\[ \int \frac {\left (-250 x^2+50 x^3\right ) \log (x) \log (3+x)+\left (375 x+50 x^2-25 x^3+\left (-750 x-250 x^2\right ) \log (x)\right ) \log ^2(3+x)}{\left (-54000+14400 x+4320 x^2-1728 x^3+144 x^4\right ) \log ^2(x)} \, dx \]
Optimal antiderivative \[ \frac {25 x^{2} \ln \! \left (3+x \right )^{2}}{144 \left (5-x \right )^{2} \ln \! \left (x \right )} \]
command
integrate((((-250*x**2-750*x)*ln(x)-25*x**3+50*x**2+375*x)*ln(3+x)**2+(50*x**3-250*x**2)*ln(x)*ln(3+x))/(144*x**4-1728*x**3+4320*x**2+14400*x-54000)/ln(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
\[ \frac {25 x^{2} \log {\left (x + 3 \right )}^{2}}{144 x^{2} \log {\left (x \right )} - 1440 x \log {\left (x \right )} + 3600 \log {\left (x \right )}} \]