\[ \boxed { \begin {array}{rl} x'(t) &=-x(t) y(t)^2+x(t)+y(t) \\ y'(t) &=x(t)^2 y(t)-x(t)-y(t) \\ z'(t) &=y(t)^2-x(t)^2 \end {array} } \]
Mathematica: cpu = 0.266534 (sec), leaf count = 65 \[ \text {DSolve}\left [\left \{x'(t)=-x(t) y(t)^2+x(t)+y(t),y'(t)=x(t)^2 y(t)-x(t)-y(t),z'(t)=y(t)^2-x(t)^2\right \},\{x(t),y(t),z(t)\},t\right ] \]
Maple: cpu = 0.608 (sec), leaf count = 304 \[ \left \{ [ \left \{ x \left ( t \right ) =0 \right \} , \left \{ y \left ( t \right ) =0 \right \} , \left \{ z \left ( t \right ) ={\it \_C1} \right \} ],[ \left \{ x \left ( t \right ) ={\it ODESolStruc} \left ( { \it \_a},[ \left \{ \left ( {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) \right ) {\it \_b} \left ( {\it \_a} \right ) -{\frac {1}{2\,{{\it \_a}}^{2}} \left ( 4\,{\it \_b} \left ( {\it \_a} \right ) {{\it \_a}}^{4}-4\,{{\it \_a}}^{5}+2\,{\it \_a}\, \left ( { \it \_b} \left ( {\it \_a} \right ) \right ) ^{2}-4\,{{\it \_a}}^{2}{ \it \_b} \left ( {\it \_a} \right ) +3\,{{\it \_a}}^{3}-{\it \_b} \left ( {\it \_a} \right ) +{\it \_a}-\sqrt {-4\,{\it \_b} \left ( {\it \_a} \right ) {{\it \_a}}^{7}+4\,{{\it \_a}}^{8}+8\,{{\it \_a}}^{4} \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}-16\,{{\it \_a }}^{5}{\it \_b} \left ( {\it \_a} \right ) +9\,{{\it \_a}}^{6}-4\, \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{3}{\it \_a}+12\, \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}{{\it \_a}}^{2 }-14\,{{\it \_a}}^{3}{\it \_b} \left ( {\it \_a} \right ) +6\,{{\it \_a} }^{4}+ \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}-2\,{ \it \_a}\,{\it \_b} \left ( {\it \_a} \right ) +{{\it \_a}}^{2}} \right ) }=0 \right \} , \left \{ {\it \_a}=x \left ( t \right ) ,{\it \_b } \left ( {\it \_a} \right ) ={\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right \} , \left \{ t=\int \! \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{-1}\,{\rm d}{\it \_a}+{\it \_C2},x \left ( t \right ) ={\it \_a} \right \} ] \right ) \right \} , \left \{ y \left ( t \right ) ={\frac {-2\, \left ( x \left ( t \right ) \right ) ^{4}+2\, \left ( x \left ( t \right ) \right ) ^{3}{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) - \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) x \left ( t \right ) + \left ( x \left ( t \right ) \right ) ^{2}-2\,x \left ( t \right ) {\frac {\rm d}{{\rm d}t}} x \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{2}}{- \left ( x \left ( t \right ) \right ) ^{3}-x \left ( t \right ) +{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) }} \right \} , \left \{ z \left ( t \right ) =\int \!-{\frac { \left ( x \left ( t \right ) \right ) ^{5}-2\, \left ( x \left ( t \right ) \right ) ^{3}+2\, \left ( x \left ( t \right ) \right ) ^{2}{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) -{\frac {{\rm d}^{2}}{{\rm d}{t}^ {2}}}x \left ( t \right ) }{ \left ( x \left ( t \right ) \right ) ^{3}+x \left ( t \right ) -{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) }} \,{\rm d}t+{\it \_C1} \right \} ] \right \} \]