ODE
\[ x (y(x)+x) y'(x)+y(x)^2=0 \] ODE Classification
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]
Book solution method
Homogeneous equation
Mathematica ✓
cpu = 0.300084 (sec), leaf count = 75
\[\left \{\left \{y(x)\to \frac {e^{2 c_1}-\sqrt {e^{2 c_1} \left (x^2+e^{2 c_1}\right )}}{x}\right \},\left \{y(x)\to \frac {\sqrt {e^{2 c_1} \left (x^2+e^{2 c_1}\right )}+e^{2 c_1}}{x}\right \}\right \}\]
Maple ✓
cpu = 0.174 (sec), leaf count = 44
\[\left [y \left (x \right ) = \frac {1+\sqrt {x^{2} \textit {\_C1} +1}}{\textit {\_C1} x}, y \left (x \right ) = -\frac {-1+\sqrt {x^{2} \textit {\_C1} +1}}{\textit {\_C1} x}\right ]\] Mathematica raw input
DSolve[y[x]^2 + x*(x + y[x])*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (E^(2*C[1]) - Sqrt[E^(2*C[1])*(E^(2*C[1]) + x^2)])/x}, {y[x] -> (E^(2*C[1]) + Sqrt[E^(2*C[1])*(E^(2*C[1]) + x^2)])/x}}
Maple raw input
dsolve(x*(x+y(x))*diff(y(x),x)+y(x)^2 = 0, y(x))
Maple raw output
[y(x) = 1/_C1*(1+(_C1*x^2+1)^(1/2))/x, y(x) = -1/_C1*(-1+(_C1*x^2+1)^(1/2))/x]