\[ \int \frac {2500 e^{10}-4 x^2}{x^4+e^{20} \left (390625+1250 x+x^2\right )+e^{10} \left (1250 x^2+2 x^3\right )} \, dx \]
Optimal antiderivative \[ \frac {4}{\left (1+\frac {625}{x}\right ) {\mathrm e}^{10}+x} \]
command
integrate((2500*exp(5)^2-4*x^2)/((x^2+1250*x+390625)*exp(5)^4+(2*x^3+1250*x^2)*exp(5)^2+x^4),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {4 \, x}{x^{2} + x e^{10} + 625 \, e^{10}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int -\frac {4 \, {\left (x^{2} - 625 \, e^{10}\right )}}{x^{4} + {\left (x^{2} + 1250 \, x + 390625\right )} e^{20} + 2 \, {\left (x^{3} + 625 \, x^{2}\right )} e^{10}}\,{d x} \]________________________________________________________________________________________