\[ \int \frac {25 x+25 x^3 \log (3)+10 x^2 \log (3) \log (625)+x \log (3) \log ^2(625)+\left (-25 x+25 x^2+25 x^3 \log (3)+\left (-5+10 x+10 x^2 \log (3)\right ) \log (625)+(1+x \log (3)) \log ^2(625)\right ) \log \left (\frac {-5+5 x+5 x^2 \log (3)+(1+x \log (3)) \log (625)}{5 x+\log (625)}\right )}{-50 x+50 x^2+50 x^3 \log (3)+\left (-10+20 x+20 x^2 \log (3)\right ) \log (625)+(2+2 x \log (3)) \log ^2(625)} \, dx \]
Optimal antiderivative \[ \frac {\ln \left (x \ln \left (3\right )+1-\frac {1}{\frac {4 \ln \left (5\right )}{5}+x}\right ) x}{2} \]
command
Int[(25*x + 25*x^3*Log[3] + 10*x^2*Log[3]*Log[625] + x*Log[3]*Log[625]^2 + (-25*x + 25*x^2 + 25*x^3*Log[3] + (-5 + 10*x + 10*x^2*Log[3])*Log[625] + (1 + x*Log[3])*Log[625]^2)*Log[(-5 + 5*x + 5*x^2*Log[3] + (1 + x*Log[3])*Log[625])/(5*x + Log[625])])/(-50*x + 50*x^2 + 50*x^3*Log[3] + (-10 + 20*x + 20*x^2*Log[3])*Log[625] + (2 + 2*x*Log[3])*Log[625]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \frac {\log (625) \sqrt {25+\log ^2(3) \log ^2(625)+10 \log (3) (10-\log (625))} \text {arctanh}\left (\frac {10 x \log (3)+5+\log (3) \log (625)}{\sqrt {25+\log ^2(3) \log ^2(625)+10 \log (3) (10-\log (625))}}\right )}{50 \log (3)}+\frac {(5-4 \log (5)) \sqrt {25+\log ^2(3) \log ^2(625)+10 \log (3) (10-\log (625))} \text {arctanh}\left (\frac {10 x \log (3)+5+\log (3) \log (625)}{\sqrt {25+\log ^2(3) \log ^2(625)+10 \log (3) (10-\log (625))}}\right )}{50 \log (3)}-\frac {\sqrt {25+\log ^2(3) \log ^2(625)+10 \log (3) (10-\log (625))} \text {arctanh}\left (\frac {10 x \log (3)+5+\log (3) \log (625)}{\sqrt {25+\log ^2(3) \log ^2(625)+10 \log (3) (10-\log (625))}}\right )}{10 \log (3)}+\frac {1}{10} x \log (625) \log \left (-\frac {-5 x^2 \log (3)-x (5+\log (3) \log (625))+5-\log (625)}{5 x+\log (625)}\right )+\frac {1}{10} x (5-\log (625)) \log \left (-\frac {-5 x^2 \log (3)-x (5+\log (3) \log (625))+5-\log (625)}{5 x+\log (625)}\right )+\frac {\log (625) (5+\log (3) \log (625)) \log \left (-5 x^2 \log (3)-x (5+\log (3) \log (625))+5-\log (625)\right )}{100 \log (3)}+\frac {(5-4 \log (5)) (5+\log (3) \log (625)) \log \left (-5 x^2 \log (3)-x (5+\log (3) \log (625))+5-\log (625)\right )}{100 \log (3)}-\frac {(5+\log (3) \log (625)) \log \left (-5 x^2 \log (3)-x (5+\log (3) \log (625))+5-\log (625)\right )}{20 \log (3)}+\frac {x}{2}-\frac {1}{50} \log ^2(625) \log (5 x+\log (625))-\frac {1}{10} x \log (625)-\frac {1}{10} x (5-\log (625))-\frac {1}{50} (5-4 \log (5)) \log (625) \log (5 x+\log (625))+\frac {1}{10} \log (625) \log (5 x+\log (625)) \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \text {\$Aborted} \]________________________________________________________________________________________