\[ \int \frac {\left (1+e^5-2 x\right ) \log ^2\left (1+e^5-2 x\right )+e^{\frac {x}{\log \left (1+e^5-2 x\right )}} \left (-2 x^2 \log \left (\frac {3 x}{2}\right )+\left (-x-e^5 x+2 x^2\right ) \log \left (1+e^5-2 x\right ) \log \left (\frac {3 x}{2}\right )\right )}{\left (x+e^5 x-2 x^2\right ) \log ^2\left (1+e^5-2 x\right ) \log \left (\frac {3 x}{2}\right )} \, dx \]
Optimal antiderivative \[ \ln \! \left (\ln \! \left (\frac {3 x}{2}\right ) {\mathrm e}^{-{\mathrm e}^{\frac {x}{\ln \left ({\mathrm e}^{5}+1-2 x \right )}}}\right ) \]
command
integrate((((-x*exp(5)+2*x^2-x)*log(3/2*x)*log(exp(5)+1-2*x)-2*x^2*log(3/2*x))*exp(x/log(exp(5)+1-2*x))+(exp(5)+1-2*x)*log(exp(5)+1-2*x)^2)/(x*exp(5)-2*x^2+x)/log(3/2*x)/log(exp(5)+1-2*x)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ -e^{\left (\frac {x}{\log \left (-2 \, x + e^{5} + 1\right )}\right )} + \log \left (-\log \left (2\right ) + \log \left (3 \, x\right )\right ) \]