\[ \int \frac {-8 x-40 x^2+(-1-10 x) \log ^2(x)+\left (-4-20 x+(1+5 x) \log ^2(x)\right ) \log \left (x+5 x^2\right )}{\left (2 x^2+10 x^3\right ) \log ^2(x)+\left (x+5 x^2\right ) \log ^2(x) \log \left (x+5 x^2\right )} \, dx \]
Optimal antiderivative \[ \frac {4}{\ln \! \left (x \right )}-4-\ln \! \left (2+\frac {\ln \! \left (5 x^{2}+x \right )}{x}\right ) \]
command
Int[(-8*x - 40*x^2 + (-1 - 10*x)*Log[x]^2 + (-4 - 20*x + (1 + 5*x)*Log[x]^2)*Log[x + 5*x^2])/((2*x^2 + 10*x^3)*Log[x]^2 + (x + 5*x^2)*Log[x]^2*Log[x + 5*x^2]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {-8 x-40 x^2+(-1-10 x) \log ^2(x)+\left (-4-20 x+(1+5 x) \log ^2(x)\right ) \log \left (x+5 x^2\right )}{\left (2 x^2+10 x^3\right ) \log ^2(x)+\left (x+5 x^2\right ) \log ^2(x) \log \left (x+5 x^2\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \log (x)-\log (2 x+\log (x (5 x+1)))+\frac {4}{\log (x)} \]