\[ \int \frac {e^9 \left (-10-100 x^3\right )+e^{10} \left (-25-50 x-250 x^3-125 x^4\right )}{4 e^8 x^2+e^9 \left (20 x^2+20 x^3\right )+e^{10} \left (25 x^2+50 x^3+25 x^4\right )} \, dx \]
Optimal antiderivative \[ \frac {-5 x^{3}+1}{\left (\frac {2 \,{\mathrm e}^{4} {\mathrm e}^{-5}}{5}+x +1\right ) x} \]
command
integrate(((-125*x^4-250*x^3-50*x-25)*exp(5)^2+(-100*x^3-10)*exp(4)*exp(5))/((25*x^4+50*x^3+25*x^2)*exp(5)^2+(20*x^3+20*x^2)*exp(4)*exp(5)+4*x^2*exp(4)^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -5 \, x - \frac {{\left (25 \, x e^{2} + 20 \, x e + 4 \, x - 5 \, e^{2}\right )} e^{\left (-1\right )}}{5 \, x^{2} e + 5 \, x e + 2 \, x} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________