ODE
\[ x (y(x)+x) y'(x)=x^2+y(x)^2 \] ODE Classification
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]
Book solution method
Homogeneous equation
Mathematica ✓
cpu = 0.0218283 (sec), leaf count = 30
\[\left \{\left \{y(x)\to 2 x W\left (\frac {e^{\frac {1}{2} \left (c_1-1\right )}}{2 \sqrt {x}}\right )+x\right \}\right \}\]
Maple ✓
cpu = 0.011 (sec), leaf count = 31
\[ \left \{ -2\,\ln \left ( {\frac {y \left ( x \right ) -x}{x}} \right ) -{\frac {y \left ( x \right ) }{x}}-\ln \left ( x \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[x*(x + y[x])*y'[x] == x^2 + y[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> x + 2*x*ProductLog[E^((-1 + C[1])/2)/(2*Sqrt[x])]}}
Maple raw input
dsolve(x*(x+y(x))*diff(y(x),x) = x^2+y(x)^2, y(x),'implicit')
Maple raw output
-2*ln((y(x)-x)/x)-y(x)/x-ln(x)-_C1 = 0