\[ \int \frac {\left (7 x+14 e^{2 x} x\right ) \log \left (-\frac {10}{x}\right )+\left (-7 e^{2 x}-7 x+\left (-7 e^{2 x} x-7 x^2\right ) \log \left (-\frac {10}{x}\right )\right ) \log \left (e^{2 x}+x\right )}{3 e^{3 x} x+3 e^x x^2} \, dx \]
Optimal antiderivative \[ \frac {7 \ln \left ({\mathrm e}^{2 x}+x \right ) \ln \left (-\frac {10}{x}\right ) {\mathrm e}^{-x}}{3} \]
command
Int[((7*x + 14*E^(2*x)*x)*Log[-10/x] + (-7*E^(2*x) - 7*x + (-7*E^(2*x)*x - 7*x^2)*Log[-10/x])*Log[E^(2*x) + x])/(3*E^(3*x)*x + 3*E^x*x^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \frac {7}{3} e^{-x} \log \left (-\frac {10}{x}\right ) \log \left (x+e^{2 x}\right ) \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {\left (7 x+14 e^{2 x} x\right ) \log \left (-\frac {10}{x}\right )+\left (-7 e^{2 x}-7 x+\left (-7 e^{2 x} x-7 x^2\right ) \log \left (-\frac {10}{x}\right )\right ) \log \left (e^{2 x}+x\right )}{3 e^{3 x} x+3 e^x x^2} \, dx \]________________________________________________________________________________________