ODE
\[ x (a+x) y'(x)=y(x) (b+c y(x)) \] ODE Classification
[_separable]
Book solution method
Separable ODE, Neither variable missing
Mathematica ✓
cpu = 0.366384 (sec), leaf count = 50
\[\left \{\left \{y(x)\to -\frac {b e^{b c_1} x^{\frac {b}{a}}}{-(a+x)^{\frac {b}{a}}+c e^{b c_1} x^{\frac {b}{a}}}\right \}\right \}\]
Maple ✓
cpu = 0.038 (sec), leaf count = 32
\[\left [y \left (x \right ) = \frac {b}{\left (a +x \right )^{\frac {b}{a}} x^{-\frac {b}{a}} \textit {\_C1} b -c}\right ]\] Mathematica raw input
DSolve[x*(a + x)*y'[x] == y[x]*(b + c*y[x]),y[x],x]
Mathematica raw output
{{y[x] -> -((b*E^(b*C[1])*x^(b/a))/(c*E^(b*C[1])*x^(b/a) - (a + x)^(b/a)))}}
Maple raw input
dsolve(x*(a+x)*diff(y(x),x) = (b+c*y(x))*y(x), y(x))
Maple raw output
[y(x) = b/((a+x)^(b/a)*x^(-b/a)*_C1*b-c)]