\[ \int \frac {e^{\frac {e^3 x+e^{2 x} x^2 \log \left (e^4+64 x\right )}{e^3}} \left (e^7+64 e^3 x+64 e^{2 x} x^2+e^{2 x} \left (128 x^2+128 x^3+e^4 \left (2 x+2 x^2\right )\right ) \log \left (e^4+64 x\right )\right )}{e^7+64 e^3 x} \, dx \]
Optimal antiderivative \[ {\mathrm e}^{{\mathrm e}^{2 x} x^{2} \ln \left ({\mathrm e}^{4}+64 x \right ) {\mathrm e}^{-3}+x} \]
command
Int[(E^((E^3*x + E^(2*x)*x^2*Log[E^4 + 64*x])/E^3)*(E^7 + 64*E^3*x + 64*E^(2*x)*x^2 + E^(2*x)*(128*x^2 + 128*x^3 + E^4*(2*x + 2*x^2))*Log[E^4 + 64*x]))/(E^7 + 64*E^3*x),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {e^{\frac {e^3 x+e^{2 x} x^2 \log \left (e^4+64 x\right )}{e^3}} \left (e^7+64 e^3 x+64 e^{2 x} x^2+e^{2 x} \left (128 x^2+128 x^3+e^4 \left (2 x+2 x^2\right )\right ) \log \left (e^4+64 x\right )\right )}{e^7+64 e^3 x} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ e^x \left (64 x+e^4\right )^{e^{2 x-3} x^2} \]