4.11.28 $x(a+y(x))y^{\prime }(x)=y(x)(A+Bx)$

ODE
$$x(a+y(x))y^{\prime }(x)=y(x)(A+Bx)$$ ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica ✓
cpu = 0.0177155 (sec), leaf count = 31

$$\left\{\{y(x)\to aW\left(\frac{x^{\frac{A}{a}}{e}^{\frac{Bx+c_1}{a}}}{a}\right)\}\right\}$$

Maple ✓
cpu = 0.01 (sec), leaf count = 22

$$\{Bx+A\ln \left(x\right)-y\left(x\right)-a\ln \left(y\left(x\right)\right)+\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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),'implicit')

Maple raw output

B*x+A*ln(x)-y(x)-a*ln(y(x))+_C1 = 0