\[ \int \frac {\left (-20 x+e^{3+2 x} \left (16 x+16 x^2+4 x^3\right )\right ) \log (x) \log \left (\frac {-5 x+e^{3+2 x} (2+x)}{2+x}\right )+\left (20 x+10 x^2+e^{3+2 x} \left (-8-8 x-2 x^2\right )+\left (-10 x-5 x^2+e^{3+2 x} \left (4+4 x+x^2\right )\right ) \log (x)\right ) \log ^2\left (\frac {-5 x+e^{3+2 x} (2+x)}{2+x}\right )}{\left (-10 x-5 x^2+e^{3+2 x} \left (4+4 x+x^2\right )\right ) \log ^3(x)} \, dx \]
Optimal antiderivative \[ \frac {x \ln \! \left ({\mathrm e}^{3+2 x}-\frac {5 x}{2+x}\right )^{2}}{\ln \! \left (x \right )^{2}} \]
command
integrate(((((x**2+4*x+4)*exp(3+2*x)-5*x**2-10*x)*ln(x)+(-2*x**2-8*x-8)*exp(3+2*x)+10*x**2+20*x)*ln(((2+x)*exp(3+2*x)-5*x)/(2+x))**2+((4*x**3+16*x**2+16*x)*exp(3+2*x)-20*x)*ln(x)*ln(((2+x)*exp(3+2*x)-5*x)/(2+x)))/((x**2+4*x+4)*exp(3+2*x)-5*x**2-10*x)/ln(x)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Exception raised: TypeError} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {x \log {\left (\frac {- 5 x + \left (x + 2\right ) e^{2 x + 3}}{x + 2} \right )}^{2}}{\log {\left (x \right )}^{2}} \]