\[ \sqrt {x^2+1} (y(x)-x) y'(x)-a \sqrt {\left (y(x)^2+1\right )^3}=0 \] ✓ Mathematica : cpu = 1.89272 (sec), leaf count = 69
\[\text {Solve}\left [\left \{\frac {2 a \tan ^{-1}\left (\frac {1-a \tan \left (\frac {K[1]}{2}\right )}{\sqrt {a^2-1}}\right )}{\sqrt {a^2-1}}+K[1]+\tan ^{-1}(x)=c_1,y(x)=\frac {\tan (K[1])+x}{1-x \tan (K[1])}\right \},\{K[1],y(x)\}\right ]\] ✓ Maple : cpu = 2.161 (sec), leaf count = 65
\[y \relax (x ) = \tan \left (\RootOf \left (-\arctan \relax (x )+\int _{}^{-\arctan \relax (x )+\textit {\_Z}}-\frac {-\sqrt {2}\, \sqrt {\frac {a^{2}}{\cos \left (2 \textit {\_a} \right )+1}}\, \sin \left (2 \textit {\_a} \right )+\cos \left (2 \textit {\_a} \right )-1}{2 a^{2}+\cos \left (2 \textit {\_a} \right )-1}d \textit {\_a} +c_{1}\right )\right )\]