\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) -\sqrt {{\frac {a \left ( y \left ( x \right ) \right ) ^{2}+by \left ( x \right ) +c}{a{x}^{2}+bx+c}}}=0} \]
Mathematica: cpu = 0.187524 (sec), leaf count = 269 \[ \left \{\left \{y(x)\to \frac {e^{-\sqrt {a} c_1} \left (8 a^{3/2} c e^{2 \sqrt {a} c_1} \sqrt {a x^2+b x+c}-8 a^{3/2} c \sqrt {a x^2+b x+c}+8 a^2 c x e^{2 \sqrt {a} c_1}+8 a^2 c x+2 b^3 e^{\sqrt {a} c_1}-b^3 e^{2 \sqrt {a} c_1}-2 \sqrt {a} b^2 e^{2 \sqrt {a} c_1} \sqrt {a x^2+b x+c}+2 \sqrt {a} b^2 \sqrt {a x^2+b x+c}-2 a b^2 x e^{2 \sqrt {a} c_1}-2 a b^2 x-8 a b c e^{\sqrt {a} c_1}+4 a b c e^{2 \sqrt {a} c_1}+4 a b c-b^3\right )}{a \left (16 a c-4 b^2\right )}\right \}\right \} \]
Maple: cpu = 0.078 (sec), leaf count = 124 \[ \left \{ -{1\sqrt {{\frac {a \left ( y \left ( x \right ) \right ) ^{2}+b y \left ( x \right ) +c}{a{x}^{2}+bx+c}}}\sqrt {a{x}^{2}+bx+c}\ln \left ( {\frac {1}{2} \left ( 2\,\sqrt {a{x}^{2}+bx+c}\sqrt {a}+2\,ax+b \right ) {\frac {1}{\sqrt {a}}}} \right ) {\frac {1}{\sqrt {a \left ( y \left ( x \right ) \right ) ^{2}+by \left ( x \right ) +c}}}{\frac {1}{ \sqrt {a}}}}+{1\ln \left ( {1 \left ( ay \left ( x \right ) +{\frac {b}{2 }} \right ) {\frac {1}{\sqrt {a}}}}+\sqrt {a \left ( y \left ( x \right ) \right ) ^{2}+by \left ( x \right ) +c} \right ) {\frac {1}{\sqrt {a}}}}+ {\it \_C1}=0 \right \} \]
Sage: cpu = 0 (sec), leaf count = 0 \[ \text {Maxima was unable to solve this ODE} \]