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 = 0.88013 (sec), leaf count = 32
\[\left \{\left \{y(x)\to -\frac {1}{2 x W\left (\frac {e^{\frac {9 c_1}{2^{2/3}}-1}}{x^2}\right )}\right \}\right \}\]
Maple ✓
cpu = 0.024 (sec), leaf count = 30
\[ \left \{ \ln \left ( x \right ) -{\it \_C1}-{\frac {2\,\ln \left ( xy \left ( x \right ) \right ) xy \left ( x \right ) +1}{4\,xy \left ( x \right ) }}=0 \right \} \] 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*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),'implicit')
Maple raw output
ln(x)-_C1-1/4*(2*ln(x*y(x))*x*y(x)+1)/x/y(x) = 0