ODE
\[ x (1-x y(x)) y'(x)+y(x) (x y(x)+1)=0 \] ODE Classification
[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]
Book solution method
Exact equation, integrating factor
Mathematica ✓
cpu = 0.98151 (sec), leaf count = 30
\[\left \{\left \{y(x)\to -\frac {1}{x W\left (\frac {e^{\frac {9 c_1}{2^{2/3}}-1}}{x^2}\right )}\right \}\right \}\]
Maple ✓
cpu = 0.017 (sec), leaf count = 29
\[ \left \{ \ln \left ( x \right ) -{\it \_C1}-{\frac {\ln \left ( xy \left ( x \right ) \right ) xy \left ( x \right ) +1}{2\,xy \left ( x \right ) }}=0 \right \} \] Mathematica raw input
DSolve[y[x]*(1 + x*y[x]) + x*(1 - x*y[x])*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -(1/(x*ProductLog[E^(-1 + (9*C[1])/2^(2/3))/x^2]))}}
Maple raw input
dsolve(x*(1-x*y(x))*diff(y(x),x)+(1+x*y(x))*y(x) = 0, y(x),'implicit')
Maple raw output
ln(x)-_C1-1/2*(ln(x*y(x))*x*y(x)+1)/x/y(x) = 0