\[ \boxed { \sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) -a \left ( \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+1 \right ) ^{3/2}=0} \]
Mathematica: cpu = 0 (sec), leaf count = 0 \[ \text {Hanged} \]
Maple: cpu = 3.026 (sec), leaf count = 1864 \[ \left \{ y \left ( x \right ) ={\it RootOf} \left ( -\ln \left ( x \right ) +\int ^{{\it \_Z}}\!{\frac {1}{{{\it \_g}}^{2}+1} \left ( { \it RootOf} \left ( \arctan \left ( {\it \_g} \right ) +\int ^{{\it \_Z}} \!{\frac {1}{ \left ( {{\it \_f}}^{2}{a}^{2}+{a}^{2}-1 \right ) \left ( {{\it \_f}}^{2}+1 \right ) } \left ( 1+\sqrt {{{\it \_f}}^{2}{a}^{2}+{a} ^{2}} \right ) }{d{\it \_f}}+{\it \_C1} \right ) -{\it \_g} \right ) }{d{ \it \_g}}+{\it \_C2} \right ) x,y \left ( x \right ) ={\it RootOf} \left ( -\ln \left ( x \right ) +\int ^{{\it \_Z}}\!{\it RootOf} \left ( 2\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z }+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{ \it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it Erf} \left ( 1/2\,\sqrt {2} \sqrt {{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}} \right ) \sqrt {\pi }\sqrt {2}{\it \_C1}\,{{\it \_Z}}^{3}{{\it \_f}}^{ 3}+2\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{ \it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{ \it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it \_f}\,{\it Erf} \left ( 1/2 \,\sqrt {2}\sqrt {{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z} +{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}} \right ) \sqrt {\pi }\sqrt {2}{\it \_C1}\,{{\it \_Z}}^{3}+ \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{{\frac {5}{2}}}{{\rm e}^{{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{2\,{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it Erf} \left ( {\frac {\sqrt {2}}{2} \sqrt {{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}} \right ) \sqrt {\pi }\sqrt {2}{\it \_C1}\,{{\it \_Z}}^{4}{{\it \_f}}^{ 4}+2\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{ \it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{5/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{ \it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it Erf} \left ( 1/2\,\sqrt {2} \sqrt {{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}} \right ) \sqrt {\pi }\sqrt {2}{\it \_C1}\,{{\it \_Z}}^{4}{{\it \_f}}^{ 2}+3\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{ \it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{ \it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it Erf} \left ( 1/2\,\sqrt {2} \sqrt {{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}} \right ) \sqrt {\pi }\sqrt {2}{\it \_C1}\,{{\it \_Z}}^{4}{{\it \_f}}^{ 4}+2\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{ \it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{ \it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it Erf} \left ( 1/2\,\sqrt {2} \sqrt {{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}} \right ) \sqrt {\pi }\sqrt {2}{\it \_C1}\,{{\it \_Z}}^{4}{{\it \_f}}^{ 2}-2\,{\it \_C1}\,{{\it \_Z}}^{4}+2\,{\it \_C1}\,{{\it \_f}}^{4}- \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{{\frac {5}{2}}}{{\rm e}^{{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{2\,{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}\sqrt {\pi }\sqrt {2}{\it \_C1}\,{{\it \_Z}}^{4}{{\it \_f}}^{4}-2\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{ \it \_f}}^{2}+1 \right ) }} \right ) ^{5/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{ {\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}\sqrt {\pi }\sqrt {2}{\it \_C1}\,{{\it \_Z}}^{4}{{\it \_f}}^{2}-3\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{ {\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{ {\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{ \it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}\sqrt {\pi }\sqrt {2}{\it \_C1}\,{{\it \_Z}}^{4}{{\it \_f }}^{4}-2\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z} +{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{ \it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}\sqrt {\pi }\sqrt {2}{\it \_C1} \,{{\it \_Z}}^{4}{{\it \_f}}^{2}-2\, \left ( {\frac { \left ( {\it \_Z} \,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{ \frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}\sqrt {\pi }\sqrt {2}{\it \_C1}\,{{\it \_Z}}^{3}{{\it \_f}}^{3}-2\, \left ( { \frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/ 2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+ {\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it \_f}\,\sqrt {\pi }\sqrt {2}{\it \_C1}\,{{\it \_Z}}^{ 3}- \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{{\frac {5}{2}}}{{\rm e}^{{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{2\,{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}\sqrt {\pi }\sqrt {2}{\it \_C1}\,{{\it \_Z}}^{4}+ \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z} +{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{{\frac {3}{2}}}{{\rm e}^{{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{2\,{{\it \_Z}} ^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}\sqrt {\pi }\sqrt {2}{\it \_C1}\,{{\it \_Z}}^{4}+2\,{\it \_C1}\,{{\it \_Z}}^{4}{{\it \_f}}^{8}-4 \,{\it \_C1}\,{{\it \_Z}}^{4}{{\it \_f}}^{6}+8\,{\it \_C1}\,{{\it \_Z} }^{3}{{\it \_f}}^{7}-12\,{\it \_C1}\,{{\it \_Z}}^{3}{{\it \_f}}^{5}+12 \,{\it \_C1}\,{{\it \_Z}}^{2}{{\it \_f}}^{6}+4\,{\it \_C1}\,{{\it \_Z} }^{4}{{\it \_f}}^{2}-12\,{\it \_C1}\,{{\it \_Z}}^{2}{{\it \_f}}^{4}+8 \,{\it \_C1}\,{\it \_Z}\,{{\it \_f}}^{5}+4\,{\it \_C1}\,{{\it \_Z}}^{3 }{\it \_f}-4\,{\it \_C1}\,{\it \_Z}\,{{\it \_f}}^{3}+ \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{ {\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{{\frac {5 }{2}}}{{\rm e}^{{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{ \it \_f} \right ) ^{2}}{2\,{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{\it Erf} \left ( {\frac {\sqrt {2}}{2}\sqrt {{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{ {\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}} \right ) \sqrt { \pi }\sqrt {2}{\it \_C1}\,{{\it \_Z}}^{4}- \left ( {\frac { \left ( { \it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z} }^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{{\frac {3}{2}}}{ {\rm e}^{{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f } \right ) ^{2}}{2\,{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} }}{\it Erf} \left ( {\frac {\sqrt {2}}{2}\sqrt {{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2 } \left ( {{\it \_f}}^{2}+1 \right ) }}}} \right ) \sqrt {\pi }\sqrt {2}{ \it \_C1}\,{{\it \_Z}}^{4}+4\, \left ( {\frac { \left ( {\it \_Z}\,{{ \it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{ \frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{{\it \_Z}} ^{4}{{\it \_f}}^{6}+8\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^ {2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f }}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2 } \left ( {{\it \_f}}^{2}+1 \right ) }}}}{{\it \_Z}}^{4}{{\it \_f}}^{4}+ 8\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{{\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{ 2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f} }^{2}+1 \right ) }}}}{{\it \_Z}}^{3}{{\it \_f}}^{5}+4\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{ {\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{ {\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{ \it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{{\it \_Z}}^{4}{{\it \_f}}^{2}+8\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{ {\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{ {\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{ \it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{{\it \_Z}}^{3}{{\it \_f}}^{3}+4\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{ {\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{ {\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{ \it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{{\it \_Z}}^{2}{{\it \_f}}^{4}+4\, \left ( {\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{\it \_f} \right ) ^{2}}{{ {\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }} \right ) ^{3/2}{ {\rm e}^{1/2\,{\frac { \left ( {\it \_Z}\,{{\it \_f}}^{2}-{\it \_Z}+{ \it \_f} \right ) ^{2}}{{{\it \_Z}}^{2} \left ( {{\it \_f}}^{2}+1 \right ) }}}}{{\it \_Z}}^{2}{{\it \_f}}^{2} \right ) {d{\it \_f}}+{\it \_C2} \right ) x \right \} \]