\[ a y(x) y''(x)+b y'(x)^2-\frac {y(x) y'(x)}{\sqrt {c^2+x^2}}=0 \] ✓ Mathematica : cpu = 0.600543 (sec), leaf count = 211
\[\left \{\left \{y(x)\to c_2 \exp \left (\int _1^x-\frac {\left (\frac {K[2]}{\sqrt {c^2+K[2]^2}}+1\right )^{\left .\frac {1}{2}\right /a}}{\left (1-\frac {K[2]}{\sqrt {c^2+K[2]^2}}\right )^{\left .\frac {1}{2}\right /a} \int _1^{K[2]}\frac {\exp \left (\frac {\frac {1}{2} \log \left (\frac {K[1]}{\sqrt {c^2+K[1]^2}}+1\right )-\frac {1}{2} \log \left (1-\frac {K[1]}{\sqrt {c^2+K[1]^2}}\right )}{a}\right ) \left (-\sqrt {c^2+K[1]^2} a-b \sqrt {c^2+K[1]^2}\right )}{a \sqrt {c^2+K[1]^2}}dK[1]-c_1 \left (1-\frac {K[2]}{\sqrt {c^2+K[2]^2}}\right )^{\left .\frac {1}{2}\right /a}}dK[2]\right )\right \}\right \}\] ✓ Maple : cpu = 0.653 (sec), leaf count = 75
\[ \left \{ y \left ( x \right ) = \left ( \left ( {\frac {a}{a+b} \left ( {\frac {{\it \_C1}\,\sqrt [a]{2}a{x}^{1+{a}^{-1}}}{a+1}{\mbox {$_2$F$_1$}(-{\frac {1}{2\,a}},-{\frac {1}{2\,a}}-{\frac {1}{2}};\,1-{a}^{-1};\,-{\frac {{c}^{2}}{{x}^{2}}})}}+{\it \_C2} \right ) ^{-1}} \right ) ^{{\frac {a}{a+b}}} \right ) ^{-1} \right \} \]