\[ \left \{a x'(t)=(b-c) y(t) z(t),b y'(t)=(c-a) x(t) z(t),c z'(t)=(a-b) x(t) y(t)\right \} \] ✓ Mathematica : cpu = 4.13771 (sec), leaf count = 10101
\[\left \{\left \{x(t)\to \frac {\frac {\sqrt {2} b^2 \sqrt {a (a-c)} c_1 \text {sn}\left (\frac {\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}-\frac {\sqrt {2} b \sqrt {a (a-c)} c c_1 \text {sn}\left (\frac {\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}}{a},y(t)\to -\frac {\sqrt {2 c_1 b^2-2 c c_1 b+\frac {c \left (\frac {\sqrt {2} b^2 \sqrt {a (a-c)} c_1 \text {sn}\left (\frac {\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}-\frac {\sqrt {2} b \sqrt {a (a-c)} c c_1 \text {sn}\left (\frac {\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}\right ){}^2}{a}-\left (\frac {\sqrt {2} b^2 \sqrt {a (a-c)} c_1 \text {sn}\left (\frac {\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}-\frac {\sqrt {2} b \sqrt {a (a-c)} c c_1 \text {sn}\left (\frac {\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}\right ){}^2}}{\sqrt {b^2-b c}},z(t)\to \frac {\sqrt {-2 c_2 c^2+2 b c_2 c-\frac {b \left (\frac {\sqrt {2} b^2 \sqrt {a (a-c)} c_1 \text {sn}\left (\frac {\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}-\frac {\sqrt {2} b \sqrt {a (a-c)} c c_1 \text {sn}\left (\frac {\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}\right ){}^2}{a}+\left (\frac {\sqrt {2} b^2 \sqrt {a (a-c)} c_1 \text {sn}\left (\frac {\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}-\frac {\sqrt {2} b \sqrt {a (a-c)} c c_1 \text {sn}\left (\frac {\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}\right ){}^2}}{\sqrt {b-c} \sqrt {c}}\right \},\left \{x(t)\to \frac {\frac {\sqrt {2} b^2 \sqrt {a (a-c)} c_1 \text {sn}\left (\frac {\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}-\frac {\sqrt {2} b \sqrt {a (a-c)} c c_1 \text {sn}\left (\frac {\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}}{a},y(t)\to \frac {\sqrt {2 c_1 b^2-2 c c_1 b+\frac {c \left (\frac {\sqrt {2} b^2 \sqrt {a (a-c)} c_1 \text {sn}\left (\frac {\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}-\frac {\sqrt {2} b \sqrt {a (a-c)} c c_1 \text {sn}\left (\frac {\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}\right ){}^2}{a}-\left (\frac {\sqrt {2} b^2 \sqrt {a (a-c)} c_1 \text {sn}\left (\frac {\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}-\frac {\sqrt {2} b \sqrt {a (a-c)} c c_1 \text {sn}\left (\frac {\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}\right ){}^2}}{\sqrt {b^2-b c}},z(t)\to -\frac {\sqrt {-2 c_2 c^2+2 b c_2 c-\frac {b \left (\frac {\sqrt {2} b^2 \sqrt {a (a-c)} c_1 \text {sn}\left (\frac {\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}-\frac {\sqrt {2} b \sqrt {a (a-c)} c c_1 \text {sn}\left (\frac {\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}\right ){}^2}{a}+\left (\frac {\sqrt {2} b^2 \sqrt {a (a-c)} c_1 \text {sn}\left (\frac {\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}-\frac {\sqrt {2} b \sqrt {a (a-c)} c c_1 \text {sn}\left (\frac {\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}\right ){}^2}}{\sqrt {b-c} \sqrt {c}}\right \},\left \{x(t)\to \frac {\frac {\sqrt {2} b^2 \sqrt {a (a-c)} c_1 \text {sn}\left (\frac {-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}-\frac {\sqrt {2} b \sqrt {a (a-c)} c c_1 \text {sn}\left (\frac {-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}}{a},y(t)\to -\frac {\sqrt {2 c_1 b^2-2 c c_1 b+\frac {c \left (\frac {\sqrt {2} b^2 \sqrt {a (a-c)} c_1 \text {sn}\left (\frac {-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}-\frac {\sqrt {2} b \sqrt {a (a-c)} c c_1 \text {sn}\left (\frac {-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}\right ){}^2}{a}-\left (\frac {\sqrt {2} b^2 \sqrt {a (a-c)} c_1 \text {sn}\left (\frac {-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}-\frac {\sqrt {2} b \sqrt {a (a-c)} c c_1 \text {sn}\left (\frac {-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}\right ){}^2}}{\sqrt {b^2-b c}},z(t)\to -\frac {\sqrt {-2 c_2 c^2+2 b c_2 c-\frac {b \left (\frac {\sqrt {2} b^2 \sqrt {a (a-c)} c_1 \text {sn}\left (\frac {-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}-\frac {\sqrt {2} b \sqrt {a (a-c)} c c_1 \text {sn}\left (\frac {-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}\right ){}^2}{a}+\left (\frac {\sqrt {2} b^2 \sqrt {a (a-c)} c_1 \text {sn}\left (\frac {-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}-\frac {\sqrt {2} b \sqrt {a (a-c)} c c_1 \text {sn}\left (\frac {-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}\right ){}^2}}{\sqrt {b-c} \sqrt {c}}\right \},\left \{x(t)\to \frac {\frac {\sqrt {2} b^2 \sqrt {a (a-c)} c_1 \text {sn}\left (\frac {-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}-\frac {\sqrt {2} b \sqrt {a (a-c)} c c_1 \text {sn}\left (\frac {-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}}{a},y(t)\to \frac {\sqrt {2 c_1 b^2-2 c c_1 b+\frac {c \left (\frac {\sqrt {2} b^2 \sqrt {a (a-c)} c_1 \text {sn}\left (\frac {-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}-\frac {\sqrt {2} b \sqrt {a (a-c)} c c_1 \text {sn}\left (\frac {-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}\right ){}^2}{a}-\left (\frac {\sqrt {2} b^2 \sqrt {a (a-c)} c_1 \text {sn}\left (\frac {-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}-\frac {\sqrt {2} b \sqrt {a (a-c)} c c_1 \text {sn}\left (\frac {-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}\right ){}^2}}{\sqrt {b^2-b c}},z(t)\to \frac {\sqrt {-2 c_2 c^2+2 b c_2 c-\frac {b \left (\frac {\sqrt {2} b^2 \sqrt {a (a-c)} c_1 \text {sn}\left (\frac {-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}-\frac {\sqrt {2} b \sqrt {a (a-c)} c c_1 \text {sn}\left (\frac {-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}\right ){}^2}{a}+\left (\frac {\sqrt {2} b^2 \sqrt {a (a-c)} c_1 \text {sn}\left (\frac {-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}-\frac {\sqrt {2} b \sqrt {a (a-c)} c c_1 \text {sn}\left (\frac {-\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} t}{\sqrt {b} \sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} t}{\sqrt {b-c}}+\frac {\sqrt {2} \sqrt {a} \sqrt {a-c} c \sqrt {c_2} c_3}{\sqrt {b} \sqrt {b-c}}-\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a-c} \sqrt {c_2} c_3}{\sqrt {b-c}}}{a}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{(a-c) \sqrt {b (b-c) c_1}}\right ){}^2}}{\sqrt {b-c} \sqrt {c}}\right \}\right \}\] ✓ Maple : cpu = 1.4 (sec), leaf count = 1117
\[\left \{[\{x \left (t \right ) = 0\}, \{y \left (t \right ) = 0\}, \{z \left (t \right ) = c_{1}\}], [\{x \left (t \right ) = 0\}, \{y \left (t \right ) = c_{1}\}, \{z \left (t \right ) = 0\}], [\{x \left (t \right ) = c_{1}\}, \{y \left (t \right ) = 0\}, \{z \left (t \right ) = 0\}], \left [\left \{x \left (t \right ) = \RootOf \left (c_{3}+t -\left (\int _{}^{\textit {\_Z}}-\frac {2 \left (a -c \right ) \left (a -b \right ) b c}{\sqrt {\left (a -c \right ) \left (a -b \right ) \left (-4 a^{4} \textit {\_a}^{4}+8 \textit {\_a}^{4} a^{3} b +8 \textit {\_a}^{4} a^{3} c -4 \textit {\_a}^{4} a^{2} b^{2}-16 \textit {\_a}^{4} a^{2} b c -4 \textit {\_a}^{4} a^{2} c^{2}+8 \textit {\_a}^{4} a \,b^{2} c +8 \textit {\_a}^{4} a b \,c^{2}-4 \textit {\_a}^{4} b^{2} c^{2}+16 c_{2} \textit {\_a}^{2} a^{4}-32 c_{2} \textit {\_a}^{2} a^{3} b -32 c_{2} \textit {\_a}^{2} a^{3} c +16 c_{2} \textit {\_a}^{2} a^{2} b^{2}+64 