[_exact]
Book solution method
Exact equation
Mathematica ✓
cpu = 0.0690978 (sec), leaf count = 33
Maple ✓
cpu = 0.034 (sec), leaf count = 15
DSolve[E^y[x] + E^x*y[x] + (E^x + E^y[x]*x)*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]/E^x - ProductLog[E^(-x + C[1]/E^x)*x]}}
Maple raw input
dsolve((exp(x)+x*exp(y(x)))*diff(y(x),x)+y(x)*exp(x)+exp(y(x)) = 0, y(x),'implicit')
Maple raw output
y(x)*exp(x)+x*exp(y(x))+_C1 = 0