\[ y'(x)=\frac {a x^4+a x^3+a x^3 \log (x+1)-x^2 y(x)^2-x y(x)^2+y(x)-x y(x)^2 \log (x+1)}{x} \] ✓ Mathematica : cpu = 0.181987 (sec), leaf count = 80
DSolve[Derivative[1][y][x] == (a*x^3 + a*x^4 + a*x^3*Log[1 + x] + y[x] - x*y[x]^2 - x^2*y[x]^2 - x*Log[1 + x]*y[x]^2)/x,y[x],x]
\[\left \{\left \{y(x)\to \sqrt {a} x \tanh \left (\frac {1}{12} \left (4 \sqrt {a} x^3+3 \sqrt {a} x^2+6 \sqrt {a} x^2 \log (x+1)+6 \sqrt {a} x-6 \sqrt {a} \log (x+1)+12 \sqrt {a} c_1\right )\right )\right \}\right \}\] ✓ Maple : cpu = 0.083 (sec), leaf count = 48
dsolve(diff(y(x),x) = (y(x)+x^3*a*ln(1+x)+a*x^4+a*x^3-x*y(x)^2*ln(1+x)-x^2*y(x)^2-x*y(x)^2)/x,y(x))
\[y \left (x \right ) = \tanh \left (\frac {\sqrt {a}\, \left (6 \ln \left (1+x \right ) x^{2}+4 x^{3}+3 x^{2}-6 \ln \left (1+x \right )+12 c_{1}+6 x +9\right )}{12}\right ) x \sqrt {a}\]