\[ -a y(x) y'(x)-a x+y(x) \sqrt {y'(x)^2+1}=0 \] ✓ Mathematica : cpu = 0.33402 (sec), leaf count = 212
DSolve[-(a*x) - a*y[x]*Derivative[1][y][x] + y[x]*Sqrt[1 + Derivative[1][y][x]^2] == 0,y[x],x]
\[\left \{\left \{y(x)\to -\frac {\sqrt {a^6 \left (-x^2\right )+3 a^4 x^2-3 a^2 x^2+2 a^2 x e^{a^2 c_1-c_1}-2 x e^{a^2 c_1-c_1}+e^{2 a^2 c_1-2 c_1}+x^2}}{\sqrt {a^6-3 a^4+3 a^2-1}}\right \},\left \{y(x)\to \frac {\sqrt {a^6 \left (-x^2\right )+3 a^4 x^2-3 a^2 x^2+2 a^2 x e^{a^2 c_1-c_1}-2 x e^{a^2 c_1-c_1}+e^{2 a^2 c_1-2 c_1}+x^2}}{\sqrt {a^6-3 a^4+3 a^2-1}}\right \}\right \}\] ✓ Maple : cpu = 0.258 (sec), leaf count = 215
dsolve(y(x)*(diff(y(x),x)^2+1)^(1/2)-a*y(x)*diff(y(x),x)-a*x=0,y(x))
\[-{\mathrm e}^{\int _{}^{\frac {-a^{2} x +\sqrt {\left (a^{2}-1\right ) y \left (x \right )^{2}+a^{2} x^{2}}}{\left (a^{2}-1\right ) y \left (x \right )}}\frac {\left (a \sqrt {\textit {\_a}^{2}+1}-\textit {\_a} \right ) a}{\sqrt {\textit {\_a}^{2}+1}\, \left (\textit {\_a} a -\sqrt {\textit {\_a}^{2}+1}\right ) \left (\textit {\_a}^{2} a -\sqrt {\textit {\_a}^{2}+1}\, \textit {\_a} +a \right )}d \textit {\_a}} c_{1}+x = 0\]