\[ y'(x)=\frac {x^2 \log (2 x)+2 x y(x) \log (2 x)+y(x)^2 \log (2 x)-\log (x)+\log (2 x)}{\log (x)} \] ✓ Mathematica : cpu = 0.318638 (sec), leaf count = 19
DSolve[Derivative[1][y][x] == (-Log[x] + Log[2*x] + x^2*Log[2*x] + 2*x*Log[2*x]*y[x] + Log[2*x]*y[x]^2)/Log[x],y[x],x]
\[\{\{y(x)\to -x+\tan (\log (2) \text {li}(x)+x+c_1)\}\}\] ✓ Maple : cpu = 1.019 (sec), leaf count = 25
dsolve(diff(y(x),x) = (-ln(x)+2*ln(2*x)*x*y(x)+ln(2*x)+ln(2*x)*y(x)^2+ln(2*x)*x^2)/ln(x),y(x))
\[y \left (x \right ) = -x -\tan \left (\ln \left (2\right ) \Ei \left (1, -\ln \left (x \right )\right )+c_{1}-x \right )\]