\, \arctan \left (\frac {\frac {10 \,2^{\frac {1}{3}} \sqrt {3}\, 7^{\frac {1}{3}}}{21}-\frac {2^{\frac {1}{3}} \sqrt {3}\, x 7^{\frac {1}{3}}}{7}+\frac {\left (3 x^{2}+4\right )^{\frac {1}{3}} \sqrt {3}}{3}}{\left (3 x^{2}+4\right )^{\frac {1}{3}}}\right ) 14^{\frac {2}{3}}}{196}+\frac {\ln \left (-10 \,14^{\frac {1}{3}}+3 \,14^{\frac {1}{3}} x +14 \left (3 x^{2}+4\right )^{\frac {1}{3}}\right ) 14^{\frac {2}{3}}}{196}-\frac {\ln \left (100 \,14^{\frac {2}{3}}-60 \,14^{\frac {2}{3}} x +9 \,14^{\frac {2}{3}} x^{2}+\left (140 \,14^{\frac {1}{3}}-42 \,14^{\frac {1}{3}} x \right ) \left (3 x^{2}+4\right )^{\frac {1}{3}}+196 \left (3 x^{2}+4\right )^{\frac {2}{3}}\right ) 14^{\frac {2}{3}}}{392} \]
command
Integrate[(2 + 3*x)/((4 + 3*x^2)^(1/3)*(-12 + 52*x + 9*x^2)),x]
Mathematica 13.1 output
\[ -\frac {2 \sqrt {3} \text {ArcTan}\left (\frac {10 \sqrt [3]{14}-3 \sqrt [3]{14} x+7 \sqrt [3]{4+3 x^2}}{7 \sqrt {3} \sqrt [3]{4+3 x^2}}\right )-2 \log \left (-10 \sqrt [3]{14}+3 \sqrt [3]{14} x+14 \sqrt [3]{4+3 x^2}\right )+\log \left (100\ 14^{2/3}-60\ 14^{2/3} x+9\ 14^{2/3} x^2+196 \left (4+3 x^2\right )^{2/3}+14 (10-3 x) \sqrt [3]{56+42 x^2}\right )}{28 \sqrt [3]{14}} \]
Mathematica 12.3 output
\[ \int \frac {2+3 x}{\sqrt [3]{4+3 x^2} \left (-12+52 x+9 x^2\right )} \, dx \]________________________________________________________________________________________