\[ \int \frac {\left (12 x+18 x^2+e^x \left (12+12 x+6 x^2\right )+\left (6 x+6 e^x x\right ) \log (x)\right ) \log \left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )+\left (3 e^x x+3 x^2+\left (3 e^x+3 x\right ) \log (x)\right ) \log ^2\left (25 e^x x^2+25 x^3+\left (50 e^x x+50 x^2\right ) \log (x)+\left (25 e^x+25 x\right ) \log ^2(x)\right )}{2 e^x x+2 x^2+\left (2 e^x+2 x\right ) \log (x)} \, dx \]
Optimal antiderivative \[ \frac {3 x \ln \! \left (5 \left (x +\ln \! \left (x \right )\right ) \left (5 x +5 \ln \! \left (x \right )\right ) \left (x +{\mathrm e}^{x}\right )\right )^{2}}{2} \]
command
integrate((((3*exp(x)+3*x)*ln(x)+3*exp(x)*x+3*x**2)*ln((25*exp(x)+25*x)*ln(x)**2+(50*exp(x)*x+50*x**2)*ln(x)+25*exp(x)*x**2+25*x**3)**2+((6*exp(x)*x+6*x)*ln(x)+(6*x**2+12*x+12)*exp(x)+18*x**2+12*x)*ln((25*exp(x)+25*x)*ln(x)**2+(50*exp(x)*x+50*x**2)*ln(x)+25*exp(x)*x**2+25*x**3))/((2*exp(x)+2*x)*ln(x)+2*exp(x)*x+2*x**2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Exception raised: TypeError} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {3 x \log {\left (25 x^{3} + 25 x^{2} e^{x} + \left (25 x + 25 e^{x}\right ) \log {\left (x \right )}^{2} + \left (50 x^{2} + 50 x e^{x}\right ) \log {\left (x \right )} \right )}^{2}}{2} \]