\[ \sqrt {x^2+1} (y(x)-x) y'(x)-a \sqrt {\left (y(x)^2+1\right )^3}=0 \] ✓ Mathematica : cpu = 2.97555 (sec), leaf count = 69
DSolve[-(a*Sqrt[(1 + y[x]^2)^3]) + Sqrt[1 + x^2]*(-x + y[x])*Derivative[1][y][x] == 0,y[x],x]
\[\text {Solve}\left [\left \{\frac {2 a \arctan \left (\frac {1-a \tan \left (\frac {K[1]}{2}\right )}{\sqrt {a^2-1}}\right )}{\sqrt {a^2-1}}+K[1]+\arctan (x)=c_1,y(x)=\frac {\tan (K[1])+x}{1-x \tan (K[1])}\right \},\{K[1],y(x)\}\right ]\] ✓ Maple : cpu = 1.826 (sec), leaf count = 103
dsolve((y(x)-x)*(x^2+1)^(1/2)*diff(y(x),x)-a*((y(x)^2+1)^3)^(1/2) = 0,y(x))
\[\frac {\arctan \left (\frac {\cos \left (\arctan \left (x \right )-\arctan \left (y \left (x \right )\right )\right )}{\sqrt {a^{2}-1}}\right ) \cos \left (\arctan \left (x \right )-\arctan \left (y \left (x \right )\right )\right ) \sqrt {2}\, \sqrt {\frac {a^{2}}{1+\cos \left (2 \arctan \left (x \right )-2 \arctan \left (y \left (x \right )\right )\right )}}+\arctan \left (\frac {\sqrt {a^{2}-1}\, \tan \left (\arctan \left (x \right )-\arctan \left (y \left (x \right )\right )\right )}{a}\right ) a -\sqrt {a^{2}-1}\, \left (c_{1}-\arctan \left (y \left (x \right )\right )\right )}{\sqrt {a^{2}-1}} = 0\]