\[ y'(x)=\frac {-2 a^{3/2} b x^2 y(x)^2+2 a^{3/2} c y(x)^2+a^{5/2} y(x)^4+\sqrt {a} b^2 x^4-2 \sqrt {a} b c x^2+\sqrt {a} c^2+b x^3}{a x^2 y(x)} \] ✓ Mathematica : cpu = 0.970533 (sec), leaf count = 236
DSolve[Derivative[1][y][x] == (Sqrt[a]*c^2 - 2*Sqrt[a]*b*c*x^2 + b*x^3 + Sqrt[a]*b^2*x^4 + 2*a^(3/2)*c*y[x]^2 - 2*a^(3/2)*b*x^2*y[x]^2 + a^(5/2)*y[x]^4)/(a*x^2*y[x]),y[x],x]
\[\left \{\left \{y(x)\to -\frac {\sqrt {2 a^{5/2} b x^2-2 a^{5/2} c+4 a^3 b^2 x^3-4 a^3 b c x+a^2 x+4 \sqrt {a} b^2 c_1 x^2-4 \sqrt {a} b c c_1+2 b c_1 x}}{\sqrt {2} \sqrt {2 a^{3/2} b c_1+a^{7/2}+2 a^4 b x}}\right \},\left \{y(x)\to \frac {\sqrt {2 a^{5/2} b x^2-2 a^{5/2} c+4 a^3 b^2 x^3-4 a^3 b c x+a^2 x+4 \sqrt {a} b^2 c_1 x^2-4 \sqrt {a} b c c_1+2 b c_1 x}}{\sqrt {2} \sqrt {2 a^{3/2} b c_1+a^{7/2}+2 a^4 b x}}\right \}\right \}\] ✓ Maple : cpu = 0.297 (sec), leaf count = 97
dsolve(diff(y(x),x) = (b*x^3+c^2*a^(1/2)-2*c*b*x^2*a^(1/2)+2*c*y(x)^2*a^(3/2)+b^2*x^4*a^(1/2)-2*y(x)^2*a^(3/2)*b*x^2+a^(5/2)*y(x)^4)/a/x^2/y(x),y(x))
\[y \left (x \right ) = -\frac {2 \sqrt {a^{\frac {3}{2}} \left (\left (x c_{1}+1\right ) \left (b \,x^{2}-c \right ) \sqrt {a}+\frac {x}{2}\right ) \left (x c_{1}+1\right )}}{a^{\frac {3}{2}} \left (2 x c_{1}+2\right )}\]