\[ \left (a \left (x^2+y(x)^2\right )^{3/2}-x^2\right ) y'(x)^2+a \left (x^2+y(x)^2\right )^{3/2}+2 x y(x) y'(x)-y(x)^2=0 \] ✓ Mathematica : cpu = 24.0023 (sec), leaf count = 305
DSolve[-y[x]^2 + a*(x^2 + y[x]^2)^(3/2) + 2*x*y[x]*Derivative[1][y][x] + (-x^2 + a*(x^2 + y[x]^2)^(3/2))*Derivative[1][y][x]^2 == 0,y[x],x]
\[\left \{\text {Solve}\left [\arctan \left (\frac {x}{y(x)}\right )-\frac {2 \sqrt {a \left (x^2+y(x)^2\right )^2 \left (-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}\right )} \arctan \left (\frac {\sqrt {a} \sqrt {x^2+y(x)^2}}{\sqrt {-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}}}\right )}{\sqrt {a} \left (x^2+y(x)^2\right ) \sqrt {-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}}}=c_1,y(x)\right ],\text {Solve}\left [\frac {2 \sqrt {a \left (x^2+y(x)^2\right )^2 \left (-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}\right )} \arctan \left (\frac {\sqrt {a} \sqrt {x^2+y(x)^2}}{\sqrt {-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}}}\right )}{\sqrt {a} \left (x^2+y(x)^2\right ) \sqrt {-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}}}+\arctan \left (\frac {x}{y(x)}\right )=c_1,y(x)\right ]\right \}\] ✓ Maple : cpu = 6.133 (sec), leaf count = 139
dsolve((a*(y(x)^2+x^2)^(3/2)-x^2)*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)+a*(y(x)^2+x^2)^(3/2)-y(x)^2=0,y(x))
\[y \left (x \right ) = \frac {x}{\tan \left (\operatorname {RootOf}\left (-\textit {\_Z} -\left (\int _{}^{\frac {x^{2} \left (\tan \left (\textit {\_Z} \right )^{2}+1\right )}{\tan \left (\textit {\_Z} \right )^{2}}}\frac {\sqrt {-a \,\textit {\_a}^{\frac {5}{2}} \left (\sqrt {\textit {\_a}}\, a -1\right )}\, \left (\sqrt {\textit {\_a}}\, a +1\right )}{2 \textit {\_a}^{2} \left (\textit {\_a} \,a^{2}-1\right )}d \textit {\_a} \right )+c_{1}\right )\right )}\]