\[ \int \frac {e^{\frac {-2 x+3 x^2+2 x^3+8 e^8 \log \left (x^2\right )}{2+x}} \left (-8 x+24 x^2+30 x^3+8 x^4+e^8 (64+32 x)-16 e^8 x \log \left (x^2\right )\right )}{4 x+4 x^2+x^3} \, dx \]
Optimal antiderivative \[ 2 \,{\mathrm e}^{\frac {8 \,{\mathrm e}^{8} \ln \left (x^{2}\right )}{2+x}-x +2 x^{2}} \]
command
integrate((-16*x*exp(8)*log(x^2)+(32*x+64)*exp(8)+8*x^4+30*x^3+24*x^2-8*x)*exp((8*exp(8)*log(x^2)+2*x^3+3*x^2-2*x)/(2+x))/(x^3+4*x^2+4*x),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ 2 \, e^{\left (\frac {2 \, x^{3}}{x + 2} + \frac {3 \, x^{2}}{x + 2} + \frac {8 \, e^{8} \log \left (x^{2}\right )}{x + 2} - \frac {2 \, x}{x + 2}\right )} \]