\[ \int \frac {\left (20-48 x+12 x^2\right ) \log ^4(3)+\left (5 x-6 x^2+x^3\right ) \log ^4(3) \log (5)+\left (5 x-6 x^2+x^3\right ) \log ^4(3) \log \left (-5 x+6 x^2-x^3\right )}{\left (5 x-6 x^2+x^3\right ) \log (5)+\left (5 x-6 x^2+x^3\right ) \log \left (-5 x+6 x^2-x^3\right )} \, dx \]
Optimal antiderivative \[ \left (x +4 \ln \! \left (\ln \! \left (\left (-1+x \right ) x \left (5-x \right )\right )+\ln \! \left (5\right )\right )\right ) \ln \! \left (3\right )^{4} \]
command
Int[((20 - 48*x + 12*x^2)*Log[3]^4 + (5*x - 6*x^2 + x^3)*Log[3]^4*Log[5] + (5*x - 6*x^2 + x^3)*Log[3]^4*Log[-5*x + 6*x^2 - x^3])/((5*x - 6*x^2 + x^3)*Log[5] + (5*x - 6*x^2 + x^3)*Log[-5*x + 6*x^2 - x^3]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {\left (20-48 x+12 x^2\right ) \log ^4(3)+\left (5 x-6 x^2+x^3\right ) \log ^4(3) \log (5)+\left (5 x-6 x^2+x^3\right ) \log ^4(3) \log \left (-5 x+6 x^2-x^3\right )}{\left (5 x-6 x^2+x^3\right ) \log (5)+\left (5 x-6 x^2+x^3\right ) \log \left (-5 x+6 x^2-x^3\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ 4 \log ^4(3) \log \left (\log \left (-x \left (x^2-6 x+5\right )\right )+\log (5)\right )+x \log ^4(3) \]