\[ y'(x)=\frac {(y(x)+1) (2 y(x)+1)}{x (2 x y(x)-2 y(x)+x-2)} \] ✓ Mathematica : cpu = 1.92831 (sec), leaf count = 149
DSolve[Derivative[1][y][x] == ((1 + y[x])*(1 + 2*y[x]))/(x*(-2 + x - 2*y[x] + 2*x*y[x])),y[x],x]
\[\text {Solve}\left [\frac {2^{2/3} \left (x \log \left (-\frac {6\ 2^{2/3} (y(x)+1)}{2 (x-1) y(x)+x-2}\right )-x \log \left (\frac {3\ 2^{2/3} (2 x y(x)+x)}{2 (x-1) y(x)+x-2}\right )+2 x y(x) \left (\log \left (-\frac {6\ 2^{2/3} (y(x)+1)}{2 (x-1) y(x)+x-2}\right )-\log \left (\frac {3\ 2^{2/3} (2 x y(x)+x)}{2 (x-1) y(x)+x-2}\right )+\log (x)+1\right )+x+x \log (x)-1\right )}{9 (2 x y(x)+x)}=c_1,y(x)\right ]\] ✓ Maple : cpu = 0.13 (sec), leaf count = 45
dsolve(diff(y(x),x) = 1/x*(1+2*y(x))*(1+y(x))/(-2*y(x)-2+x+2*x*y(x)),y(x))
\[y \left (x \right ) = \frac {-x \LambertW \left (\frac {{\mathrm e}^{-\frac {1}{x}}}{x c_{1}}\right )-2}{2 x \LambertW \left (\frac {{\mathrm e}^{-\frac {1}{x}}}{x c_{1}}\right )+2}\]