ODE
\[ x (1-2 x y(x)) y'(x)+y(x) (2 x y(x)+1)=0 \] ODE Classification
[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]
Book solution method
Change of Variable, Two new variables
Mathematica ✓
cpu = 1.06935 (sec), leaf count = 32
\[\left \{\left \{y(x)\to -\frac {1}{2 x W\left (\frac {e^{-1+\frac {9 c_1}{2^{2/3}}}}{x^2}\right )}\right \}\right \}\]
Maple ✓
cpu = 0.131 (sec), leaf count = 18
\[\left [y \left (x \right ) = -\frac {1}{2 \LambertW \left (-\frac {\textit {\_C1}}{2 x^{2}}\right ) x}\right ]\] Mathematica raw input
DSolve[y[x]*(1 + 2*x*y[x]) + x*(1 - 2*x*y[x])*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -1/2*1/(x*ProductLog[E^(-1 + (9*C[1])/2^(2/3))/x^2])}}
Maple raw input
dsolve(x*(1-2*x*y(x))*diff(y(x),x)+y(x)*(1+2*x*y(x)) = 0, y(x))
Maple raw output
[y(x) = -1/2/LambertW(-1/2*_C1/x^2)/x]