\[ \int \frac {e^{\frac {\frac {3 x}{e}+(100+25 x) \log (x)+(-4-x) \log ^3(x)}{(4+x) \log (x)}} \left (\frac {3 \left (-4 x-x^2\right )}{e}+\frac {12 x \log (x)}{e}+\left (-32-16 x-2 x^2\right ) \log ^3(x)\right )}{\left (16 x+8 x^2+x^3\right ) \log ^2(x)} \, dx \]
Optimal antiderivative \[ {\mathrm e}^{\frac {{\mathrm e}^{\ln \left (3\right )-1} x}{\ln \left (x \right ) \left (4+x \right )}-\ln \left (x \right )^{2}+25} \]
command
Int[(E^(((3*x)/E + (100 + 25*x)*Log[x] + (-4 - x)*Log[x]^3)/((4 + x)*Log[x]))*((3*(-4*x - x^2))/E + (12*x*Log[x])/E + (-32 - 16*x - 2*x^2)*Log[x]^3))/((16*x + 8*x^2 + x^3)*Log[x]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {\exp \left (\frac {\frac {3 x}{e}+(100+25 x) \log (x)+(-4-x) \log ^3(x)}{(4+x) \log (x)}\right ) \left (\frac {3 \left (-4 x-x^2\right )}{e}+\frac {12 x \log (x)}{e}+\left (-32-16 x-2 x^2\right ) \log ^3(x)\right )}{\left (16 x+8 x^2+x^3\right ) \log ^2(x)} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \exp \left (-\log ^2(x)+\frac {3 x}{e x \log (x)+4 e \log (x)}+25\right ) \]