\[ \int \frac {56 x^2-8 x^3+\left (-160 x-8 x^2-16 x^3-32 x \log (2)\right ) \log (x)+\left (100+20 x+30 x^2+\left (40+4 x+6 x^2\right ) \log (2)+4 \log ^2(2)\right ) \log ^2(x)}{16 x^2+(-40 x-8 x \log (2)) \log (x)+\left (25+10 \log (2)+\log ^2(2)\right ) \log ^2(x)} \, dx \]
Optimal antiderivative \[ \frac {2 x^{2} \left (1+x \right )}{\ln \! \left (2\right )+5-\frac {4 x}{\ln \left (x \right )}}+4 x \]
command
Integrate[(56*x^2 - 8*x^3 + (-160*x - 8*x^2 - 16*x^3 - 32*x*Log[2])*Log[x] + (100 + 20*x + 30*x^2 + (40 + 4*x + 6*x^2)*Log[2] + 4*Log[2]^2)*Log[x]^2)/(16*x^2 + (-40*x - 8*x*Log[2])*Log[x] + (25 + 10*Log[2] + Log[2]^2)*Log[x]^2),x]
Mathematica 13.1 output
\[ \int \frac {56 x^2-8 x^3+\left (-160 x-8 x^2-16 x^3-32 x \log (2)\right ) \log (x)+\left (100+20 x+30 x^2+\left (40+4 x+6 x^2\right ) \log (2)+4 \log ^2(2)\right ) \log ^2(x)}{16 x^2+(-40 x-8 x \log (2)) \log (x)+\left (25+10 \log (2)+\log ^2(2)\right ) \log ^2(x)} \, dx \]
Mathematica 12.3 output
\[ 4 x+\frac {x^2 (10+\log (4))}{(5+\log (2))^2}+\frac {2 x^3 (15+\log (8))}{3 (5+\log (2))^2}+\frac {8 x^3 \left (25-15 \log ^2(2)-4 x^2 (5+\log (2))+\log ^2(16)+x \left (5+\log ^2(2)+\log (64)\right )+\log (1024)\right )}{(-5+4 x-\log (2)) (5+\log (2))^2 (4 x-(5+\log (2)) \log (x))} \]