\[ y''(x) \left (a \sqrt {y'(x)^2+1}-x y'(x)\right )-y'(x)^2-1=0 \] ✓ Mathematica : cpu = 0.659107 (sec), leaf count = 347
\[\left \{\left \{y(x)\to \frac {-2 \sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )}+c_1 x \log \left (-c_1 \left (\sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )}+c_1 x\right )+a^2 (-x)+a x^2\right )+c_1 x \log \left (c_1 \left (\sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )}+c_1 x\right )+a^2 x+a x^2\right )+c_1 x \log (x-a)-c_1 x \log (x (x-a))-c_1 x \log (a+x)-c_1 x \log (x (a+x))}{2 x}+c_2\right \},\left \{y(x)\to c_2-\frac {-2 \sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )}+c_1 x \log \left (-c_1 \left (\sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )}+c_1 x\right )+a^2 (-x)+a x^2\right )+c_1 x \log \left (c_1 \left (\sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )}+c_1 x\right )+a^2 x+a x^2\right )+c_1 (-x) \log (x-a)-c_1 x \log (x (x-a))+c_1 x \log (a+x)-c_1 x \log (x (a+x))}{2 x}\right \}\right \}\] ✓ Maple : cpu = 1.071 (sec), leaf count = 96
\[ \left \{ y \left ( x \right ) =\int \!{\frac {1}{{a}^{3}-a{x}^{2}} \left ( -{\it \_C1}\,{a}^{2}-x\sqrt {{a}^{2} \left ( {{\it \_C1}}^{2}+{a}^{2}-{x}^{2} \right ) } \right ) }\,{\rm d}x+{\it \_C2},y \left ( x \right ) =\int \!{\frac {1}{{a}^{3}-a{x}^{2}} \left ( -{\it \_C1}\,{a}^{2}+x\sqrt {{a}^{2} \left ( {{\it \_C1}}^{2}+{a}^{2}-{x}^{2} \right ) } \right ) }\,{\rm d}x+{\it \_C2} \right \} \]