\[ \boxed { ay \left ( x \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +b \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}-{\frac {y \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }{\sqrt {{c}^{2}+{x}^{2}}}}=0} \]
Mathematica: cpu = 0.662084 (sec), leaf count = 126 \[ \left \{\left \{y(x)\to c_2 \left (-a^2 \left (x \left (\sqrt {c^2+x^2}+x\right )^{\frac {1}{a}}+c_1\right )+a \left (\sqrt {c^2+x^2}+x\right )^{\frac {1}{a}} \left (\sqrt {c^2+x^2}-b x\right )+b \sqrt {c^2+x^2} \left (\sqrt {c^2+x^2}+x\right )^{\frac {1}{a}}+c_1\right ){}^{\frac {a \left (a^2-1\right )}{(a-1) (a+1) (a+b)}}\right \}\right \} \]
Maple: cpu = 3.074 (sec), leaf count = 79 \[ \left \{ y \left ( x \right ) = \left ( \left ( {\frac {a}{a+b} \left ( -{ {\it \_C1}\,\sqrt [a]{2}{x}^{{a}^{-1}+1} {\mbox {$_2$F$_1$}(-{\frac {1}{2\,a}},-{\frac {1}{2\,a}}-{\frac {1}{2}};\,1-{a}^{-1};\,-{\frac {{c}^{2}}{{x}^{2}}})} \left ( -{a}^{-1}-1 \right ) ^{-1}}+{\it \_C2} \right ) ^{-1}} \right ) ^ {{\frac {a}{a+b}}} \right ) ^{-1} \right \} \]