\[ y'(x)=x \sqrt {a^2+2 a x+x^2+4 y(x)}-\frac {a}{2}-\frac {x}{2} \] ✓ Mathematica : cpu = 0.491214 (sec), leaf count = 109
DSolve[Derivative[1][y][x] == -1/2*a - x/2 + x*Sqrt[a^2 + 2*a*x + x^2 + 4*y[x]],y[x],x]
\[\text {Solve}\left [-\frac {1}{2} \sqrt {a^2+2 a x+x^2+4 y(x)}-\frac {1}{2} a \log \left (\sqrt {a^2+2 a x+x^2+4 y(x)}+a+x\right )+\frac {1}{2} a \tanh ^{-1}\left (\frac {2 a+2 x}{2 \sqrt {a^2+2 a x+x^2+4 y(x)}}\right )+\frac {1}{4} a \log (y(x))+\frac {x^2}{2}=c_1,y(x)\right ]\] ✓ Maple : cpu = 0.354 (sec), leaf count = 28
dsolve(diff(y(x),x) = -1/2*x-1/2*a+x*(x^2+2*a*x+a^2+4*y(x))^(1/2),y(x))
\[c_{1}+x^{2}+\frac {1}{2}-\sqrt {x^{2}+2 a x +a^{2}+4 y \left (x \right )} = 0\]