ODE
\[ y'(x) (a+b x+y(x))+a-b x-y(x)=0 \] 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 = 0.204905 (sec), leaf count = 49
\[\left \{\left \{y(x)\to \frac {2 a W\left (-e^{\frac {(b+1)^2 x}{2 a}-1+c_1}\right )+a (-b)+a-b (b+1) x}{b+1}\right \}\right \}\]
Maple ✓
cpu = 0.233 (sec), leaf count = 91
\[\left [y \left (x \right ) = \frac {-b^{2} x +2 \LambertW \left (\frac {{\mathrm e}^{-\frac {\textit {\_C1} \,b^{2}}{2 a}} {\mathrm e}^{\frac {b^{2} x}{2 a}} {\mathrm e}^{-\frac {\textit {\_C1} b}{a}} {\mathrm e}^{\frac {b}{2}} {\mathrm e}^{\frac {b x}{a}} {\mathrm e}^{-\frac {\textit {\_C1}}{2 a}} {\mathrm e}^{-\frac {1}{2}} {\mathrm e}^{\frac {x}{2 a}}}{2 a}\right ) a -a b -b x +a}{b +1}\right ]\] Mathematica raw input
DSolve[a - b*x - y[x] + (a + b*x + y[x])*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (a - a*b - b*(1 + b)*x + 2*a*ProductLog[-E^(-1 + ((1 + b)^2*x)/(2*a) + C[1])])/(1 + b)}}
Maple raw input
dsolve((a+b*x+y(x))*diff(y(x),x)+a-b*x-y(x) = 0, y(x))
Maple raw output
[y(x) = (-b^2*x+2*LambertW(1/2*exp(-1/2/a*_C1*b^2)*exp(1/2/a*b^2*x)/exp(1/a*_C1*b)*exp(1/2*b)*exp(1/a*b*x)*exp(-1/2/a*_C1)*exp(-1/2)*exp(1/2/a*x)/a)*a-a*b-b*x+a)/(b+1)]