\[ y'(x)=\frac {x^3 \left (\sqrt {9 x^4-4 y(x)^3}+3 x+3\right )}{(x+1) y(x)^2} \] ✓ Mathematica : cpu = 1.91134 (sec), leaf count = 372
DSolve[Derivative[1][y][x] == (x^3*(3 + 3*x + Sqrt[9*x^4 - 4*y[x]^3]))/((1 + x)*y[x]^2),y[x],x]
\[\left \{\left \{y(x)\to \frac {\sqrt [3]{-4 x^6+12 x^5-24 x^4-8 x^3+24 x^3 \log (x+1)+24 c_1 x^3+30 x^2-36 x^2 \log (x+1)-36 c_1 x^2-132 x-36 \log ^2(x+1)+72 x \log (x+1)+132 \log (x+1)+72 c_1 x-72 c_1 \log (x+1)-121-36 c_1{}^2+132 c_1}}{2^{2/3}}\right \},\left \{y(x)\to -\frac {\sqrt [3]{-1} \sqrt [3]{-4 x^6+12 x^5-24 x^4-8 x^3+24 x^3 \log (x+1)+24 c_1 x^3+30 x^2-36 x^2 \log (x+1)-36 c_1 x^2-132 x-36 \log ^2(x+1)+72 x \log (x+1)+132 \log (x+1)+72 c_1 x-72 c_1 \log (x+1)-121-36 c_1{}^2+132 c_1}}{2^{2/3}}\right \},\left \{y(x)\to \frac {(-1)^{2/3} \sqrt [3]{-4 x^6+12 x^5-24 x^4-8 x^3+24 x^3 \log (x+1)+24 c_1 x^3+30 x^2-36 x^2 \log (x+1)-36 c_1 x^2-132 x-36 \log ^2(x+1)+72 x \log (x+1)+132 \log (x+1)+72 c_1 x-72 c_1 \log (x+1)-121-36 c_1{}^2+132 c_1}}{2^{2/3}}\right \}\right \}\] ✓ Maple : cpu = 0.164 (sec), leaf count = 48
dsolve(diff(y(x),x) = x^3*(3*x+3+(9*x^4-4*y(x)^3)^(1/2))/(1+x)/y(x)^2,y(x))
\[\int _{\textit {\_b}}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {9 x^{4}-4 \textit {\_a}^{3}}}d \textit {\_a} -\frac {x^{3}}{3}+\frac {x^{2}}{2}-x +\ln \left (1+x \right )-c_{1} = 0\]