\[ \int \frac {e^{2 x} \left (200+100 x-50 x^2\right )+e^x \left (-4+18 x-6 x^2\right )+\left (e^{2 x} \left (-200 x+50 x^2\right )+e^x \left (4 x-9 x^2+2 x^3\right )\right ) \log \left (-4 x+9 x^2-2 x^3+e^x \left (200 x-50 x^2\right )\right )}{\left (4 x-9 x^2+2 x^3+e^x \left (-200 x+50 x^2\right )\right ) \log ^2\left (-4 x+9 x^2-2 x^3+e^x \left (200 x-50 x^2\right )\right )} \, dx \]
Optimal antiderivative \[ 1+\frac {{\mathrm e}^{x}}{\ln \! \left (2 \left (x +25 \,{\mathrm e}^{x}-\frac {1}{2}\right ) \left (-x +4\right ) x \right )} \]
command
Int[(E^(2*x)*(200 + 100*x - 50*x^2) + E^x*(-4 + 18*x - 6*x^2) + (E^(2*x)*(-200*x + 50*x^2) + E^x*(4*x - 9*x^2 + 2*x^3))*Log[-4*x + 9*x^2 - 2*x^3 + E^x*(200*x - 50*x^2)])/((4*x - 9*x^2 + 2*x^3 + E^x*(-200*x + 50*x^2))*Log[-4*x + 9*x^2 - 2*x^3 + E^x*(200*x - 50*x^2)]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {e^{2 x} \left (200+100 x-50 x^2\right )+e^x \left (-4+18 x-6 x^2\right )+\left (e^{2 x} \left (-200 x+50 x^2\right )+e^x \left (4 x-9 x^2+2 x^3\right )\right ) \log \left (-4 x+9 x^2-2 x^3+e^x \left (200 x-50 x^2\right )\right )}{\left (4 x-9 x^2+2 x^3+e^x \left (-200 x+50 x^2\right )\right ) \log ^2\left (-4 x+9 x^2-2 x^3+e^x \left (200 x-50 x^2\right )\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {e^x}{\log \left (-\left (\left (-2 x-50 e^x+1\right ) (4-x) x\right )\right )} \]