\[ \left \{a t x'(t)=b c (y(t)-z(t)),b t y'(t)=a c (z(t)-x(t)),c t z'(t)=a b (x(t)-y(t))\right \} \] ✓ Mathematica : cpu = 0.0250552 (sec), leaf count = 1148
DSolve[{a*t*Derivative[1][x][t] == b*c*(y[t] - z[t]), b*t*Derivative[1][y][t] == a*c*(-x[t] + z[t]), c*t*Derivative[1][z][t] == a*b*(x[t] - y[t])},{x[t], y[t], z[t]},t]
\[\left \{\left \{x(t)\to \frac {\left (2 a^2 t^{i \sqrt {a^2+b^2+c^2}}+b^2 \left (t^{2 i \sqrt {a^2+b^2+c^2}}+1\right )+c^2 \left (t^{2 i \sqrt {a^2+b^2+c^2}}+1\right )\right ) c_1 t^{-i \sqrt {a^2+b^2+c^2}}}{2 \left (a^2+b^2+c^2\right )}-\frac {b \left (t^{i \sqrt {a^2+b^2+c^2}}-1\right ) \left (a b \left (t^{i \sqrt {a^2+b^2+c^2}}-1\right )+i c \sqrt {a^2+b^2+c^2} \left (t^{i \sqrt {a^2+b^2+c^2}}+1\right )\right ) c_2 t^{-i \sqrt {a^2+b^2+c^2}}}{2 a \left (a^2+b^2+c^2\right )}-\frac {c \left (t^{i \sqrt {a^2+b^2+c^2}}-1\right ) \left (a c \left (t^{i \sqrt {a^2+b^2+c^2}}-1\right )-i b \sqrt {a^2+b^2+c^2} \left (t^{i \sqrt {a^2+b^2+c^2}}+1\right )\right ) c_3 t^{-i \sqrt {a^2+b^2+c^2}}}{2 a \left (a^2+b^2+c^2\right )},y(t)\to -\frac {a \left (t^{i \sqrt {a^2+b^2+c^2}}-1\right ) \left (a b \left (t^{i \sqrt {a^2+b^2+c^2}}-1\right )-i c \sqrt {a^2+b^2+c^2} \left (t^{i \sqrt {a^2+b^2+c^2}}+1\right )\right ) c_1 t^{-i \sqrt {a^2+b^2+c^2}}}{2 b \left (a^2+b^2+c^2\right )}+\frac {\left (2 b^2 t^{i \sqrt {a^2+b^2+c^2}}+a^2 \left (t^{2 i \sqrt {a^2+b^2+c^2}}+1\right )+c^2 \left (t^{2 i \sqrt {a^2+b^2+c^2}}+1\right )\right ) c_2 t^{-i \sqrt {a^2+b^2+c^2}}}{2 \left (a^2+b^2+c^2\right )}-\frac {c \left (t^{i \sqrt {a^2+b^2+c^2}}-1\right ) \left (b c \left (t^{i \sqrt {a^2+b^2+c^2}}-1\right )+i a \sqrt {a^2+b^2+c^2} \left (t^{i \sqrt {a^2+b^2+c^2}}+1\right )\right ) c_3 t^{-i \sqrt {a^2+b^2+c^2}}}{2 b \left (a^2+b^2+c^2\right )},z(t)\to -\frac {a \left (t^{i \sqrt {a^2+b^2+c^2}}-1\right ) \left (a c \left (t^{i \sqrt {a^2+b^2+c^2}}-1\right )+i b \sqrt {a^2+b^2+c^2} \left (t^{i \sqrt {a^2+b^2+c^2}}+1\right )\right ) c_1 t^{-i \sqrt {a^2+b^2+c^2}}}{2 c \left (a^2+b^2+c^2\right )}-\frac {b \left (t^{i \sqrt {a^2+b^2+c^2}}-1\right ) \left (b c \left (t^{i \sqrt {a^2+b^2+c^2}}-1\right )-i a \sqrt {a^2+b^2+c^2} \left (t^{i \sqrt {a^2+b^2+c^2}}+1\right )\right ) c_2 t^{-i \sqrt {a^2+b^2+c^2}}}{2 c \left (a^2+b^2+c^2\right )}+\frac {\left (2 c^2 t^{i \sqrt {a^2+b^2+c^2}}+a^2 \left (t^{2 i \sqrt {a^2+b^2+c^2}}+1\right )+b^2 \left (t^{2 i \sqrt {a^2+b^2+c^2}}+1\right )\right ) c_3 t^{-i \sqrt {a^2+b^2+c^2}}}{2 \left (a^2+b^2+c^2\right )}\right \}\right \}\] ✓ Maple : cpu = 0.141 (sec), leaf count = 308
dsolve({a*t*diff(x(t),t) = b*c*(y(t)-z(t)), b*t*diff(y(t),t) = c*a*(z(t)-x(t)), c*t*diff(z(t),t) = a*b*(x(t)-y(t))})
\[\left \{x \left (t \right ) = c_{1}+c_{2} \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right )+c_{3} \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ), y \left (t \right ) = \frac {\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) \sqrt {a^{2}+b^{2}+c^{2}}\, c_{2} a c -\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) c_{3} a^{2} b -\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) \sqrt {a^{2}+b^{2}+c^{2}}\, c_{3} a c -\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) c_{2} a^{2} b +c_{1} b^{3}+c_{1} b \,c^{2}}{b \left (b^{2}+c^{2}\right )}, z \left (t \right ) = \frac {c_{1} c^{3}+\left (\left (-c_{2} \sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right )-c_{3} \cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right )\right ) a^{2}+c_{1} b^{2}\right ) c +\left (\sin \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) c_{3}-\cos \left (\sqrt {a^{2}+b^{2}+c^{2}}\, \ln \left (t \right )\right ) c_{2}\right ) b \sqrt {a^{2}+b^{2}+c^{2}}\, a}{\left (b^{2}+c^{2}\right ) c}\right \}\]