\[ y'(x)=\frac {x^6 \sqrt {4 x^2 y(x)+1}+\frac {x}{2}+\frac {1}{2}}{x^3 (x+1)} \] ✓ Mathematica : cpu = 0.737065 (sec), leaf count = 206
DSolve[Derivative[1][y][x] == (1/2 + x/2 + x^6*Sqrt[1 + 4*x^2*y[x]])/(x^3*(1 + x)),y[x],x]
\[\left \{\left \{y(x)\to -\frac {1}{4 x^2}+\frac {1}{4} \log ^2\left (x^2 \cosh \left (-\frac {x^4}{2}+\frac {2 x^3}{3}-x^2+2 x+2 c_1\right )+2 x \cosh \left (-\frac {x^4}{2}+\frac {2 x^3}{3}-x^2+2 x+2 c_1\right )+\cosh \left (-\frac {x^4}{2}+\frac {2 x^3}{3}-x^2+2 x+2 c_1\right )-x^2 \sinh \left (-\frac {x^4}{2}+\frac {2 x^3}{3}-x^2+2 x+2 c_1\right )-2 x \sinh \left (-\frac {x^4}{2}+\frac {2 x^3}{3}-x^2+2 x+2 c_1\right )-\sinh \left (-\frac {x^4}{2}+\frac {2 x^3}{3}-x^2+2 x+2 c_1\right )\right )\right \}\right \}\] ✓ Maple : cpu = 0.638 (sec), leaf count = 43
dsolve(diff(y(x),x) = 1/2*(x+1+2*x^6*(4*x^2*y(x)+1)^(1/2))/x^3/(1+x),y(x))
\[c_{1}+2 \ln \left (1+x \right )-\frac {\sqrt {4 x^{2} y \left (x \right )+1}}{x}-2 x +x^{2}-\frac {2 x^{3}}{3}+\frac {x^{4}}{2} = 0\]