\[ a x \sqrt {y'(x)^2+1}+x y'(x)-y(x)=0 \] ✓ Mathematica : cpu = 1.40118 (sec), leaf count = 395
\[\left \{\text {Solve}\left [\frac {a \left (-\log \left (\frac {\left (a^2-1\right ) \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2-\frac {i y(x)}{x}-1\right )}{a^3 \left (\frac {y(x)}{x}-i\right )}\right )+\log \left (-\frac {\left (a^2-1\right ) \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2+\frac {i y(x)}{x}-1\right )}{a^3 \left (\frac {y(x)}{x}+i\right )}\right )+\log \left (\frac {y(x)^2}{x^2}+1\right )\right )-2 i \tan ^{-1}\left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )}{2 \left (a^2-1\right )}=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ],\text {Solve}\left [\frac {2 i \tan ^{-1}\left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \left (\log \left (-\frac {\left (a^2-1\right ) \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2-\frac {i y(x)}{x}-1\right )}{a^3 \left (\frac {y(x)}{x}-i\right )}\right )-\log \left (\frac {\left (a^2-1\right ) \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2+\frac {i y(x)}{x}-1\right )}{a^3 \left (\frac {y(x)}{x}+i\right )}\right )+\log \left (\frac {y(x)^2}{x^2}+1\right )\right )}{2 \left (a^2-1\right )}=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ]\right \}\] ✓ Maple : cpu = 0.176 (sec), leaf count = 223
\[ \left \{ x-{{\it \_C1}{{\rm e}^{{\frac {1}{a}{\it Arcsinh} \left ( {\frac {1}{ \left ( {a}^{2}-1 \right ) x} \left ( \sqrt {-{a}^{2}{x}^{2}+{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}a+y \left ( x \right ) \right ) } \right ) }}}{\frac {1}{\sqrt {{\frac {1}{ \left ( {a}^{2}-1 \right ) ^{2}{x}^{2}} \left ( -{a}^{2}{x}^{2}+{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}+2\,\sqrt {-{a}^{2}{x}^{2}+{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}ay \left ( x \right ) +{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2} \right ) }}}}}=0,x-{{\it \_C1}{{\rm e}^{-{\frac {1}{a}{\it Arcsinh} \left ( {\frac {1}{ \left ( {a}^{2}-1 \right ) x} \left ( \sqrt {-{a}^{2}{x}^{2}+{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}a-y \left ( x \right ) \right ) } \right ) }}}{\frac {1}{\sqrt {-{\frac {1}{ \left ( {a}^{2}-1 \right ) ^{2}{x}^{2}} \left ( {a}^{2}{x}^{2}-{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}+2\,\sqrt {-{a}^{2}{x}^{2}+{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}ay \left ( x \right ) -{x}^{2}- \left ( y \left ( x \right ) \right ) ^{2} \right ) }}}}}=0 \right \} \]