\[ \int \frac {\left (-8 x+4 x^2\right ) \log (-2+x) \log \left (\frac {1}{2} (1+2 x)\right )+\left (-2 x-4 x^2+\left (2+3 x-2 x^2\right ) \log (-2+x)\right ) \log ^2\left (\frac {1}{2} (1+2 x)\right )}{\left (-2 x^2-3 x^3+2 x^4\right ) \log ^3(-2+x)} \, dx \]
Optimal antiderivative \[ \frac {\ln \! \left (\frac {1}{2}+x \right )^{2}}{x \ln \! \left (-2+x \right )^{2}} \]
command
integrate((((-2*x**2+3*x+2)*ln(-2+x)-4*x**2-2*x)*ln(1/2+x)**2+(4*x**2-8*x)*ln(-2+x)*ln(1/2+x))/(2*x**4-3*x**3-2*x**2)/ln(-2+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
\[ \frac {\log {\left (x + \frac {1}{2} \right )}^{2}}{x \log {\left (x - 2 \right )}^{2}} \]