\[ \left (a x+\sqrt {y(x)^2+1}\right ) y'(x)+a y(x)+\sqrt {x^2+1}=0 \] ✓ Mathematica : cpu = 0.281508 (sec), leaf count = 53
DSolve[Sqrt[1 + x^2] + a*y[x] + (a*x + Sqrt[1 + y[x]^2])*Derivative[1][y][x] == 0,y[x],x]
\[\text {Solve}\left [a x y(x)+\frac {1}{2} \sqrt {x^2+1} x+\frac {1}{2} \left (y(x) \sqrt {y(x)^2+1}+\sinh ^{-1}(y(x))\right )+\frac {1}{2} \sinh ^{-1}(x)=c_1,y(x)\right ]\] ✓ Maple : cpu = 0.039 (sec), leaf count = 41
dsolve(((y(x)^2+1)^(1/2)+a*x)*diff(y(x),x)+(x^2+1)^(1/2)+a*y(x) = 0,y(x))
\[\frac {x \sqrt {x^{2}+1}}{2}+\frac {\arcsinh \left (x \right )}{2}+a x y \left (x \right )+\frac {y \left (x \right ) \sqrt {y \left (x \right )^{2}+1}}{2}+\frac {\arcsinh \left (y \left (x \right )\right )}{2}+c_{1} = 0\]