\[ y'(x)-\sqrt {\frac {a y(x)^2+b y(x)+c}{a x^2+b x+c}}=0 \] ✓ Mathematica : cpu = 0.339405 (sec), leaf count = 269
DSolve[-Sqrt[(c + b*y[x] + a*y[x]^2)/(c + b*x + a*x^2)] + Derivative[1][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \frac {e^{-\sqrt {a} c_1} \left (-8 a^{3/2} c \sqrt {a x^2+b x+c}+8 a^{3/2} c e^{2 \sqrt {a} c_1} \sqrt {a x^2+b x+c}+8 a^2 c x+8 a^2 c x e^{2 \sqrt {a} c_1}+2 b^3 e^{\sqrt {a} c_1}-b^3 e^{2 \sqrt {a} c_1}+2 \sqrt {a} b^2 \sqrt {a x^2+b x+c}-2 \sqrt {a} b^2 e^{2 \sqrt {a} c_1} \sqrt {a x^2+b x+c}-2 a b^2 x-2 a b^2 x e^{2 \sqrt {a} c_1}+4 a b c-8 a b c e^{\sqrt {a} c_1}+4 a b c e^{2 \sqrt {a} c_1}-b^3\right )}{a \left (16 a c-4 b^2\right )}\right \}\right \}\] ✓ Maple : cpu = 0.098 (sec), leaf count = 124
dsolve(diff(y(x),x)-((a*y(x)^2+b*y(x)+c)/(a*x^2+b*x+c))^(1/2) = 0,y(x))
\[-\frac {\sqrt {\frac {a y \left (x \right )^{2}+b y \left (x \right )+c}{a \,x^{2}+b x +c}}\, \sqrt {a \,x^{2}+b x +c}\, \ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right )}{\sqrt {a y \left (x \right )^{2}+b y \left (x \right )+c}\, \sqrt {a}}+\frac {\ln \left (\sqrt {a y \left (x \right )^{2}+b y \left (x \right )+c}+\frac {2 a y \left (x \right )+b}{2 \sqrt {a}}\right )}{\sqrt {a}}+c_{1} = 0\]