\[ \int \frac {e^{\frac {1}{-400 x^5+\left (2000 x^4+400 x^5\right ) \log (x)}} (-5+4 x+(-20-5 x) \log (x))}{200 x^7+\left (-2000 x^6-400 x^7\right ) \log (x)+\left (5000 x^5+2000 x^6+200 x^7\right ) \log ^2(x)} \, dx \]
Optimal antiderivative \[ 2 \,{\mathrm e}^{-\frac {1}{400 x^{4} \left (x +\left (-x -5\right ) \ln \left (x \right )\right )}} \]
command
Int[(E^(-400*x^5 + (2000*x^4 + 400*x^5)*Log[x])^(-1)*(-5 + 4*x + (-20 - 5*x)*Log[x]))/(200*x^7 + (-2000*x^6 - 400*x^7)*Log[x] + (5000*x^5 + 2000*x^6 + 200*x^7)*Log[x]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ 2 e^{-\frac {1}{400 x^4 (x+x (-\log (x))-5 \log (x))}} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {e^{\frac {1}{-400 x^5+\left (2000 x^4+400 x^5\right ) \log (x)}} (-5+4 x+(-20-5 x) \log (x))}{200 x^7+\left (-2000 x^6-400 x^7\right ) \log (x)+\left (5000 x^5+2000 x^6+200 x^7\right ) \log ^2(x)} \, dx \]________________________________________________________________________________________