\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) =y \left ( t \right ) -z \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) = \left ( x \left ( t \right ) \right ) ^{2}+y \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}z \left ( t \right ) = \left ( x \left ( t \right ) \right ) ^{2}+z \left ( t \right ) \right \} } \]
Mathematica: cpu = 0.052507 (sec), leaf count = 308 \[ \left \{\left \{x(t)\to e^{-c_3} \left (e^{c_3} c_1+e^t\right ),y(t)\to c_2 \left (e^{-c_3} \left (e^{c_3} c_1+e^t\right )-c_1\right )+\left (e^{-c_3} \left (e^{c_3} c_1+e^t\right )-c_1\right ) \left (-\frac {c_1^2}{e^{-c_3} \left (e^{c_3} c_1+e^t\right )-c_1}+e^{-c_3} \left (e^{c_3} c_1+e^t\right )+2 c_1 \log \left (e^{-c_3} \left (e^{c_3} c_1+e^t\right )-c_1\right )\right ),z(t)\to -e^{-c_3} \left (e^{c_3} c_1+e^t\right )+c_2 \left (e^{-c_3} \left (e^{c_3} c_1+e^t\right )-c_1\right )+\left (e^{-c_3} \left (e^{c_3} c_1+e^t\right )-c_1\right ) \left (-\frac {c_1^2}{e^{-c_3} \left (e^{c_3} c_1+e^t\right )-c_1}+e^{-c_3} \left (e^{c_3} c_1+e^t\right )+2 c_1 \log \left (e^{-c_3} \left (e^{c_3} c_1+e^t\right )-c_1\right )\right )+c_1\right \}\right \} \]
Maple: cpu = 0.047 (sec), leaf count = 45 \[ \left \{ [ \left \{ x \left ( t \right ) ={\it \_C2}+{\it \_C3}\,{{\rm e} ^{t}} \right \} , \left \{ y \left ( t \right ) = \left ( \int \! \left ( x \left ( t \right ) \right ) ^{2}{{\rm e}^{-t}}\,{\rm d}t+{\it \_C1} \right ) {{\rm e}^{t}} \right \} , \left \{ z \left ( t \right ) =-{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +y \left ( t \right ) \right \} ] \right \} \]