\[ \boxed { \left ( \sqrt { \left ( y \left ( x \right ) \right ) ^{2}+1}+ax \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +\sqrt {{x}^{2}+1}+ay \left ( x \right ) =0} \]
Mathematica: cpu = 0.063008 (sec), leaf count = 53 \[ \text {Solve}\left [a x y(x)+\frac {1}{2} \sqrt {x^2+1} x+\frac {1}{2} \left (y(x) \sqrt {y(x)^2+1}+\sinh ^{-1}(y(x))\right )+\frac {1}{2} \sinh ^{-1}(x)=c_1,y(x)\right ] \]
Maple: cpu = 0.031 (sec), leaf count = 41 \[ \left \{ {\frac {x}{2}\sqrt {{x}^{2}+1}}+{\frac {{\it Arcsinh} \left ( x \right ) }{2}}+axy \left ( x \right ) +{\frac {y \left ( x \right ) }{2} \sqrt { \left ( y \left ( x \right ) \right ) ^{2}+1}}+{\frac {{\it Arcsinh} \left ( y \left ( x \right ) \right ) }{2}}+{\it \_C1}=0 \right \} \]