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