\[ \left \{a x'(t)=(b-c) y(t) z(t),b y'(t)=(c-a) x(t) z(t),c z'(t)=(a-b) x(t) y(t)\right \} \] ✓ Mathematica : cpu = 5.53857 (sec), leaf count = 1461
\[\left \{\left \{x(t)\to \frac {\sqrt {2} b \sqrt {a (a-c)} (c-b) c_1 \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} \left (c_3-t\right )}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{a (c-a) \sqrt {b (b-c) c_1}},y(t)\to -\frac {\sqrt {2} \sqrt {-b (b-c) c_1 \left (\text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} \left (c_3-t\right )}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right ){}^2-1\right )}}{\sqrt {b (b-c)}},z(t)\to \frac {\sqrt {2} \sqrt {\frac {(b-c) \left (b (b-a) c_1 \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} \left (c_3-t\right )}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right ){}^2+c (c-a) c_2\right )}{c-a}}}{\sqrt {b-c} \sqrt {c}}\right \},\left \{x(t)\to \frac {\sqrt {2} b \sqrt {a (a-c)} (c-b) c_1 \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} \left (c_3-t\right )}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{a (c-a) \sqrt {b (b-c) c_1}},y(t)\to \frac {\sqrt {2} \sqrt {-b (b-c) c_1 \left (\text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} \left (c_3-t\right )}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right ){}^2-1\right )}}{\sqrt {b (b-c)}},z(t)\to -\frac {\sqrt {2} \sqrt {\frac {(b-c) \left (b (b-a) c_1 \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} \left (c_3-t\right )}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right ){}^2+c (c-a) c_2\right )}{c-a}}}{\sqrt {b-c} \sqrt {c}}\right \},\left \{x(t)\to \frac {\sqrt {2} b \sqrt {a (a-c)} (c-b) c_1 \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} \left (t-c_3\right )}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{a (c-a) \sqrt {b (b-c) c_1}},y(t)\to -\frac {\sqrt {2} \sqrt {-b (b-c) c_1 \left (\text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} \left (t-c_3\right )}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right ){}^2-1\right )}}{\sqrt {b (b-c)}},z(t)\to -\frac {\sqrt {2} \sqrt {\frac {(b-c) \left (b (b-a) c_1 \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} \left (t-c_3\right )}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right ){}^2+c (c-a) c_2\right )}{c-a}}}{\sqrt {b-c} \sqrt {c}}\right \},\left \{x(t)\to \frac {\sqrt {2} b \sqrt {a (a-c)} (c-b) c_1 \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} \left (t-c_3\right )}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right )}{a (c-a) \sqrt {b (b-c) c_1}},y(t)\to \frac {\sqrt {2} \sqrt {-b (b-c) c_1 \left (\text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} \left (t-c_3\right )}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right ){}^2-1\right )}}{\sqrt {b (b-c)}},z(t)\to \frac {\sqrt {2} \sqrt {\frac {(b-c) \left (b (b-a) c_1 \text {sn}\left (\frac {\sqrt {2} \sqrt {a-c} \sqrt {b-c} \sqrt {c_2} \left (t-c_3\right )}{\sqrt {a} \sqrt {b}}|-\frac {(a-b) b c_1}{(a-c) c c_2}\right ){}^2+c (c-a) c_2\right )}{c-a}}}{\sqrt {b-c} \sqrt {c}}\right \}\right \}\]
✓ Maple : cpu = 0.81 (sec), leaf count = 1117
\[ \left \{ [ \left \{ x \left ( t \right ) =0 \right \} , \left \{ y \left ( t \right ) =0 \right \} , \left \{ z \left ( t \right ) ={\it \_C1} \right \} ],[ \left \{ x \left ( t \right ) =0 \right \} , \left \{ y \left ( t \right ) ={\it \_C1} \right \} , \left \{ z \left ( t \right ) =0 \right \} ],[ \left \{ x \left ( t \right ) ={\it \_C1} \right \} , \left \{ y \left ( t \right ) =0 \right \} , \left \{ z \left ( t \right ) =0 \right \} ],[ \left \{ x \left ( t \right ) ={\it RootOf} \left ( -\int ^{{\it \_Z}}\!