\[ \boxed { \left \{ {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) =ax \left ( t \right ) +by \left ( t \right ) ,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) =cx \left ( t \right ) +by \left ( t \right ) \right \} } \]
Mathematica: cpu = 0.045506 (sec), leaf count = 696 \[ \left \{\left \{x(t)\to \frac {c_1 \left (a \left (-e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}\right )+a e^{\frac {1}{2} t \left (\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}+b e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}+\sqrt {a^2-2 a b+b^2+4 b c} e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}-b e^{\frac {1}{2} t \left (\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}+\sqrt {a^2-2 a b+b^2+4 b c} e^{\frac {1}{2} t \left (\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}\right )}{2 \sqrt {a^2-2 a b+b^2+4 b c}}-\frac {b c_2 \left (e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}-e^{\frac {1}{2} t \left (\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}\right )}{\sqrt {a^2-2 a b+b^2+4 b c}},y(t)\to \frac {c_2 \left (a e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}-a e^{\frac {1}{2} t \left (\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}-b e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}+\sqrt {a^2-2 a b+b^2+4 b c} e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}+b e^{\frac {1}{2} t \left (\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}+\sqrt {a^2-2 a b+b^2+4 b c} e^{\frac {1}{2} t \left (\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}\right )}{2 \sqrt {a^2-2 a b+b^2+4 b c}}-\frac {c c_1 \left (e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}-e^{\frac {1}{2} t \left (\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )}\right )}{\sqrt {a^2-2 a b+b^2+4 b c}}\right \}\right \} \]
Maple: cpu = 0.046 (sec), leaf count = 237 \[ \left \{ \left \{ x \left ( t \right ) ={\it \_C1}\,{{\rm e}^{{\frac {t }{2} \left ( a+b+\sqrt {{a}^{2}-2\,ab+{b}^{2}+4\,bc} \right ) }}}+{\it \_C2}\,{{\rm e}^{-{\frac {t}{2} \left ( -a-b+\sqrt {{a}^{2}-2\,ab+{b}^{ 2}+4\,bc} \right ) }}},y \left ( t \right ) = \left ( {\frac {1}{2}}+{ \frac {1}{b} \left ( {\frac {1}{2}\sqrt {{a}^{2}-2\,ab+{b}^{2}+4\,bc}}- {\frac {a}{2}} \right ) } \right ) {\it \_C1}\,{{\rm e}^{{\frac {t}{2} \left ( a+b+\sqrt {{a}^{2}-2\,ab+{b}^{2}+4\,bc} \right ) }}}+ \left ( { \frac {1}{2}{{\rm e}^{-{\frac {t}{2} \left ( -a-b+\sqrt {{a}^{2}-2\,ab+ {b}^{2}+4\,bc} \right ) }}}}+{\frac {1}{b} \left ( -{\frac {1}{2}\sqrt { {a}^{2}-2\,ab+{b}^{2}+4\,bc}{{\rm e}^{-{\frac {t}{2} \left ( -a-b+ \sqrt {{a}^{2}-2\,ab+{b}^{2}+4\,bc} \right ) }}}}-{\frac {a}{2}{{\rm e} ^{-{\frac {t}{2} \left ( -a-b+\sqrt {{a}^{2}-2\,ab+{b}^{2}+4\,bc} \right ) }}}} \right ) } \right ) {\it \_C2} \right \} \right \} \]