\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) =1/2\, \left ( x \left ( t \right ) \right ) ^{2}-1/24\,y \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =2\,x \left ( t \right ) y \left ( t \right ) -3\,z \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}z \left ( t \right ) =3\,x \left ( t \right ) z \left ( t \right ) -1/6\, \left ( y \left ( t \right ) \right ) ^{2} \right \} } \]
Mathematica: cpu = 72.632223 (sec), leaf count = 66 \[ \text {DSolve}\left [\left \{x'(t)=\frac {x(t)^2}{2}-\frac {y(t)}{24},y'(t)=2 x(t) y(t)-3 z(t),z'(t)=3 x(t) z(t)-\frac {y(t)^2}{6}\right \},\{x(t),y(t),z(t)\},t\right ] \]
Maple: cpu = 0.904 (sec), leaf count = 376 \[ \left \{ [ \left \{ y \left ( t \right ) =0 \right \} , \left \{ x \left ( t \right ) =2\, \left ( 2\,{\it \_C1}-t \right ) ^{-1} \right \} , \left \{ z \left ( t \right ) =0 \right \} ],[ \left \{ y \left ( t \right ) =256\, \left ( {\it \_C1}\,t+{\it \_C2} \right ) ^{-4} \right \} , \left \{ x \left ( t \right ) ={\frac {1}{2\,y \left ( t \right ) } \left ( {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) -{\frac {\sqrt {3}}{3} \left ( y \left ( t \right ) \right ) ^{{\frac {3}{2}}}} \right ) },x \left ( t \right ) ={\frac {1}{2\,y \left ( t \right ) } \left ( {\frac {\rm d}{ {\rm d}t}}y \left ( t \right ) +{\frac {\sqrt {3}}{3} \left ( y \left ( t \right ) \right ) ^{{\frac {3}{2}}}} \right ) } \right \} , \left \{ z \left ( t \right ) ={\frac {2\,x \left ( t \right ) y \left ( t \right ) }{ 3}}-{\frac {{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) }{3}} \right \} ],[ \left \{ y \left ( t \right ) ={\it ODESolStruc} \left ( { {\rm e}^{\int \!{\it \_g} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}+ {\it \_C2}}},[ \left \{ {\frac {\rm d}{{\rm d}{\it \_f}}}{\it \_g} \left ( {\it \_f} \right ) =-{\frac {\sqrt {15} \left ( {\it \_g} \left ( {\it \_f} \right ) \right ) ^{3}}{20\,{{\it \_f}}^{2}}\sqrt {-{ \frac {{{\it \_f}}^{2} \left ( 3\,{{\it \_f}}^{2}{\it \_g} \left ( {\it \_f} \right ) +12\,{\it \_f}-5\,{\it \_g} \left ( {\it \_f} \right ) \right ) \left ( {\it \_g} \left ( {\it \_f} \right ) {\it \_f}+4 \right ) ^{2}}{ \left ( {\it \_g} \left ( {\it \_f} \right ) \right ) ^{3 }}}}}+{\frac { \left ( {\it \_g} \left ( {\it \_f} \right ) \right ) ^{2} }{2}}+{\frac {{\it \_g} \left ( {\it \_f} \right ) }{{\it \_f}}} \right \} , \left \{ {\it \_f}={{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \left ( y \left ( t \right ) \right ) ^{-{\frac {3}{2}}}},{\it \_g} \left ( {\it \_f} \right ) =-2\,{ \left ( y \left ( t \right ) \right ) ^{3/2} \left ( -2\,{\frac { \left ( {\frac {{\rm d}^{2}}{ {\rm d}{t}^{2}}}y \left ( t \right ) \right ) y \left ( t \right ) }{{ \frac {\rm d}{{\rm d}t}}y \left ( t \right ) }}+3\,{\frac {\rm d}{ {\rm d}t}}y \left ( t \right ) \right ) ^{-1}} \right \} , \left \{ t= \int \!{\frac {{\it \_g} \left ( {\it \_f} \right ) }{{\it \_f}}{\frac { 1}{\sqrt {{{\rm e}^{\int \!{\it \_g} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}+{\it \_C2}}}}}}}\,{\rm d}{\it \_f}+{\it \_C1},y \left ( t \right ) ={{\rm e}^{\int \!{\it \_g} \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}+{\it \_C2}}} \right \} ] \right ) \right \} , \left \{ x \left ( t \right ) ={\frac {-2\, \left ( {\frac {{\rm d}^{3}} {{\rm d}{t}^{3}}}y \left ( t \right ) \right ) y \left ( t \right ) +3\, \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}y \left ( t \right ) \right ) {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) }{-12\, \left ( { \frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}y \left ( t \right ) \right ) y \left ( t \right ) +15\, \left ( {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) ^{2}}} \right \} , \left \{ z \left ( t \right ) ={ \frac {-4\, \left ( {\frac {{\rm d}^{3}}{{\rm d}{t}^{3}}}y \left ( t \right ) \right ) \left ( y \left ( t \right ) \right ) ^{2}+18\, \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}y \left ( t \right ) \right ) y \left ( t \right ) {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) -15\, \left ( {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) ^{3}}{-36\, \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}y \left ( t \right ) \right ) y \left ( t \right ) +45\, \left ( {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) ^{2}}} \right \} ] \right \} \]