\[ y'(x)=\frac {x^3 \sqrt {4 a x-y(x)^2}+2 a x+2 a}{(x+1) y(x)} \] ✓ Mathematica : cpu = 2.55579 (sec), leaf count = 217
DSolve[Derivative[1][y][x] == (2*a + 2*a*x + x^3*Sqrt[4*a*x - y[x]^2])/((1 + x)*y[x]),y[x],x]
\[\left \{\left \{y(x)\to -\frac {1}{6} \sqrt {144 a x-4 x^6+12 x^5-33 x^4+36 x^3+24 x^3 \log (x+1)-24 c_1 x^3-36 x^2-36 x^2 \log (x+1)+36 c_1 x^2-36 \log ^2(x+1)+72 x \log (x+1)-72 c_1 x+72 c_1 \log (x+1)-36 c_1{}^2}\right \},\left \{y(x)\to \frac {1}{6} \sqrt {144 a x-4 x^6+12 x^5-33 x^4+36 x^3+24 x^3 \log (x+1)-24 c_1 x^3-36 x^2-36 x^2 \log (x+1)+36 c_1 x^2-36 \log ^2(x+1)+72 x \log (x+1)-72 c_1 x+72 c_1 \log (x+1)-36 c_1{}^2}\right \}\right \}\] ✓ Maple : cpu = 0.312 (sec), leaf count = 39
dsolve(diff(y(x),x) = (2*a*x+2*a+x^3*(-y(x)^2+4*a*x)^(1/2))/(1+x)/y(x),y(x))
\[-\sqrt {-y \left (x \right )^{2}+4 a x}-\frac {x^{3}}{3}+\frac {x^{2}}{2}-x +\ln \left (1+x \right )-c_{1} = 0\]