\[ \int \frac {-16 x^2+8 x^3+8 x^4+e^2 \left (2+8 x+8 x^2\right )+e \left (-9-4 x-16 x^2-16 x^3\right )}{2 x^2+8 x^3+8 x^4+e^2 \left (2+8 x+8 x^2\right )+e \left (-4 x-16 x^2-16 x^3\right )} \, dx \]
Optimal antiderivative \[ \frac {3}{2 \left (\frac {x}{3}-\frac {{\mathrm e}}{3}\right ) \left (2+\frac {1}{x}\right )}+3+x \]
command
integrate(((8*x^2+8*x+2)*exp(1)^2+(-16*x^3-16*x^2-4*x-9)*exp(1)+8*x^4+8*x^3-16*x^2)/((8*x^2+8*x+2)*exp(1)^2+(-16*x^3-16*x^2-4*x)*exp(1)+8*x^4+8*x^3+2*x^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ x + \frac {9 \, x}{2 \, {\left (2 \, x^{2} - 2 \, x e + x - e\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________