-2\,{\frac {bc \left ( a-c \right ) \left ( a-b \right ) }{\sqrt {bc \left ( a-c \right ) \left ( a-b \right ) \left ( -4\,{{\it \_a}}^{4}{a}^{4}+8\,{{\it \_a}}^{4}{a}^{3}b+8\,{{\it \_a}}^{4}{a}^{3}c-4\,{{\it \_a}}^{4}{a}^{2}{b}^{2}-16\,{{\it \_a}}^{4}{a}^{2}bc-4\,{{\it \_a}}^{4}{a}^{2}{c}^{2}+8\,{{\it \_a}}^{4}a{b}^{2}c+8\,{{\it \_a}}^{4}ab{c}^{2}-4\,{{\it \_a}}^{4}{b}^{2}{c}^{2}+16\,{\it \_C2}\,{{\it \_a}}^{2}{a}^{4}-32\,{\it \_C2}\,{{\it \_a}}^{2}{a}^{3}b-32\,{\it \_C2}\,{{\it \_a}}^{2}{a}^{3}c+16\,{\it \_C2}\,{{\it \_a}}^{2}{a}^{2}{b}^{2}+64\,{\it \_C2}\,{{\it \_a}}^{2}{a}^{2}bc+16\,{\it \_C2}\,{{\it \_a}}^{2}{a}^{2}{c}^{2}-32\,{\it \_C2}\,{{\it \_a}}^{2}a{b}^{2}c-32\,{\it \_C2}\,{{\it \_a}}^{2}ab{c}^{2}+16\,{\it \_C2}\,{{\it \_a}}^{2}{b}^{2}{c}^{2}-16\,{{\it \_C2}}^{2}{a}^{4}+32\,{{\it \_C2}}^{2}{a}^{3}b+32\,{{\it \_C2}}^{2}{a}^{3}c-16\,{{\it \_C2}}^{2}{a}^{2}{b}^{2}-64\,{{\it \_C2}}^{2}{a}^{2}bc-16\,{{\it \_C2}}^{2}{a}^{2}{c}^{2}+32\,{{\it \_C2}}^{2}a{b}^{2}c+32\,{{\it \_C2}}^{2}ab{c}^{2}-16\,{{\it \_C2}}^{2}{b}^{2}{c}^{2}+{\it \_C1}\,bc \right ) }}}{d{\it \_a}}+t+{\it \_C3} \right ) ,x \left ( t \right ) ={\it RootOf} \left ( -\int ^{{\it \_Z}}\!2\,{\frac {bc \left ( a-c \right ) \left ( a-b \right ) }{\sqrt {bc \left ( a-c \right ) \left ( a-b \right ) \left ( -4\,{{\it \_a}}^{4}{a}^{4}+8\,{{\it \_a}}^{4}{a}^{3}b+8\,{{\it \_a}}^{4}{a}^{3}c-4\,{{\it \_a}}^{4}{a}^{2}{b}^{2}-16\,{{\it \_a}}^{4}{a}^{2}bc-4\,{{\it \_a}}^{4}{a}^{2}{c}^{2}+8\,{{\it \_a}}^{4}a{b}^{2}c+8\,{{\it \_a}}^{4}ab{c}^{2}-4\,{{\it \_a}}^{4}{b}^{2}{c}^{2}+16\,{\it \_C2}\,{{\it \_a}}^{2}{a}^{4}-32\,{\it \_C2}\,{{\it \_a}}^{2}{a}^{3}b-32\,{\it \_C2}\,{{\it \_a}}^{2}{a}^{3}c+16\,{\it \_C2}\,{{\it \_a}}^{2}{a}^{2}{b}^{2}+64\,{\it \_C2}\,{{\it \_a}}^{2}{a}^{2}bc+16\,{\it \_C2}\,{{\it \_a}}^{2}{a}^{2}{c}^{2}-32\,{\it \_C2}\,{{\it \_a}}^{2}a{b}^{2}c-32\,{\it \_C2}\,{{\it \_a}}^{2}ab{c}^{2}+16\,{\it \_C2}\,{{\it \_a}}^{2}{b}^{2}{c}^{2}-16\,{{\it \_C2}}^{2}{a}^{4}+32\,{{\it \_C2}}^{2}{a}^{3}b+32\,{{\it \_C2}}^{2}{a}^{3}c-16\,{{\it \_C2}}^{2}{a}^{2}{b}^{2}-64\,{{\it \_C2}}^{2}{a}^{2}bc-16\,{{\it \_C2}}^{2}{a}^{2}{c}^{2}+32\,{{\it \_C2}}^{2}a{b}^{2}c+32\,{{\it \_C2}}^{2}ab{c}^{2}-16\,{{\it \_C2}}^{2}{b}^{2}{c}^{2}+{\it \_C1}\,bc \right ) }}}{d{\it \_a}}+t+{\it \_C3} \right ) \right \} , \left \{ y \left ( t \right ) =-{\frac {\sqrt {2}}{2\,bx \left ( t \right ) \left ( b-c \right ) \left ( a-b \right ) }\sqrt {x \left ( t \right ) b \left ( b-c \right ) \left ( a-b \right ) \left ( \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) bc-\sqrt {4}\sqrt {b \left ( {\frac {bc \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) ^{2}}{4}}+ \left ( x \left ( t \right ) \right ) ^{2} \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{2} \left ( a-c \right ) \left ( a-b \right ) \right ) c} \right ) a}},y \left ( t \right ) ={\frac {\sqrt {2}}{2\,bx \left ( t \right ) \left ( b-c \right ) \left ( a-b \right ) }\sqrt {x \left ( t \right ) b \left ( b-c \right ) \left ( a-b \right ) \left ( \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) bc-\sqrt {4}\sqrt {b \left ( {\frac {bc \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) ^{2}}{4}}+ \left ( x \left ( t \right ) \right ) ^{2} \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{2} \left ( a-c \right ) \left ( a-b \right ) \right ) c} \right ) a}},y \left ( t \right ) =-{\frac {\sqrt {2}}{2\,bx \left ( t \right ) \left ( b-c \right ) \left ( a-b \right ) }\sqrt {x \left ( t \right ) b \left ( b-c \right ) \left ( a-b \right ) \left ( \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) bc+\sqrt {4}\sqrt {b \left ( {\frac {bc \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) ^{2}}{4}}+ \left ( x \left ( t \right ) \right ) ^{2} \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{2} \left ( a-c \right ) \left ( a-b \right ) \right ) c} \right ) a}},y \left ( t \right ) ={\frac {\sqrt {2}}{2\,bx \left ( t \right ) \left ( b-c \right ) \left ( a-b \right ) }\sqrt {x \left ( t \right ) b \left ( b-c \right ) \left ( a-b \right ) \left ( \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) bc+\sqrt {4}\sqrt {b \left ( {\frac {bc \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) ^{2}}{4}}+ \left ( x \left ( t \right ) \right ) ^{2} \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{2} \left ( a-c \right ) \left ( a-b \right ) \right ) c} \right ) a}} \right \} , \left \{ z \left ( t \right ) ={\frac {a{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) }{y \left ( t \right ) \left ( b-c \right ) }} \right \} ] \right \} \]