\[ y'(x)=\frac {\sqrt {a^2+2 a x+x^2+4 y(x)}-\frac {a x}{2}-\frac {a}{2}-\frac {x^2}{2}-\frac {x}{2}}{x+1} \] ✓ Mathematica : cpu = 0.309522 (sec), leaf count = 46
DSolve[Derivative[1][y][x] == (-1/2*a - x/2 - (a*x)/2 - x^2/2 + Sqrt[a^2 + 2*a*x + x^2 + 4*y[x]])/(1 + x),y[x],x]
\[\left \{\left \{y(x)\to \frac {1}{4} \left (-a^2-2 a x-x^2+4 \log ^2(x+1)-8 c_1 \log (x+1)+4 c_1{}^2\right )\right \}\right \}\] ✓ Maple : cpu = 0.394 (sec), leaf count = 33
dsolve(diff(y(x),x) = -1/2*(x^2+x+a*x+a-2*(x^2+2*a*x+a^2+4*y(x))^(1/2))/(1+x),y(x))
\[c_{1}+\frac {a}{2}+2 \ln \left (1+x \right )-\sqrt {x^{2}+2 a x +a^{2}+4 y \left (x \right )} = 0\]