\[ \left \{(x(t)-y(t)) (x(t)-z(t)) x'(t)=f(t),(y(t)-x(t)) (y(t)-z(t)) y'(t)=f(t),(z(t)-x(t)) (z(t)-y(t)) z'(t)=f(t)\right \} \] ✓ Mathematica : cpu = 0.139265 (sec), leaf count = 2168
DSolve[{(x[t] - y[t])*(x[t] - z[t])*Derivative[1][x][t] == f[t], (-x[t] + y[t])*(y[t] - z[t])*Derivative[1][y][t] == f[t], (-x[t] + z[t])*(-y[t] + z[t])*Derivative[1][z][t] == f[t]},{x[t], y[t], z[t]},t]
\[\left \{\left \{x(t)\to \frac {c_1}{3}+\frac {\sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (3 c_2-c_1{}^2\right )}{3 \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}},y(t)\to \frac {1}{2} \left (\frac {2 c_1}{3}-\sqrt {\left (-\frac {2 c_1}{3}+\frac {\sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (3 c_2-c_1{}^2\right )}{3 \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}\right ){}^2-4 \left (\left (\frac {c_1}{3}+\frac {\sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (3 c_2-c_1{}^2\right )}{3 \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}\right ){}^2-c_1 \left (\frac {c_1}{3}+\frac {\sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (3 c_2-c_1{}^2\right )}{3 \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}\right )+c_2\right )}-\frac {\sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (3 c_2-c_1{}^2\right )}{3 \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}\right ),z(t)\to -\frac {\sqrt [3]{2} c_1{}^2}{3 \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}+\frac {c_1{}^2}{3\ 2^{2/3} \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}+\frac {c_1}{3}+\frac {1}{2} \sqrt {\left (-\frac {2 c_1}{3}+\frac {\sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (3 c_2-c_1{}^2\right )}{3 \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}\right ){}^2-4 \left (\left (\frac {c_1}{3}+\frac {\sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (3 c_2-c_1{}^2\right )}{3 \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}\right ){}^2-c_1 \left (\frac {c_1}{3}+\frac {\sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (3 c_2-c_1{}^2\right )}{3 \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}\right )+c_2\right )}-\frac {\sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}{6 \sqrt [3]{2}}+\frac {\sqrt [3]{2} c_2}{\sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}-\frac {c_2}{2^{2/3} \sqrt [3]{2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]+\sqrt {4 \left (3 c_2-c_1{}^2\right ){}^3+\left (2 c_1{}^3-9 c_2 c_1+27 c_3+27 \int _1^tf(K[1])dK[1]\right ){}^2}}}\right \}\right \}\] ✓ Maple : cpu = 1.825 (sec), leaf count = 899
dsolve({(x(t)-y(t))*(x(t)-z(t))*diff(x(t),t) = f(t), (y(t)-x(t))*(y(t)-z(t))*diff(y(t),t) = f(t), (z(t)-x(t))*(z(t)-y(t))*diff(z(t),t) = f(t)})
\[\text {Expression too large to display}\]