\[ \boxed { \left ( a \left ( \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2} \right ) ^{3/2}-{x}^{2} \right ) \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+2\,xy \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +a \left ( \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2} \right ) ^{3/2}- \left ( y \left ( x \right ) \right ) ^{2}=0} \]
Mathematica: cpu = 4.686095 (sec), leaf count = 725 \[ \left \{\text {Solve}\left [\tan ^{-1}\left (\frac {x}{y(x)}\right )-\frac {i \sqrt {a} \left (x^2+y(x)^2\right ) \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )} \left (\sqrt {2} \left (\log \left (\frac {a^{3/2} \left (3 i \sqrt {2} a \sqrt {x^2+y(x)^2}+4 \sqrt {a} \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}-i \sqrt {2}\right )}{4 a \sqrt {x^2+y(x)^2}+4}\right )-\log \left (\frac {-3 i \sqrt {2} a^{3/2} \sqrt {x^2+y(x)^2}-4 a \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}+i \sqrt {2} \sqrt {a}}{4 a \sqrt {x^2+y(x)^2}+4}\right )\right )+2 \log \left (\frac {-2 i a \sqrt {x^2+y(x)^2}+2 \sqrt {a} \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}+i}{\sqrt {a}}\right )\right )}{2 \sqrt {a \left (x^2+y(x)^2\right )^2 \left (\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )\right )}}=c_1,y(x)\right ],\text {Solve}\left [\tan ^{-1}\left (\frac {x}{y(x)}\right )+\frac {i \sqrt {a} \left (x^2+y(x)^2\right ) \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )} \left (\sqrt {2} \left (\log \left (\frac {a^{3/2} \left (3 i \sqrt {2} a \sqrt {x^2+y(x)^2}+4 \sqrt {a} \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}-i \sqrt {2}\right )}{4 a \sqrt {x^2+y(x)^2}+4}\right )-\log \left (\frac {-3 i \sqrt {2} a^{3/2} \sqrt {x^2+y(x)^2}-4 a \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}+i \sqrt {2} \sqrt {a}}{4 a \sqrt {x^2+y(x)^2}+4}\right )\right )+2 \log \left (\frac {-2 i a \sqrt {x^2+y(x)^2}+2 \sqrt {a} \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}+i}{\sqrt {a}}\right )\right )}{2 \sqrt {a \left (x^2+y(x)^2\right )^2 \left (\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )\right )}}=c_1,y(x)\right ]\right \} \]
Maple: cpu = 40.935 (sec), leaf count = 135 \[ \left \{ y \left ( x \right ) ={x \left ( \tan \left ( {\it RootOf} \left ( -{\it \_Z}+\int ^{{\frac {{x}^{2} \left ( \left ( \tan \left ( { \it \_Z} \right ) \right ) ^{2}+1 \right ) }{ \left ( \tan \left ( {\it \_Z} \right ) \right ) ^{2}}}}\!-{\frac {1}{2\,{{\it \_a}}^{2} \left ( { \it \_a}\,{a}^{2}-1 \right ) } \left ( \sqrt {{\it \_a}}a+1 \right ) \sqrt {-{{\it \_a}}^{{\frac {5}{2}}}a \left ( \sqrt {{\it \_a}}a-1 \right ) }}{d{\it \_a}}+{\it \_C1} \right ) \right ) \right ) ^{-1}},y \left ( x \right ) ={x \left ( \tan \left ( {\it RootOf} \left ( -{\it \_Z }+\int ^{{\frac {{x}^{2} \left ( \left ( \tan \left ( {\it \_Z} \right ) \right ) ^{2}+1 \right ) }{ \left ( \tan \left ( {\it \_Z} \right ) \right ) ^{2}}}}\!{\frac {1}{2\,{{\it \_a}}^{2} \left ( {\it \_a}\,{a}^ {2}-1 \right ) } \left ( \sqrt {{\it \_a}}a+1 \right ) \sqrt {-{{\it \_a} }^{{\frac {5}{2}}}a \left ( \sqrt {{\it \_a}}a-1 \right ) }}{d{\it \_a}} +{\it \_C1} \right ) \right ) \right ) ^{-1}} \right \} \]