\[ a x \sqrt {y'(x)^2+1}+x y'(x)-y(x)=0 \] ✓ Mathematica : cpu = 0.800949 (sec), leaf count = 369
\[\left \{\text {Solve}\left [\frac {a \left (2 \log \left (x-a^2 x\right )-\log \left (\frac {\left (a^2-1\right ) \left (y(x)+i x \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2-1\right )\right )}{a^3 (x+i y(x))}\right )+\log \left (\frac {i \left (a^2-1\right ) \left (x \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2-1\right )+i y(x)\right )}{a^3 (x-i y(x))}\right )+\log \left (\frac {y(x)^2}{x^2}+1\right )\right )-2 i \tan ^{-1}\left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )}{2 \left (a^2-1\right )}=c_1,y(x)\right ],\text {Solve}\left [\frac {2 i \tan ^{-1}\left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \left (2 \log \left (x-a^2 x\right )-\log \left (\frac {\left (a^2-1\right ) \left (y(x)-i x \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2-1\right )\right )}{a^3 (x-i y(x))}\right )+\log \left (-\frac {i \left (a^2-1\right ) \left (x \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2-1\right )-i y(x)\right )}{a^3 (x+i y(x))}\right )+\log \left (\frac {y(x)^2}{x^2}+1\right )\right )}{2 \left (a^2-1\right )}=c_1,y(x)\right ]\right \}\]
✓ Maple : cpu = 1.29 (sec), leaf count = 223
\[ \left \{ x-{{\it \_C1}{{\rm e}^{{\frac {1}{a}{\it Arcsinh} \left ( {\frac {1}{ \left ( {a}^{2}-1 \right ) x} \left ( \sqrt {-{a}^{2}{x}^{2}+{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}a+y \left ( x \right ) \right ) } \right ) }}}{\frac {1}{\sqrt {{\frac {1}{ \left ( {a}^{2}-1 \right ) ^{2}{x}^{2}} \left ( -{a}^{2}{x}^{2}+{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}+2\,\sqrt {-{a}^{2}{x}^{2}+{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}ay \left ( x \right ) +{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2} \right ) }}}}}=0,x-{{\it \_C1}{{\rm e}^{-{\frac {1}{a}{\it Arcsinh} \left ( {\frac {1}{ \left ( {a}^{2}-1 \right ) x} \left ( \sqrt {-{a}^{2}{x}^{2}+{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}a-y \left ( x \right ) \right ) } \right ) }}}{\frac {1}{\sqrt {-{\frac {1}{ \left ( {a}^{2}-1 \right ) ^{2}{x}^{2}} \left ( {a}^{2}{x}^{2}-{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}+2\,\sqrt {-{a}^{2}{x}^{2}+{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}ay \left ( x \right ) -{x}^{2}- \left ( y \left ( x \right ) \right ) ^{2} \right ) }}}}}=0 \right \} \]