\[ \left (a \left (x^2+y(x)^2\right )^{3/2}-x^2\right ) y'(x)^2+a \left (x^2+y(x)^2\right )^{3/2}+2 x y(x) y'(x)-y(x)^2=0 \] ✓ Mathematica : cpu = 6.21129 (sec), leaf count = 713
\[\left \{\text {Solve}\left [\tan ^{-1}\left (\frac {x}{y(x)}\right )-\frac {i \sqrt {a \left (\left (x^2+y(x)^2\right )^{5/2}-a \left (x^2+y(x)^2\right )^3\right )} \left (\sqrt {2} \left (\log \left (\frac {a^{3/2} \left (3 i \sqrt {2} a \sqrt {x^2+y(x)^2}+4 \sqrt {a} \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}-i \sqrt {2}\right )}{4 a \sqrt {x^2+y(x)^2}+4}\right )-\log \left (\frac {-3 i \sqrt {2} a^{3/2} \sqrt {x^2+y(x)^2}-4 a \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}+i \sqrt {2} \sqrt {a}}{4 a \sqrt {x^2+y(x)^2}+4}\right )\right )+2 \log \left (\frac {-2 i a \sqrt {x^2+y(x)^2}+2 \sqrt {a} \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}+i}{\sqrt {a}}\right )\right )}{2 \sqrt {a} \left (x^2+y(x)^2\right ) \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}}=c_1,y(x)\right ],\text {Solve}\left [\tan ^{-1}\left (\frac {x}{y(x)}\right )+\frac {i \sqrt {a \left (\left (x^2+y(x)^2\right )^{5/2}-a \left (x^2+y(x)^2\right )^3\right )} \left (\sqrt {2} \left (\log \left (\frac {a^{3/2} \left (3 i \sqrt {2} a \sqrt {x^2+y(x)^2}+4 \sqrt {a} \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}-i \sqrt {2}\right )}{4 a \sqrt {x^2+y(x)^2}+4}\right )-\log \left (\frac {-3 i \sqrt {2} a^{3/2} \sqrt {x^2+y(x)^2}-4 a \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}+i \sqrt {2} \sqrt {a}}{4 a \sqrt {x^2+y(x)^2}+4}\right )\right )+2 \log \left (\frac {-2 i a \sqrt {x^2+y(x)^2}+2 \sqrt {a} \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}+i}{\sqrt {a}}\right )\right )}{2 \sqrt {a} \left (x^2+y(x)^2\right ) \sqrt {\sqrt {x^2+y(x)^2}-a \left (x^2+y(x)^2\right )}}=c_1,y(x)\right ]\right \}\] ✓ Maple : cpu = 7.271 (sec), leaf count = 135
\[y \relax (x ) = \frac {x}{\tan \left (\RootOf \left (-\textit {\_Z} +\int _{}^{\frac {x^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )+1\right )}{\tan \left (\textit {\_Z} \right )^{2}}}-\frac {\sqrt {-\textit {\_a}^{\frac {5}{2}} a \left (\sqrt {\textit {\_a}}\, a -1\right )}\, \left (\sqrt {\textit {\_a}}\, a +1\right )}{2 \textit {\_a}^{2} \left (\textit {\_a} \,a^{2}-1\right )}d \textit {\_a} +c_{1}\right )\right )}\]