\[ \int \frac {e^3 x^2-5 e^8 x^3+e^5 \left (-e^6+10 e^{11} x-25 e^{16} x^2\right )+\left (-e^3 x^2+e^5 \left (-e^6+10 e^{11} x-25 e^{16} x^2\right )\right ) \log (x)}{\left (-e^3 x^3+5 e^8 x^4+e^5 \left (e^6 x-10 e^{11} x^2+25 e^{16} x^3\right )\right ) \log (x)} \, dx \]
Optimal antiderivative \[ \ln \! \left (\frac {\frac {x}{5 x \,{\mathrm e}^{8}-{\mathrm e}^{3}}+\frac {{\mathrm e}^{5}}{x}}{\ln \! \left (x \right )}\right ) \]
command
Int[(E^3*x^2 - 5*E^8*x^3 + E^5*(-E^6 + 10*E^11*x - 25*E^16*x^2) + (-(E^3*x^2) + E^5*(-E^6 + 10*E^11*x - 25*E^16*x^2))*Log[x])/((-(E^3*x^3) + 5*E^8*x^4 + E^5*(E^6*x - 10*E^11*x^2 + 25*E^16*x^3))*Log[x]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {e^3 x^2-5 e^8 x^3+e^5 \left (-e^6+10 e^{11} x-25 e^{16} x^2\right )+\left (-e^3 x^2+e^5 \left (-e^6+10 e^{11} x-25 e^{16} x^2\right )\right ) \log (x)}{\left (-e^3 x^3+5 e^8 x^4+e^5 \left (e^6 x-10 e^{11} x^2+25 e^{16} x^3\right )\right ) \log (x)} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \log \left (-x^2-5 e^{13} x+e^8\right )-\log (x)-\log \left (1-5 e^5 x\right )-\log (\log (x)) \]