\[ \left \{x'(t)=\text {a1} x(t)+\text {b1} y(t)+\text {c1},y'(t)=\text {a2} x(t)+\text {b2} y(t)+\text {c2}\right \} \] ✓ Mathematica : cpu = 1.20018 (sec), leaf count = 2062
\[\left \{\left \{x(t)\to -\frac {\text {b1} e^{-\frac {1}{2} \left (\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t} \left (\frac {2 \left (\left (\text {a1}-\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) \text {c2}-2 \text {a2} \text {c1}\right ) e^{\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}} t}}{-\text {a1}-\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}}-\frac {2 \left (2 \text {a2} \text {c1}+\left (-\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) \text {c2}\right )}{\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}}\right ) \left (e^{\frac {1}{2} \left (\text {a1}+\text {b2}-\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}-e^{\frac {1}{2} \left (\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}\right )}{2 \left (\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}\right )}-\frac {\text {b1} c_2 \left (e^{\frac {1}{2} \left (\text {a1}+\text {b2}-\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}-e^{\frac {1}{2} \left (\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}\right )}{\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}}+\frac {e^{-\frac {1}{2} \left (\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t} \left (\frac {2 \left (-\text {a1} \text {c1}+\text {b2} \text {c1}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}} \text {c1}-2 \text {b1} \text {c2}\right ) e^{\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}} t}}{-\text {a1}-\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}}-\frac {2 \left (\text {a1} \text {c1}-\text {b2} \text {c1}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}} \text {c1}+2 \text {b1} \text {c2}\right )}{\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}}\right ) \left (-e^{\frac {1}{2} \left (\text {a1}+\text {b2}-\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t} \text {a1}+e^{\frac {1}{2} \left (\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t} \text {a1}+\text {b2} e^{\frac {1}{2} \left (\text {a1}+\text {b2}-\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}} e^{\frac {1}{2} \left (\text {a1}+\text {b2}-\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}-\text {b2} e^{\frac {1}{2} \left (\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}} e^{\frac {1}{2} \left (\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}\right )}{4 \left (\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}\right )}+\frac {\left (-e^{\frac {1}{2} \left (\text {a1}+\text {b2}-\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t} \text {a1}+e^{\frac {1}{2} \left (\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t} \text {a1}+\text {b2} e^{\frac {1}{2} \left (\text {a1}+\text {b2}-\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}} e^{\frac {1}{2} \left (\text {a1}+\text {b2}-\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}-\text {b2} e^{\frac {1}{2} \left (\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}} e^{\frac {1}{2} \left (\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}\right ) c_1}{2 \sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}},y(t)\to -\frac {\text {a2} e^{-\frac {1}{2} \left (\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t} \left (\frac {2 \left (-\text {a1} \text {c1}+\text {b2} \text {c1}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}} \text {c1}-2 \text {b1} \text {c2}\right ) e^{\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}} t}}{-\text {a1}-\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}}-\frac {2 \left (\text {a1} \text {c1}-\text {b2} \text {c1}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}} \text {c1}+2 \text {b1} \text {c2}\right )}{\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}}\right ) \left (e^{\frac {1}{2} \left (\text {a1}+\text {b2}-\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}-e^{\frac {1}{2} \left (\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}\right )}{2 \left (\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}\right )}-\frac {\text {a2} c_1 \left (e^{\frac {1}{2} \left (\text {a1}+\text {b2}-\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}-e^{\frac {1}{2} \left (\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}\right )}{\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}}+\frac {e^{-\frac {1}{2} \left (\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t} \left (\frac {2 \left (\left (\text {a1}-\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) \text {c2}-2 \text {a2} \text {c1}\right ) e^{\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}} t}}{-\text {a1}-\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}}-\frac {2 \left (2 \text {a2} \text {c1}+\left (-\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) \text {c2}\right )}{\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}}\right ) \left (e^{\frac {1}{2} \left (\text {a1}+\text {b2}-\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t} \text {a1}-e^{\frac {1}{2} \left (\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t} \text {a1}-\text {b2} e^{\frac {1}{2} \left (\text {a1}+\text {b2}-\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}} e^{\frac {1}{2} \left (\text {a1}+\text {b2}-\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}+\text {b2} e^{\frac {1}{2} \left (\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}} e^{\frac {1}{2} \left (\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}\right )}{4 \left (\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}\right )}+\frac {\left (e^{\frac {1}{2} \left (\text {a1}+\text {b2}-\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t} \text {a1}-e^{\frac {1}{2} \left (\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t} \text {a1}-\text {b2} e^{\frac {1}{2} \left (\text {a1}+\text {b2}-\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}} e^{\frac {1}{2} \left (\text {a1}+\text {b2}-\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}+\text {b2} e^{\frac {1}{2} \left (\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}} e^{\frac {1}{2} \left (\text {a1}+\text {b2}+\sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}\right ) t}\right ) c_2}{2 \sqrt {\text {a1}^2-2 \text {b2} \text {a1}+\text {b2}^2+4 \text {a2} \text {b1}}}\right \}\right \}\]
✓ Maple : cpu = 0.133 (sec), leaf count = 334
\[ \left \{ \left \{ x \left ( t \right ) ={{\rm e}^{ \left ( {\frac {{\it a1}}{2}}+{\frac {{\it b2}}{2}}+{\frac {1}{2}\sqrt {{{\it a1}}^{2}-2\,{\it a1}\,{\it b2}+4\,{\it a2}\,{\it b1}+{{\it b2}}^{2}}} \right ) t}}{\it \_C2}+{{\rm e}^{ \left ( {\frac {{\it a1}}{2}}+{\frac {{\it b2}}{2}}-{\frac {1}{2}\sqrt {{{\it a1}}^{2}-2\,{\it a1}\,{\it b2}+4\,{\it a2}\,{\it b1}+{{\it b2}}^{2}}} \right ) t}}{\it \_C1}+{\frac {{\it c2}\,{\it b1}-{\it b2}\,{\it c1}}{{\it a1}\,{\it b2}-{\it a2}\,{\it b1}}},y \left ( t \right ) ={\frac {1}{ \left ( 2\,{\it a1}\,{\it b2}-2\,{\it a2}\,{\it b1} \right ) {\it b1}} \left ( -{\frac {{\it a1}\, \left ( 2\,{\it a1}\,{\it b2}-2\,{\it a2}\,{\it b1} \right ) }{{\it a1}\,{\it b2}-{\it a2}\,{\it b1}} \left ( {{\rm e}^{{\frac {t}{2} \left ( {\it a1}+{\it b2}+\sqrt {{{\it a1}}^{2}-2\,{\it a1}\,{\it b2}+4\,{\it a2}\,{\it b1}+{{\it b2}}^{2}} \right ) }}}{\it \_C2}\, \left ( {\it a1}\,{\it b2}-{\it a2}\,{\it b1} \right ) +{{\rm e}^{{\frac {t}{2} \left ( {\it a1}+{\it b2}-\sqrt {{{\it a1}}^{2}-2\,{\it a1}\,{\it b2}+4\,{\it a2}\,{\it b1}+{{\it b2}}^{2}} \right ) }}}{\it \_C1}\, \left ( {\it a1}\,{\it b2}-{\it a2}\,{\it b1} \right ) +{\it c2}\,{\it b1}-{\it b2}\,{\it c1} \right ) }+{\frac {{\it \_C2}\, \left ( 2\,{\it a1}\,{\it b2}-2\,{\it a2}\,{\it b1} \right ) }{2} \left ( {\it a1}+{\it b2}+\sqrt {{{\it a1}}^{2}-2\,{\it a1}\,{\it b2}+4\,{\it a2}\,{\it b1}+{{\it b2}}^{2}} \right ) {{\rm e}^{{\frac {t}{2} \left ( {\it a1}+{\it b2}+\sqrt {{{\it a1}}^{2}-2\,{\it a1}\,{\it b2}+4\,{\it a2}\,{\it b1}+{{\it b2}}^{2}} \right ) }}}}+{\frac {{\it \_C1}\, \left ( 2\,{\it a1}\,{\it b2}-2\,{\it a2}\,{\it b1} \right ) }{2} \left ( {\it a1}+{\it b2}-\sqrt {{{\it a1}}^{2}-2\,{\it a1}\,{\it b2}+4\,{\it a2}\,{\it b1}+{{\it b2}}^{2}} \right ) {{\rm e}^{{\frac {t}{2} \left ( {\it a1}+{\it b2}-\sqrt {{{\it a1}}^{2}-2\,{\it a1}\,{\it b2}+4\,{\it a2}\,{\it b1}+{{\it b2}}^{2}} \right ) }}}}-{\it c1}\, \left ( 2\,{\it a1}\,{\it b2}-2\,{\it a2}\,{\it b1} \right ) \right ) } \right \} \right \} \]