\[ y'(x)=-\frac {\sqrt {a} x^3 \left (-2 \sqrt {a x^4+8 y(x)}+\sqrt {a} x+\sqrt {a}\right )}{2 (x+1)} \] ✓ Mathematica : cpu = 0.521124 (sec), leaf count = 128
DSolve[Derivative[1][y][x] == -1/2*(Sqrt[a]*x^3*(Sqrt[a] + Sqrt[a]*x - 2*Sqrt[a*x^4 + 8*y[x]]))/(1 + x),y[x],x]
\[\left \{\left \{y(x)\to \frac {1}{72} \left (16 a x^6-48 a x^5+123 a x^4-96 a x^3 \log (x+1)-96 a c_1 x^3-72 a x^2+144 a x^2 \log (x+1)+144 a c_1 x^2+432 a x+144 a \log ^2(x+1)-288 a x \log (x+1)-432 a \log (x+1)-288 a c_1 x+288 a c_1 \log (x+1)+324 a+144 a c_1{}^2-432 a c_1\right )\right \}\right \}\] ✓ Maple : cpu = 0.906 (sec), leaf count = 41
dsolve(diff(y(x),x) = -1/2*x^3*(a^(1/2)*x+a^(1/2)-2*(a*x^4+8*y(x))^(1/2))*a^(1/2)/(1+x),y(x))
\[\frac {\sqrt {a \,x^{4}+8 y \left (x \right )}}{4 \sqrt {a}}-\frac {x^{3}}{3}+\frac {x^{2}}{2}-x +\ln \left (1+x \right )-c_{1} = 0\]