\[ \boxed { ay \left ( x \right ) \sqrt { \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+1}-2\,xy \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( y \left ( x \right ) \right ) ^{2}-{x}^{2}=0} \]
Mathematica: cpu = 21.646749 (sec), leaf count = 110 \[ \left \{\left \{y(x)\to -\frac {\sqrt {a^2 c_1^2 \left (-x^2\right )-4 a^2 c_1 x-4 a^2+4 x^2}}{\sqrt {a^2 c_1^2-4}}\right \},\left \{y(x)\to \frac {\sqrt {a^2 c_1^2 \left (-x^2\right )-4 a^2 c_1 x-4 a^2+4 x^2}}{\sqrt {a^2 c_1^2-4}}\right \}\right \} \]
Maple: cpu = 0.671 (sec), leaf count = 1512 \[ \left \{ \int _{{\it \_b}}^{x}\!{1 \left ( 2\,{{\it \_a}}^{3}-2\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}+\sqrt {{a}^{2} \left ( { {\it \_a}}^{4}+2\,{{\it \_a}}^{2} \left ( y \left ( x \right ) \right ) ^ {2}-{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( y \left ( x \right ) \right ) ^{4} \right ) } \right ) \left ( -2\,{a}^{2}{\it \_a} \, \left ( y \left ( x \right ) \right ) ^{2}+2\,{{\it \_a}}^{5}+4\,{{ \it \_a}}^{3} \left ( y \left ( x \right ) \right ) ^{2}+2\, \left ( y \left ( x \right ) \right ) ^{4}{\it \_a}+{{\it \_a}}^{2}\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2} \left ( y \left ( x \right ) \right ) ^{2}-{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( y \left ( x \right ) \right ) ^{4} \right ) }- \left ( y \left ( x \right ) \right ) ^{2}\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2} \left ( y \left ( x \right ) \right ) ^{2}-{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( y \left ( x \right ) \right ) ^{4} \right ) } \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \left ( x \right ) }\!-{ \left ( {a}^{2}-4\,{x}^{2} \right ) {\it \_f} \left ( -2\,{a}^{2}x{ {\it \_f}}^{2}+2\,{x}^{5}+4\,{x}^{3}{{\it \_f}}^{2}+2\,x{{\it \_f}}^{4 }+{x}^{2}\sqrt {-{a}^{2} \left ( -{{\it \_f}}^{4}+{{\it \_f}}^{2}{a}^{2 }-2\,{{\it \_f}}^{2}{x}^{2}-{x}^{4} \right ) }-{{\it \_f}}^{2}\sqrt {-{ a}^{2} \left ( -{{\it \_f}}^{4}+{{\it \_f}}^{2}{a}^{2}-2\,{{\it \_f}}^{ 2}{x}^{2}-{x}^{4} \right ) } \right ) ^{-1}}-\int _{{\it \_b}}^{x}\!{1 \left ( -4\,{\it \_f}\,{\it \_a}-{\frac {{a}^{2} \left ( -4\,{{\it \_a} }^{2}{\it \_f}-4\,{{\it \_f}}^{3}+2\,{\it \_f}\,{a}^{2} \right ) }{2}{ \frac {1}{\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2}{{ \it \_f}}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) }}}} \right ) \left ( -2\,{a}^{2}{\it \_a}\,{{\it \_f}}^{2}+2\,{{\it \_a}}^ {5}+4\,{{\it \_a}}^{3}{{\it \_f}}^{2}+2\,{{\it \_f}}^{4}{\it \_a}+{{ \it \_a}}^{2}\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2}{ {\it \_f}}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) }-{{\it \_f}}^{2}\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2}{{ \it \_f}}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) } \right ) ^{-1}}-{1 \left ( 2\,{{\it \_a}}^{3}-2\,{\it \_a}\,{{\it \_f}} ^{2}+\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2}{{\it \_f }}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) } \right ) \left ( -4\,{a}^{2}{\it \_a}\,{\it \_f}+8\,{{\it \_a}}^{3}{\it \_f}+8 \,{\it \_a}\,{{\it \_f}}^{3}-{\frac {{{\it \_a}}^{2}{a}^{2} \left ( -4 \,{{\it \_a}}^{2}{\it \_f}-4\,{{\it \_f}}^{3}+2\,{\it \_f}\,{a}^{2} \right ) }{2}{\frac {1}{\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{ \it \_a}}^{2}{{\it \_f}}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) }}}}-2\,{\it \_f}\,\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{ {\it \_a}}^{2}{{\it \_f}}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) }+{\frac {{{\it \_f}}^{2}{a}^{2} \left ( -4\,{{\it \_a}}^{2}{ \it \_f}-4\,{{\it \_f}}^{3}+2\,{\it \_f}\,{a}^{2} \right ) }{2}{\frac { 1}{\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2}{{\it \_f}} ^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) }}}} \right ) \left ( -2\,{a}^{2}{\it \_a}\,{{\it \_f}}^{2}+2\,{{\it \_a}}^{5}+4\,{{ \it \_a}}^{3}{{\it \_f}}^{2}+2\,{{\it \_f}}^{4}{\it \_a}+{{\it \_a}}^{ 2}\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2}{{\it \_f}}^ {2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) }-{{\it \_f}}^{2} \sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2}{{\it \_f}}^{2 }+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) } \right ) ^{-2}} \,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0,\int _{{\it \_b}}^{x}\!