\[ \int \frac {e^x \left (-625 x+625 x^2\right )+e^{\frac {e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} \left (2500+2500 x+625 e^x x^2\right )}{e^{x+\frac {2 e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} x^2+e^x \left (4 x^2+4 x^3+x^4\right )+e^x \left (-4 x^2-2 x^3\right ) \log (5 x)+e^x x^2 \log ^2(5 x)+e^{\frac {e^{-x} \left (-4+e^x \left (2 x+x^2\right )\right )}{x}} \left (e^x \left (4 x^2+2 x^3\right )-2 e^x x^2 \log (5 x)\right )} \, dx \]
Optimal antiderivative \[ \frac {625}{\ln \left (5 x \right )-{\mathrm e}^{x +2-\frac {4 \,{\mathrm e}^{-x}}{x}}-2-x} \]
command
integrate(((625*exp(x)*x^2+2500*x+2500)*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x)+(625*x^2-625*x)*exp(x))/(x^2*exp(x)*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x)^2+(-2*x^2*exp(x)*log(5*x)+(2*x^3+4*x^2)*exp(x))*exp(((x^2+2*x)*exp(x)-4)/exp(x)/x)+x^2*exp(x)*log(5*x)^2+(-2*x^3-4*x^2)*exp(x)*log(5*x)+(x^4+4*x^3+4*x^2)*exp(x)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________