##### 4.10.19 $$(2 y(x)+x+1) y'(x)-2 y(x)-x+1=0$$

ODE
$(2 y(x)+x+1) y'(x)-2 y(x)-x+1=0$ ODE Classiﬁcation

[[_homogeneous, class C], _rational, [_Abel, 2nd type, class A]]

Book solution method
Equation linear in the variables, $$y'(x)=f\left ( \frac {X_1}{X_2} \right )$$

Mathematica
cpu = 0.22433 (sec), leaf count = 30

$\left \{\left \{y(x)\to \frac {1}{6} \left (4 W\left (-e^{\frac {9 x}{4}-1+c_1}\right )-3 x+1\right )\right \}\right \}$

Maple
cpu = 0.094 (sec), leaf count = 21

$\left [y \left (x \right ) = -\frac {x}{2}+\frac {2 \LambertW \left (\frac {{\mathrm e}^{-\frac {1}{4}} {\mathrm e}^{\frac {9 x}{4}} \textit {\_C1}}{4}\right )}{3}+\frac {1}{6}\right ]$ Mathematica raw input

DSolve[1 - x - 2*y[x] + (1 + x + 2*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (1 - 3*x + 4*ProductLog[-E^(-1 + (9*x)/4 + C[1])])/6}}

Maple raw input

dsolve((1+x+2*y(x))*diff(y(x),x)+1-x-2*y(x) = 0, y(x))

Maple raw output

[y(x) = -1/2*x+2/3*LambertW(1/4*exp(-1/4)*exp(9/4*x)*_C1)+1/6]