\[ \int \frac {\frac {5 \left (12-6 e^x-18 x\right )}{e^2}+\frac {25 \left (-200+297 x+18 x^2+e^x (109+6 x)\right )}{e^4}+\frac {25 \left (8-4 e^x-12 x\right ) \log \left (e^x+x\right )}{e^4}}{9 e^x+9 x+\frac {5 \left (e^x (-300-18 x)-300 x-18 x^2\right )}{e^2}+\frac {25 \left (2500 x+309 x^2+9 x^3+e^x \left (2500+309 x+9 x^2\right )\right )}{e^4}+\left (\frac {5 \left (12 e^x+12 x\right )}{e^2}+\frac {25 \left (e^x (-200-12 x)-200 x-12 x^2\right )}{e^4}\right ) \log \left (e^x+x\right )+\frac {25 \left (4 e^x+4 x\right ) \log ^2\left (e^x+x\right )}{e^4}} \, dx \]
Optimal antiderivative \[ \ln \left (x +\left (x -{\mathrm e}^{2-\ln \left (5\right )}+\frac {50}{3}-\frac {2 \ln \left ({\mathrm e}^{x}+x \right )}{3}\right )^{2}\right ) \]
command
Int[((5*(12 - 6*E^x - 18*x))/E^2 + (25*(-200 + 297*x + 18*x^2 + E^x*(109 + 6*x)))/E^4 + (25*(8 - 4*E^x - 12*x)*Log[E^x + x])/E^4)/(9*E^x + 9*x + (5*(E^x*(-300 - 18*x) - 300*x - 18*x^2))/E^2 + (25*(2500*x + 309*x^2 + 9*x^3 + E^x*(2500 + 309*x + 9*x^2)))/E^4 + ((5*(12*E^x + 12*x))/E^2 + (25*(E^x*(-200 - 12*x) - 200*x - 12*x^2))/E^4)*Log[E^x + x] + (25*(4*E^x + 4*x)*Log[E^x + x]^2)/E^4),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \log \left (225 x^2+15 \left (515-6 e^2\right ) x+100 \log ^2\left (x+e^x\right )-300 x \log \left (x+e^x\right )-20 \left (250-3 e^2\right ) \log \left (x+e^x\right )+\left (250-3 e^2\right )^2\right ) \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {\frac {5 \left (12-6 e^x-18 x\right )}{e^2}+\frac {25 \left (-200+297 x+18 x^2+e^x (109+6 x)\right )}{e^4}+\frac {25 \left (8-4 e^x-12 x\right ) \log \left (e^x+x\right )}{e^4}}{9 e^x+9 x+\frac {5 \left (e^x (-300-18 x)-300 x-18 x^2\right )}{e^2}+\frac {25 \left (2500 x+309 x^2+9 x^3+e^x \left (2500+309 x+9 x^2\right )\right )}{e^4}+\left (\frac {5 \left (12 e^x+12 x\right )}{e^2}+\frac {25 \left (e^x (-200-12 x)-200 x-12 x^2\right )}{e^4}\right ) \log \left (e^x+x\right )+\frac {25 \left (4 e^x+4 x\right ) \log ^2\left (e^x+x\right )}{e^4}} \, dx \]________________________________________________________________________________________