\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) -y \left ( t \right ) +z \left ( t \right ) =0,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) -x \left ( t \right ) -y \left ( t \right ) =t,{\frac {\rm d}{{\rm d}t}}z \left ( t \right ) -x \left ( t \right ) -z \left ( t \right ) =t \right \} } \]
Mathematica: cpu = 0.014002 (sec), leaf count = 226 \[ \left \{\left \{x(t)\to c_2 \left (e^t-1\right )+c_3 \left (1-e^t\right )+c_1+e^{-t} \left (1-e^t\right ) (-t-1)+e^{-t} \left (e^t-1\right ) (-t-1),y(t)\to c_3 \left (-e^t t+e^t-1\right )+c_1 \left (e^t-1\right )+c_2 \left (e^t t+1\right )+e^{-t} (-t-1) \left (-e^t t+e^t-1\right )+e^{-t} (-t-1) \left (e^t t+1\right ),z(t)\to c_3 \left (-e^t t+2 e^t-1\right )+c_1 \left (e^t-1\right )+c_2 \left (e^t t-e^t+1\right )+e^{-t} (-t-1) \left (-e^t t+2 e^t-1\right )+e^{-t} (-t-1) \left (e^t t-e^t+1\right )\right \}\right \} \]
Maple: cpu = 0.047 (sec), leaf count = 56 \[ \left \{ \left \{ x \left ( t \right ) ={\it \_C2}+{\it \_C3}\,{{\rm e}^ {t}},y \left ( t \right ) ={\it \_C3}\,{{\rm e}^{t}}t+{\it \_C1}\,{ {\rm e}^{t}}-{\it \_C2}-t-1,z \left ( t \right ) ={\it \_C3}\,{{\rm e}^{ t}}t+{\it \_C1}\,{{\rm e}^{t}}-{\it \_C3}\,{{\rm e}^{t}}-{\it \_C2}-t- 1 \right \} \right \} \]