\[ \left \{x'(t)=\text {a1} x(t)+\text {b1} y(t)+\text {c1},y'(t)=\text {a2} x(t)+\text {b2} y(t)+\text {c2}\right \} \] ✓ Mathematica : cpu = 0.837384 (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 = 0.144 (sec), leaf count = 224
\[\left \{\left \{x \left (t \right ) = c_{1} {\mathrm e}^{\frac {\left (\mathit {a1} +\mathit {b2} -\sqrt {\mathit {a1}^{2}-2 \mathit {a1} \mathit {b2} +4 \mathit {a2} \mathit {b1} +\mathit {b2}^{2}}\right ) t}{2}}+c_{2} {\mathrm e}^{\frac {\left (\mathit {a1} +\mathit {b2} +\sqrt {\mathit {a1}^{2}-2 \mathit {a1} \mathit {b2} +4 \mathit {a2} \mathit {b1} +\mathit {b2}^{2}}\right ) t}{2}}+\frac {\mathit {c2} \mathit {b1} -\mathit {b2} \mathit {c1}}{\mathit {a1} \mathit {b2} -\mathit {a2} \mathit {b1}}, y \left (t \right ) = \frac {-c_{1} \left (\mathit {a1} \mathit {b2} -\mathit {a2} \mathit {b1} \right ) \left (\mathit {a1} -\mathit {b2} +\sqrt {\mathit {a1}^{2}-2 \mathit {a1} \mathit {b2} +4 \mathit {a2} \mathit {b1} +\mathit {b2}^{2}}\right ) {\mathrm e}^{\frac {\left (\mathit {a1} +\mathit {b2} -\sqrt {\mathit {a1}^{2}-2 \mathit {a1} \mathit {b2} +4 \mathit {a2} \mathit {b1} +\mathit {b2}^{2}}\right ) t}{2}}+c_{2} \left (\mathit {a1} \mathit {b2} -\mathit {a2} \mathit {b1} \right ) \left (-\mathit {a1} +\mathit {b2} +\sqrt {\mathit {a1}^{2}-2 \mathit {a1} \mathit {b2} +4 \mathit {a2} \mathit {b1} +\mathit {b2}^{2}}\right ) {\mathrm e}^{\frac {\left (\mathit {a1} +\mathit {b2} +\sqrt {\mathit {a1}^{2}-2 \mathit {a1} \mathit {b2} +4 \mathit {a2} \mathit {b1} +\mathit {b2}^{2}}\right ) t}{2}}-2 \left (\mathit {a1} \mathit {c2} -\mathit {a2} \mathit {c1} \right ) \mathit {b1}}{2 \left (\mathit {a1} \mathit {b2} -\mathit {a2} \mathit {b1} \right ) \mathit {b1}}\right \}\right \}\]