\[ \int \frac {2 x+x^2+x^3+e^{8 x^4} \left (-8+4 x^2\right ) \log ^3\left (\frac {x}{2+x+x^2}\right )+e^{8 x^4} \left (-64 x^4-32 x^5-32 x^6\right ) \log ^4\left (\frac {x}{2+x+x^2}\right )}{2 x+x^2+x^3} \, dx \]
Optimal antiderivative \[ x -{\mathrm e}^{8 x^{4}} \ln \left (\frac {x}{x^{2}+x +2}\right )^{4} \]
command
Int[(2*x + x^2 + x^3 + E^(8*x^4)*(-8 + 4*x^2)*Log[x/(2 + x + x^2)]^3 + E^(8*x^4)*(-64*x^4 - 32*x^5 - 32*x^6)*Log[x/(2 + x + x^2)]^4)/(2*x + x^2 + x^3),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ x-\frac {e^{8 x^4} \log ^3\left (\frac {x}{x^2+x+2}\right ) \left (x^6 \log \left (\frac {x}{x^2+x+2}\right )+x^5 \log \left (\frac {x}{x^2+x+2}\right )+2 x^4 \log \left (\frac {x}{x^2+x+2}\right )\right )}{x^4 \left (x^2+x+2\right )} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {2 x+x^2+x^3+e^{8 x^4} \left (-8+4 x^2\right ) \log ^3\left (\frac {x}{2+x+x^2}\right )+e^{8 x^4} \left (-64 x^4-32 x^5-32 x^6\right ) \log ^4\left (\frac {x}{2+x+x^2}\right )}{2 x+x^2+x^3} \, dx \]________________________________________________________________________________________