\[ y'(x)=\frac {e^{-x} y(x) \left (x^2 y(x) \log (2 x)-e^x-x \log (2 x)\right )}{x} \] ✓ Mathematica : cpu = 0.406401 (sec), leaf count = 49
DSolve[Derivative[1][y][x] == (y[x]*(-E^x - x*Log[2*x] + x^2*Log[2*x]*y[x]))/(E^x*x),y[x],x]
\[\left \{\left \{y(x)\to \frac {2^{e^{-x}} x^{e^{-x}-1}}{2^{e^{-x}} x^{e^{-x}}+c_1 e^{\text {Ei}(-x)}}\right \}\right \}\] ✓ Maple : cpu = 0.152 (sec), leaf count = 34
dsolve(diff(y(x),x) = y(x)*(-exp(x)+ln(2*x)*x^2*y(x)-ln(2*x)*x)/x/exp(x),y(x))
\[y \left (x \right ) = \frac {1}{2^{-{\mathrm e}^{-x}} x^{-{\mathrm e}^{-x}+1} c_{1} {\mathrm e}^{-\Ei \left (1, x\right )}+x}\]