\[ \int \frac {(50+(100+100 x) \log (2)+(100+100 x) \log (x)) \log \left (4 e^{2 x} x^2 \log (2)+4 e^{2 x} x^2 \log (x)\right )+(-50 \log (2)-50 \log (x)) \log ^2\left (4 e^{2 x} x^2 \log (2)+4 e^{2 x} x^2 \log (x)\right )}{x^3 \log (2)+x^3 \log (x)} \, dx \]
Optimal antiderivative \[ \frac {25 \ln \! \left (4 \,{\mathrm e}^{2 x} x^{2} \left (\ln \! \left (2\right )+\ln \! \left (x \right )\right )\right )^{2}}{x^{2}} \]
command
Integrate[((50 + (100 + 100*x)*Log[2] + (100 + 100*x)*Log[x])*Log[4*E^(2*x)*x^2*Log[2] + 4*E^(2*x)*x^2*Log[x]] + (-50*Log[2] - 50*Log[x])*Log[4*E^(2*x)*x^2*Log[2] + 4*E^(2*x)*x^2*Log[x]]^2)/(x^3*Log[2] + x^3*Log[x]),x]
Mathematica 13.1 output
\[ \int \frac {(50+(100+100 x) \log (2)+(100+100 x) \log (x)) \log \left (4 e^{2 x} x^2 \log (2)+4 e^{2 x} x^2 \log (x)\right )+(-50 \log (2)-50 \log (x)) \log ^2\left (4 e^{2 x} x^2 \log (2)+4 e^{2 x} x^2 \log (x)\right )}{x^3 \log (2)+x^3 \log (x)} \, dx \]
Mathematica 12.3 output
\[ 50 \left (-2+\frac {\log ^2\left (4 e^{2 x} x^2 \log (2 x)\right )}{2 x^2}\right ) \]