ODE
\[ a+b y(x)+x y'(x)+\log \left (y'(x)\right )=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries], _dAlembert]
Book solution method
Clairaut’s equation and related types, d’Alembert’s equation (also call Lagrange’s)
Mathematica ✓
cpu = 0.149511 (sec), leaf count = 57
\[\text {Solve}\left [\frac {b W\left (x e^{-a-b y(x)}\right )+(b+1) \log \left (1-b W\left (x e^{-a-b y(x)}\right )\right )+b^2 y(x)}{b}=c_1,y(x)\right ]\]
Maple ✓
cpu = 0.012 (sec), leaf count = 88
\[ \left \{ [x \left ( {\it \_T} \right ) ={\frac {{\it \_C1}\,b{\it \_T}+{{\it \_T}}^{ \left ( b+1 \right ) ^{-1}}}{{{\it \_T}}^{ \left ( b+1 \right ) ^{-1}}b{\it \_T}}},y \left ( {\it \_T} \right ) ={\frac {-{{\it \_T}}^{ \left ( b+1 \right ) ^{-1}}-b{{\it \_T}}^{ \left ( b+1 \right ) ^{-1}}\ln \left ( {\it \_T} \right ) + \left ( -a{{\it \_T}}^{ \left ( b+1 \right ) ^{-1}}-{\it \_C1}\,{\it \_T} \right ) b}{{b}^{2}{{\it \_T}}^{ \left ( b+1 \right ) ^{-1}}}}] \right \} \] Mathematica raw input
DSolve[a + Log[y'[x]] + b*y[x] + x*y'[x] == 0,y[x],x]
Mathematica raw output
Solve[((1 + b)*Log[1 - b*ProductLog[E^(-a - b*y[x])*x]] + b*ProductLog[E^(-a - b*y[x])*x] + b^2*y[x])/b == C[1], y[x]]
Maple raw input
dsolve(ln(diff(y(x),x))+x*diff(y(x),x)+a+b*y(x) = 0, y(x),'implicit')
Maple raw output
[x(_T) = (_C1*b*_T+_T^(1/(b+1)))/(_T^(1/(b+1)))/b/_T, y(_T) = (-_T^(1/(b+1))-b*_T^(1/(b+1))*ln(_T)+(-a*_T^(1/(b+1))-_C1*_T)*b)/b^2/(_T^(1/(b+1)))]