\[ \int \frac {16+144 x+36 x^2+e^2 x^2+e \left (-8 x-20 x^2\right )}{16+16 x+4 x^2+e^2 x^2+e \left (-8 x-4 x^2\right )} \, dx \]
Optimal antiderivative \[ \frac {16 x}{\frac {4+x}{x}-{\mathrm e}+1}+2+x \]
command
integrate((x^2*exp(1)^2+(-20*x^2-8*x)*exp(1)+36*x^2+144*x+16)/(x^2*exp(1)^2+(-4*x^2-8*x)*exp(1)+4*x^2+16*x+16),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {x e^{2} - 20 \, x e + 36 \, x}{e^{2} - 4 \, e + 4} - \frac {256}{{\left (x e - 2 \, x - 4\right )} {\left (e^{2} - 4 \, e + 4\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________