\[ a y(x) y''(x)+b y'(x)^2-\frac {y(x) y'(x)}{\sqrt {c^2+x^2}}=0 \] ✓ Mathematica : cpu = 0.287329 (sec), leaf count = 111
\[\left \{\left \{y(x)\to c_2 \left (-a^2 \left (x \left (\sqrt {c^2+x^2}+x\right )^{\frac {1}{a}}+c_1\right )+a \left (\sqrt {c^2+x^2}+x\right )^{\frac {1}{a}} \left (\sqrt {c^2+x^2}-b x\right )+b \sqrt {c^2+x^2} \left (\sqrt {c^2+x^2}+x\right )^{\frac {1}{a}}+c_1\right ){}^{\frac {a}{a+b}}\right \}\right \}\]
✓ Maple : cpu = 0.121 (sec), leaf count = 75
\[ \left \{ y \left ( x \right ) = \left ( \left ( {\frac {a}{a+b} \left ( {\frac {{\it \_C1}\,\sqrt [a]{2}a{x}^{{a}^{-1}+1}}{a+1}{\mbox {$_2$F$_1$}(-{\frac {1}{2\,a}},-{\frac {1}{2\,a}}-{\frac {1}{2}};\,1-{a}^{-1};\,-{\frac {{c}^{2}}{{x}^{2}}})}}+{\it \_C2} \right ) ^{-1}} \right ) ^{{\frac {a}{a+b}}} \right ) ^{-1} \right \} \]