\[ \int e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \left (e^2 \left (1+2 x^2\right )+25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}} \left (-8 x-4 x \log \left (25 x^2\right )\right )\right ) \, dx \]
Optimal antiderivative \[ x \,{\mathrm e}^{-{\mathrm e}^{4 x \ln \left (25 x^{2}\right ) {\mathrm e}^{-2}}+x^{2}-5} \]
command
Integrate[E^(-7 + x^2 - 25^((4*x)/E^2)*(x^2)^((4*x)/E^2))*(E^2*(1 + 2*x^2) + 25^((4*x)/E^2)*(x^2)^((4*x)/E^2)*(-8*x - 4*x*Log[25*x^2])),x]
Mathematica 13.1 output
\[ \int e^{-7+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} \left (e^2 \left (1+2 x^2\right )+25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}} \left (-8 x-4 x \log \left (25 x^2\right )\right )\right ) \, dx \]
Mathematica 12.3 output
\[ e^{-5+x^2-25^{\frac {4 x}{e^2}} \left (x^2\right )^{\frac {4 x}{e^2}}} x \]