\[ \boxed { \left \{ t{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) =2\,x \left ( t \right ) -t,{t}^{3}{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =-x \left ( t \right ) +{t}^{2}y \left ( t \right ) +t,{t}^{4}{\frac {\rm d}{{\rm d}t}}z \left ( t \right ) =-x \left ( t \right ) -{t}^{2}y \left ( t \right ) +{t}^{3}z \left ( t \right ) +t \right \} } \]
Mathematica: cpu = 0.009501 (sec), leaf count = 39 \[ \left \{\left \{x(t)\to c_3 t^2+t,y(t)\to c_2 t+c_3,z(t)\to c_1 t+\frac {c_3}{t}+c_2\right \}\right \} \]
Maple: cpu = 0.063 (sec), leaf count = 37 \[ \left \{ \left \{ x \left ( t \right ) ={\it \_C3}\,{t}^{2}+t,y \left ( t \right ) =t{\it \_C2}+{\it \_C3},z \left ( t \right ) ={\frac {{\it \_C1 }\,{t}^{2}+t{\it \_C2}+{\it \_C3}}{t}} \right \} \right \} \]