\[ y'(x)=\frac {\sqrt {x^2-4 y(x)+2 x+1}+\frac {x^2}{2}+x+\frac {1}{2}}{x+1} \] ✓ Mathematica : cpu = 1.02059 (sec), leaf count = 89
DSolve[Derivative[1][y][x] == (1/2 + x + x^2/2 + Sqrt[1 + 2*x + x^2 - 4*y[x]])/(1 + x),y[x],x]
\[\text {Solve}\left [\frac {1}{2} \left (\sqrt {x^2-4 y(x)+2 x+1}+\log \left (\sqrt {x^2-4 y(x)+2 x+1}+x+1\right )-\tanh ^{-1}\left (\frac {2 x+2}{2 \sqrt {x^2-4 y(x)+2 x+1}}\right )+2 \log (x+1)\right )-\frac {1}{4} \log (y(x))=c_1,y(x)\right ]\] ✓ Maple : cpu = 0.469 (sec), leaf count = 28
dsolve(diff(y(x),x) = 1/2*(x^2+2*x+1+2*(x^2+2*x+1-4*y(x))^(1/2))/(1+x),y(x))
\[c_{1}-2 \ln \left (1+x \right )-\frac {1}{2}-\sqrt {x^{2}+2 x +1-4 y \left (x \right )} = 0\]