c_{2} \textit {\_a}^{2} a^{2} b c +16 c_{2} \textit {\_a}^{2} a^{2} c^{2}-32 c_{2} \textit {\_a}^{2} a \,b^{2} c -32 c_{2} \textit {\_a}^{2} a b \,c^{2}+16 c_{2} \textit {\_a}^{2} b^{2} c^{2}-16 c_{2}^{2} a^{4}+32 c_{2}^{2} a^{3} b +32 c_{2}^{2} a^{3} c -16 c_{2}^{2} a^{2} b^{2}-64 c_{2}^{2} a^{2} b c -16 c_{2}^{2} a^{2} c^{2}+32 c_{2}^{2} a \,b^{2} c +32 c_{2}^{2} a b \,c^{2}-16 c_{2}^{2} b^{2} c^{2}+c_{1} b c \right ) b c}}d \textit {\_a} \right )\right ), x \left (t \right ) = \RootOf \left (c_{3}+t -\left (\int _{}^{\textit {\_Z}}\frac {2 \left (a -c \right ) \left (a -b \right ) b c}{\sqrt {\left (a -c \right ) \left (a -b \right ) \left (-4 a^{4} \textit {\_a}^{4}+8 \textit {\_a}^{4} a^{3} b +8 \textit {\_a}^{4} a^{3} c -4 \textit {\_a}^{4} a^{2} b^{2}-16 \textit {\_a}^{4} a^{2} b c -4 \textit {\_a}^{4} a^{2} c^{2}+8 \textit {\_a}^{4} a \,b^{2} c +8 \textit {\_a}^{4} a b \,c^{2}-4 \textit {\_a}^{4} b^{2} c^{2}+16 c_{2} \textit {\_a}^{2} a^{4}-32 c_{2} \textit {\_a}^{2} a^{3} b -32 c_{2} \textit {\_a}^{2} a^{3} c +16 c_{2} \textit {\_a}^{2} a^{2} b^{2}+64 c_{2} \textit {\_a}^{2} a^{2} b c +16 c_{2} \textit {\_a}^{2} a^{2} c^{2}-32 c_{2} \textit {\_a}^{2} a \,b^{2} c -32 c_{2} \textit {\_a}^{2} a b \,c^{2}+16 c_{2} \textit {\_a}^{2} b^{2} c^{2}-16 c_{2}^{2} a^{4}+32 c_{2}^{2} a^{3} b +32 c_{2}^{2} a^{3} c -16 c_{2}^{2} a^{2} b^{2}-64 c_{2}^{2} a^{2} b c -16 c_{2}^{2} a^{2} c^{2}+32 c_{2}^{2} a \,b^{2} c +32 c_{2}^{2} a b \,c^{2}-16 c_{2}^{2} b^{2} c^{2}+c_{1} b c \right ) b c}}d \textit {\_a} \right )\right )\right \}, \left \{y \left (t \right ) = -\frac {\sqrt {2}\, \sqrt {\left (b -c \right ) \left (a -b \right ) \left (b c \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )-\sqrt {4}\, \sqrt {\left (\frac {b c \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )^{2}}{4}+\left (a -c \right ) \left (a -b \right ) \left (\frac {d}{d t}x \left (t \right )\right )^{2} x \left (t \right )^{2}\right ) b c}\right ) a b x \left (t \right )}}{2 \left (b -c \right ) \left (a -b \right ) b x \left (t \right )}, y \left (t \right ) = \frac {\sqrt {2}\, \sqrt {\left (b -c \right ) \left (a -b \right ) \left (b c \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )-\sqrt {4}\, \sqrt {\left (\frac {b c \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )^{2}}{4}+\left (a -c \right ) \left (a -b \right ) \left (\frac {d}{d t}x \left (t \right )\right )^{2} x \left (t \right )^{2}\right ) b c}\right ) a b x \left (t \right )}}{2 \left (b -c \right ) \left (a -b \right ) b x \left (t \right )}, y \left (t \right ) = -\frac {\sqrt {2}\, \sqrt {\left (b -c \right ) \left (a -b \right ) \left (b c \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )+\sqrt {4}\, \sqrt {\left (\frac {b c \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )^{2}}{4}+\left (a -c \right ) \left (a -b \right ) \left (\frac {d}{d t}x \left (t \right )\right )^{2} x \left (t \right )^{2}\right ) b c}\right ) a b x \left (t \right )}}{2 \left (b -c \right ) \left (a -b \right ) b x \left (t \right )}, y \left (t \right ) = \frac {\sqrt {2}\, \sqrt {\left (b -c \right ) \left (a -b \right ) \left (b c \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )+\sqrt {4}\, \sqrt {\left (\frac {b c \left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )^{2}}{4}+\left (a -c \right ) \left (a -b \right ) \left (\frac {d}{d t}x \left (t \right )\right )^{2} x \left (t \right )^{2}\right ) b c}\right ) a b x \left (t \right )}}{2 \left (b -c \right ) \left (a -b \right ) b x \left (t \right )}\right \}, \left \{z \left (t \right ) = \frac {a \left (\frac {d}{d t}x \left (t \right )\right )}{\left (b -c \right ) y \left (t \right )}\right \}\right ]\right \}\]