ODE
\[ x (a+y(x)) y'(x)=y(x) (A+B x) \] ODE Classification
[_separable]
Book solution method
Separable ODE, Neither variable missing
Mathematica ✓
cpu = 0.189765 (sec), leaf count = 31
\[\left \{\left \{y(x)\to a W\left (\frac {x^{\frac {A}{a}} e^{\frac {B x+c_1}{a}}}{a}\right )\right \}\right \}\]
Maple ✓
cpu = 0.197 (sec), leaf count = 46
\[\left [y \left (x \right ) = {\mathrm e}^{\frac {A \ln \left (x \right )+B x -a \LambertW \left (\frac {x^{\frac {A}{a}} {\mathrm e}^{\frac {B x}{a}+\frac {\textit {\_C1}}{a}}}{a}\right )+\textit {\_C1}}{a}}\right ]\] Mathematica raw input
DSolve[x*(a + y[x])*y'[x] == (A + B*x)*y[x],y[x],x]
Mathematica raw output
{{y[x] -> a*ProductLog[(E^((B*x + C[1])/a)*x^(A/a))/a]}}
Maple raw input
dsolve(x*(a+y(x))*diff(y(x),x) = y(x)*(B*x+A), y(x))
Maple raw output
[y(x) = exp((A*ln(x)+B*x-a*LambertW(1/a*x^(1/a*A)*exp(1/a*B*x+1/a*_C1))+_C1)/a)]