{1 \left ( 2\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,{{ \it \_a}}^{3}+\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2} \left ( y \left ( x \right ) \right ) ^{2}-{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( y \left ( x \right ) \right ) ^{4} \right ) } \right ) \left ( 2\,{a}^{2}{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,{{\it \_a}}^{5}-4\,{{\it \_a}}^{3} \left ( y \left ( x \right ) \right ) ^{2}-2\, \left ( y \left ( x \right ) \right ) ^{4}{\it \_a}+{{\it \_a}}^{2}\sqrt {{a}^{2} \left ( {{\it \_a} }^{4}+2\,{{\it \_a}}^{2} \left ( y \left ( x \right ) \right ) ^{2}-{a}^{ 2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( y \left ( x \right ) \right ) ^{4} \right ) }- \left ( y \left ( x \right ) \right ) ^{2}\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2} \left ( y \left ( x \right ) \right ) ^{2}-{a}^{2} \left ( y \left ( x \right ) \right ) ^{2} + \left ( y \left ( x \right ) \right ) ^{4} \right ) } \right ) ^{-1}} \,{\rm d}{\it \_a}+\int ^{y \left ( x \right ) }\!{ \left ( {a}^{2}-4\,{x }^{2} \right ) {\it \_f} \left ( 2\,{a}^{2}x{{\it \_f}}^{2}-2\,{x}^{5}-4 \,{x}^{3}{{\it \_f}}^{2}-2\,x{{\it \_f}}^{4}+{x}^{2}\sqrt {-{a}^{2} \left ( -{{\it \_f}}^{4}+{{\it \_f}}^{2}{a}^{2}-2\,{{\it \_f}}^{2}{x}^ {2}-{x}^{4} \right ) }-{{\it \_f}}^{2}\sqrt {-{a}^{2} \left ( -{{\it \_f }}^{4}+{{\it \_f}}^{2}{a}^{2}-2\,{{\it \_f}}^{2}{x}^{2}-{x}^{4} \right ) } \right ) ^{-1}}-\int _{{\it \_b}}^{x}\!-{1 \left ( 2\,{\it \_a }\,{{\it \_f}}^{2}-2\,{{\it \_a}}^{3}+\sqrt {{a}^{2} \left ( {{\it \_a} }^{4}+2\,{{\it \_a}}^{2}{{\it \_f}}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2 }{a}^{2} \right ) } \right ) \left ( 4\,{a}^{2}{\it \_a}\,{\it \_f}-8\,{ {\it \_a}}^{3}{\it \_f}-8\,{\it \_a}\,{{\it \_f}}^{3}-{\frac {{{\it \_a}}^{2}{a}^{2} \left ( -4\,{{\it \_a}}^{2}{\it \_f}-4\,{{\it \_f}}^{3 }+2\,{\it \_f}\,{a}^{2} \right ) }{2}{\frac {1}{\sqrt {{a}^{2} \left ( { {\it \_a}}^{4}+2\,{{\it \_a}}^{2}{{\it \_f}}^{2}+{{\it \_f}}^{4}-{{ \it \_f}}^{2}{a}^{2} \right ) }}}}-2\,{\it \_f}\,\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{\it \_a}}^{2}{{\it \_f}}^{2}+{{\it \_f}}^{4}-{{ \it \_f}}^{2}{a}^{2} \right ) }+{\frac {{{\it \_f}}^{2}{a}^{2} \left ( - 4\,{{\it \_a}}^{2}{\it \_f}-4\,{{\it \_f}}^{3}+2\,{\it \_f}\,{a}^{2} \right ) }{2}{\frac {1}{\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{ \it \_a}}^{2}{{\it \_f}}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) }}}} \right ) \left ( 2\,{a}^{2}{\it \_a}\,{{\it \_f}}^{2}-2\, {{\it \_a}}^{5}-4\,{{\it \_a}}^{3}{{\it \_f}}^{2}-2\,{{\it \_f}}^{4}{ \it \_a}+{{\it \_a}}^{2}\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{ \it \_a}}^{2}{{\it \_f}}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) }-{{\it \_f}}^{2}\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{ \it \_a}}^{2}{{\it \_f}}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) } \right ) ^{-2}}+{1 \left ( 4\,{\it \_f}\,{\it \_a}-{\frac {{a }^{2} \left ( -4\,{{\it \_a}}^{2}{\it \_f}-4\,{{\it \_f}}^{3}+2\,{\it \_f}\,{a}^{2} \right ) }{2}{\frac {1}{\sqrt {{a}^{2} \left ( {{\it \_a}} ^{4}+2\,{{\it \_a}}^{2}{{\it \_f}}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2} {a}^{2} \right ) }}}} \right ) \left ( 2\,{a}^{2}{\it \_a}\,{{\it \_f}}^ {2}-2\,{{\it \_a}}^{5}-4\,{{\it \_a}}^{3}{{\it \_f}}^{2}-2\,{{\it \_f} }^{4}{\it \_a}+{{\it \_a}}^{2}\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2 \,{{\it \_a}}^{2}{{\it \_f}}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2 } \right ) }-{{\it \_f}}^{2}\sqrt {{a}^{2} \left ( {{\it \_a}}^{4}+2\,{{ \it \_a}}^{2}{{\it \_f}}^{2}+{{\it \_f}}^{4}-{{\it \_f}}^{2}{a}^{2} \right ) } \right ) ^{-1}}\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0 \right \} \]