\[ y'(x)=\frac {x^3 \sqrt {a^2+2 a x+x^2+4 y(x)}-\frac {a x}{2}-\frac {a}{2}-\frac {x^2}{2}-\frac {x}{2}}{x+1} \] ✓ Mathematica : cpu = 0.775273 (sec), leaf count = 116
DSolve[Derivative[1][y][x] == (-1/2*a - x/2 - (a*x)/2 - x^2/2 + x^3*Sqrt[a^2 + 2*a*x + x^2 + 4*y[x]])/(1 + x),y[x],x]
\[\text {Solve}\left [-\frac {1}{2} \sqrt {a^2+2 a x+x^2+4 y(x)}-\frac {1}{2} a \log \left (\sqrt {a^2+2 a x+x^2+4 y(x)}+a+x\right )+\frac {1}{2} a \tanh ^{-1}\left (\frac {a+x}{\sqrt {a^2+2 a x+x^2+4 y(x)}}\right )+\frac {1}{4} a \log (y(x))+\frac {x^3}{3}-\frac {x^2}{2}+x-\log (x+1)=c_1,y(x)\right ]\] ✓ Maple : cpu = 0.589 (sec), leaf count = 43
dsolve(diff(y(x),x) = 1/2*(-x^2-x-a*x-a+2*x^3*(x^2+2*a*x+a^2+4*y(x))^(1/2))/(1+x),y(x))
\[c_{1}+\frac {2 x^{3}}{3}-x^{2}+2 x -2 \ln \left (1+x \right )-\sqrt {x^{2}+2 a x +a^{2}+4 y \left (x \right )} = 0\]