\[ \int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{\left (4 x+x^2\right ) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (\left (8 x+2 x^2\right ) \log (x)+\left (8 x^2+22 x^3+5 x^4\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )} \, dx \]
Optimal antiderivative \[ \ln \! \left (5 x +\frac {\frac {2+\frac {\ln \left (x \right )}{\ln \left (x^{2} \left (4+x \right )\right )}}{\ln \left (x \right )}+2 x}{x}\right ) \]
command
Integrate[((-8 - 3*x)*Log[x]^2 + (-4 - x)*Log[x]^2*Log[4*x^2 + x^3] + (-8 - 2*x + (-8 - 2*x)*Log[x] + (20*x^2 + 5*x^3)*Log[x]^2)*Log[4*x^2 + x^3]^2)/((4*x + x^2)*Log[x]^2*Log[4*x^2 + x^3] + ((8*x + 2*x^2)*Log[x] + (8*x^2 + 22*x^3 + 5*x^4)*Log[x]^2)*Log[4*x^2 + x^3]^2),x]
Mathematica 13.1 output
\[ \text {\$Aborted} \]
Mathematica 12.3 output
\[ \log (2+5 x)-\log (x (2+5 x))-\log (\log (x))-\log \left (\log \left (x^2 (4+x)\right )\right )+\log \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right ) \]