\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =x \left ( t \right ) y \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +{\frac {\rm d}{{\rm d}t}}z \left ( t \right ) =x \left ( t \right ) z \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +{\frac {\rm d}{{\rm d}t}}z \left ( t \right ) =y \left ( t \right ) z \left ( t \right ) \right \} } \]
Mathematica: cpu = 126.589075 (sec), leaf count = 55 \[ \text {DSolve}\left [\left \{x'(t)+y'(t)=x(t) y(t),y'(t)+z'(t)=y(t) z(t),x'(t)+z'(t)=x(t) z(t)\right \},\{x(t),y(t),z(t)\},t\right ] \]
Maple: cpu = 1.529 (sec), leaf count = 17743 \[ \text {Too large to display} \]