\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) -x \left ( t \right ) +3\,y \left ( t \right ) ={{\rm e}^{t}}-1,{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +2\,x \left ( t \right ) +y \left ( t \right ) ={{\rm e}^{2\,t}}+t \right \} } \]
Mathematica: cpu = 0.087511 (sec), leaf count = 118 \[ \left \{\left \{x(t)\to \frac {5}{72} \left (c_1 e^{-7 t/5}+\frac {12 \left (5712 t+833 e^t+2352 e^{2 t}-5508\right )}{20825}\right )+\frac {1}{5} \left (t-e^t+e^{2 t}+1\right ),y(t)\to \frac {5}{48} \left (c_1 e^{-7 t/5}+\frac {12 \left (5712 t+833 e^t+2352 e^{2 t}-5508\right )}{20825}\right )+\frac {1}{5} \left (-t+e^t-e^{2 t}-1\right )\right \}\right \} \]
Maple: cpu = 0.031 (sec), leaf count = 51 \[ \left \{ \left \{ x \left ( t \right ) ={\frac {5\,{{\rm e}^{2\,t}}}{17} }-{\frac {{{\rm e}^{t}}}{6}}+{\frac {3\,t}{7}}-{\frac {1}{49}}+{ {\rm e}^{-{\frac {7\,t}{5}}}}{\it \_C1},y \left ( t \right ) =-{\frac {{ {\rm e}^{2\,t}}}{17}}+{\frac {{{\rm e}^{t}}}{4}}+{\frac {t}{7}}-{ \frac {26}{49}}+{\frac {3\,{\it \_C1}}{2}{{\rm e}^{-{\frac {7\,t}{5}}} }} \right \} \right \} \]