\[ y''(x) \left (a \sqrt {y'(x)^2+1}-x y'(x)\right )-y'(x)^2-1=0 \] ✓ Mathematica : cpu = 0.956735 (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 = 1.288 (sec), leaf count = 96
\[\left \{y \left (x \right ) = c_{2}+\int \frac {-c_{1} a^{2}-\sqrt {\left (a^{2}-x^{2}+c_{1}^{2}\right ) a^{2}}\, x}{a^{3}-a \,x^{2}}d x, y \left (x \right ) = c_{2}+\int \frac {-c_{1} a^{2}+\sqrt {\left (a^{2}-x^{2}+c_{1}^{2}\right ) a^{2}}\, x}{a^{3}-a \,x^{2}}d x\right \}\]