\[ \int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx \]
Optimal antiderivative \[ \frac {x}{x \left (1+\frac {x}{5}\right )+5 \left (\ln \! \left (\ln \! \left (x \right )\right )-5-\ln \! \left (3\right )\right ) \ln \! \left (x \right )} \]
command
Integrate[(500 - 5*x^2 + 125*Log[3] + (-625 - 125*Log[3])*Log[x] + (-125 + 125*Log[x])*Log[Log[x]])/(25*x^2 + 10*x^3 + x^4 + (-1250*x - 250*x^2 + (-250*x - 50*x^2)*Log[3])*Log[x] + (15625 + 6250*Log[3] + 625*Log[3]^2)*Log[x]^2 + ((250*x + 50*x^2)*Log[x] + (-6250 - 1250*Log[3])*Log[x]^2)*Log[Log[x]] + 625*Log[x]^2*Log[Log[x]]^2),x]
Mathematica 13.1 output
\[ \int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx \]
Mathematica 12.3 output
\[ \frac {5 x}{x (5+x)+25 \log (x) \left (-5+\log \left (\frac {\log (x)}{3}\right )\right )} \]