\[ \int \frac {5 x-121 x^2+324 x^3+e^4 (-81+324 x)+\left (-18 x^2+72 x^3+e^4 (-18+72 x)\right ) \log \left (-x+4 x^2\right )+\left (-x^2+4 x^3+e^4 (-1+4 x)\right ) \log ^2\left (-x+4 x^2\right )}{-81 x^2+324 x^3+\left (-18 x^2+72 x^3\right ) \log \left (-x+4 x^2\right )+\left (-x^2+4 x^3\right ) \log ^2\left (-x+4 x^2\right )} \, dx \]
Optimal antiderivative \[ x +5-\frac {{\mathrm e}^{4}}{x}+\frac {5}{\ln \! \left (4 x^{2}-x \right )+9} \]
command
Int[(5*x - 121*x^2 + 324*x^3 + E^4*(-81 + 324*x) + (-18*x^2 + 72*x^3 + E^4*(-18 + 72*x))*Log[-x + 4*x^2] + (-x^2 + 4*x^3 + E^4*(-1 + 4*x))*Log[-x + 4*x^2]^2)/(-81*x^2 + 324*x^3 + (-18*x^2 + 72*x^3)*Log[-x + 4*x^2] + (-x^2 + 4*x^3)*Log[-x + 4*x^2]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {5 x-121 x^2+324 x^3+e^4 (-81+324 x)+\left (-18 x^2+72 x^3+e^4 (-18+72 x)\right ) \log \left (-x+4 x^2\right )+\left (-x^2+4 x^3+e^4 (-1+4 x)\right ) \log ^2\left (-x+4 x^2\right )}{-81 x^2+324 x^3+\left (-18 x^2+72 x^3\right ) \log \left (-x+4 x^2\right )+\left (-x^2+4 x^3\right ) \log ^2\left (-x+4 x^2\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ x-\frac {e^4}{x}+\frac {5}{\log (-((1-4 x) x))+9} \]