\[ y'(x)=\frac {x^3 \sqrt {a^2+2 a x+x^2+4 y(x)}-\frac {a x}{2}-\frac {a}{2}-\frac {x^2}{2}-\frac {x}{2}}{x+1} \] ✓ Mathematica : cpu = 0.755298 (sec), leaf count = 56
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]
\[\left \{\left \{y(x)\to \frac {1}{4} \left (-a^2-2 a x-x^2+\frac {1}{9} \left (2 x^3-3 x^2+6 x-6 \log (-x-1)-6 c_1\right ){}^2\right )\right \}\right \}\] ✓ Maple : cpu = 0.418 (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))
\[c_{1}+\frac {2 x^{3}}{3}-x^{2}-2 \ln \left (1+x \right )+2 x -\sqrt {x^{2}+2 a x +a^{2}+4 y \left (x \right )} = 0\]