\[ y'(x)=\frac {x^2 \left (2 x \sqrt {x^3-6 y(x)}+x+1\right )}{2 (x+1)} \] ✓ Mathematica : cpu = 0.30446 (sec), leaf count = 101
DSolve[Derivative[1][y][x] == (x^2*(1 + x + 2*x*Sqrt[x^3 - 6*y[x]]))/(2*(1 + x)),y[x],x]
\[\left \{\left \{y(x)\to \frac {1}{24} \left (-4 x^6+12 x^5-33 x^4+40 x^3+24 x^3 \log (x+1)+24 c_1 x^3-36 x^2-36 x^2 \log (x+1)-36 c_1 x^2-36 \log ^2(x+1)+72 x \log (x+1)+72 c_1 x-72 c_1 \log (x+1)-36 c_1{}^2\right )\right \}\right \}\] ✓ Maple : cpu = 0.495 (sec), leaf count = 37
dsolve(diff(y(x),x) = 1/2*x^2*(x+1+2*x*(x^3-6*y(x))^(1/2))/(1+x),y(x))
\[c_{1}-x^{3}+\frac {3 x^{2}}{2}-3 x +3 \ln \left (1+x \right )-\frac {1}{2}-\sqrt {x^{3}-6 y \left (x \right )} = 0\]