\[ \boxed { \begin {array}{rl} (x(t)-y(t)) (x(t)-z(t)) x'(t) &= f(t)\\ (y(t)-x(t)) (y(t)-z(t)) y'(t) &=f(t)\\ (z(t)-x(t)) (z(t)-y(t)) z'(t) &=f(t) \end {array} } \]
Mathematica: cpu = 0.011501 (sec), leaf count = 76 \[ \text {DSolve}\left [\left \{(x(t)-y(t)) (x(t)-z(t)) x'(t)=f(t),(y(t)-x(t)) (y(t)-z(t)) y'(t)=f(t),(z(t)-x(t)) (z(t)-y(t)) z'(t)=f(t)\right \},\{x(t),y(t),z(t)\},t\right ] \]
Maple: cpu = 1.030 (sec), leaf count = 1121 \[ \left \{ [ \left \{ x \left ( t \right ) =\int \!6\,{\frac {f \left ( t \right ) }{{{\it \_C1}}^{3}+11664\,{{\it \_C2}}^{2}-23328\,{\it \_C2} \,\int \!f \left ( t \right ) \,{\rm d}t+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2}} \left ( {{\it \_C1}}^{4}+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2}{\it \_C1}- 23328\,\int \!f \left ( t \right ) \,{\rm d}t{\it \_C1}\,{\it \_C2}+ 11664\,{\it \_C1}\,{{\it \_C2}}^{2}+ \left ( \left ( 1+108\,\sqrt {{ \frac { \left ( \int \!f \left ( t \right ) \,{\rm d}t-{\it \_C2} \right ) ^{2}}{{{\it \_C1}}^{3}+11664\,{{\it \_C2}}^{2}-23328\,{\it \_C2}\,\int \!f \left ( t \right ) \,{\rm d}t+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2}}}} \right ) \left ( {{\it \_C1}}^{3}+11664\,{{\it \_C2}}^{2}-23328\,{\it \_C2}\,\int \!f \left ( t \right ) \,{\rm d}t+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2} \right ) ^{2} \right ) ^{2/3} \right ) {\frac {1} {\sqrt [3]{ \left ( 1+108\,\sqrt {{\frac { \left ( \int \!f \left ( t \right ) \,{\rm d}t-{\it \_C2} \right ) ^{2}}{{{\it \_C1}}^{3}+11664\,{ {\it \_C2}}^{2}-23328\,{\it \_C2}\,\int \!f \left ( t \right ) \,{\rm d} t+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2}}}} \right ) \left ( {{\it \_C1}}^{3}+11664\,{{\it \_C2}}^{2}-23328\,{\it \_C2}\,\int \!f \left ( t \right ) \,{\rm d}t+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2} \right ) ^{2}}}}}\,{\rm d}t+{ \it \_C3},x \left ( t \right ) =\int \!3\,{\frac {f \left ( t \right ) }{{ {\it \_C1}}^{3}+11664\,{{\it \_C2}}^{2}-23328\,{\it \_C2}\,\int \!f \left ( t \right ) \,{\rm d}t+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2}} \left ( i\sqrt {3}{{\it \_C1}}^{4}+11664\,i \sqrt {3} \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2}{ \it \_C1}-23328\,i\sqrt {3}\int \!f \left ( t \right ) \,{\rm d}t{\it \_C1}\,{\it \_C2}+11664\,i\sqrt {3}{\it \_C1}\,{{\it \_C2}}^{2}-i \sqrt {3} \left ( \left ( 1+108\,\sqrt {{\frac { \left ( \int \!f \left ( t \right ) \,{\rm d}t-{\it \_C2} \right ) ^{2}}{{{\it \_C1}}^{3} +11664\,{{\it \_C2}}^{2}-23328\,{\it \_C2}\,\int \!f \left ( t \right ) \,{\rm d}t+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2}}}} \right ) \left ( {{\it \_C1}}^{3}+11664\,{{\it \_C2}}^ {2}-23328\,{\it \_C2}\,\int \!f \left ( t \right ) \,{\rm d}t+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2} \right ) ^{2} \right ) ^{2/3}-{{\it \_C1}}^{4}-11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2}{\it \_C1}+23328\,\int \!f \left ( t \right ) \,{\rm d}t{\it \_C1}\,{\it \_C2}-11664\,{\it \_C1}\,{{\it \_C2}}^{2}- \left ( \left ( 1+108\,\sqrt {{\frac { \left ( \int \!f \left ( t \right ) \,{\rm d}t-{\it \_C2} \right ) ^{2}}{{{\it \_C1}}^{3} +11664\,{{\it \_C2}}^{2}-23328\,{\it \_C2}\,\int \!f \left ( t \right ) \,{\rm d}t+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2}}}} \right ) \left ( {{\it \_C1}}^{3}+11664\,{{\it \_C2}}^ {2}-23328\,{\it \_C2}\,\int \!f \left ( t \right ) \,{\rm d}t+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2} \right ) ^{2} \right ) ^{2/3} \right ) {\frac {1}{\sqrt [3]{ \left ( 1+108\,\sqrt {{ \frac { \left ( \int \!f \left ( t \right ) \,{\rm d}t-{\it \_C2} \right ) ^{2}}{{{\it \_C1}}^{3}+11664\,{{\it \_C2}}^{2}-23328\,{\it \_C2}\,\int \!f \left ( t \right ) \,{\rm d}t+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2}}}} \right ) \left ( {{\it \_C1}}^{3}+11664\,{{\it \_C2}}^{2}-23328\,{\it \_C2}\,\int \!f \left ( t \right ) \,{\rm d}t+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2} \right ) ^{2}}}}}\,{\rm d}t+{\it \_C3},x \left ( t \right ) =\int \!-3\,{\frac {f \left ( t \right ) }{{{\it \_C1} }^{3}+11664\,{{\it \_C2}}^{2}-23328\,{\it \_C2}\,\int \!f \left ( t \right ) \,{\rm d}t+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d} t \right ) ^{2}} \left ( i\sqrt {3}{{\it \_C1}}^{4}+11664\,i\sqrt {3} \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2}{\it \_C1}- 23328\,i\sqrt {3}\int \!f \left ( t \right ) \,{\rm d}t{\it \_C1}\,{\it \_C2}+11664\,i\sqrt {3}{\it \_C1}\,{{\it \_C2}}^{2}-i\sqrt {3} \left ( \left ( 1+108\,\sqrt {{\frac { \left ( \int \!f \left ( t \right ) \,{\rm d}t-{\it \_C2} \right ) ^{2}}{{{\it \_C1}}^{3}+11664\,{{\it \_C2 }}^{2}-23328\,{\it \_C2}\,\int \!f \left ( t \right ) \,{\rm d}t+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2}}}} \right ) \left ( {{\it \_C1}}^{3}+11664\,{{\it \_C2}}^{2}-23328\,{\it \_C2}\, \int \!f \left ( t \right ) \,{\rm d}t+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2} \right ) ^{2} \right ) ^{2/3}+{{\it \_C1}}^{4}+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2}{\it \_C1}-23328\,\int \!f \left ( t \right ) \,{\rm d}t{ \it \_C1}\,{\it \_C2}+11664\,{\it \_C1}\,{{\it \_C2}}^{2}+ \left ( \left ( 1+108\,\sqrt {{\frac { \left ( \int \!f \left ( t \right ) \,{\rm d}t-{\it \_C2} \right ) ^{2}}{{{\it \_C1}}^{3}+11664\,{{\it \_C2 }}^{2}-23328\,{\it \_C2}\,\int \!f \left ( t \right ) \,{\rm d}t+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2}}}} \right ) \left ( {{\it \_C1}}^{3}+11664\,{{\it \_C2}}^{2}-23328\,{\it \_C2}\, \int \!f \left ( t \right ) \,{\rm d}t+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2} \right ) ^{2} \right ) ^{2/3} \right ) { \frac {1}{\sqrt [3]{ \left ( 1+108\,\sqrt {{\frac { \left ( \int \!f \left ( t \right ) \,{\rm d}t-{\it \_C2} \right ) ^{2}}{{{\it \_C1}}^{3} +11664\,{{\it \_C2}}^{2}-23328\,{\it \_C2}\,\int \!f \left ( t \right ) \,{\rm d}t+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2}}}} \right ) \left ( {{\it \_C1}}^{3}+11664\,{{\it \_C2}}^ {2}-23328\,{\it \_C2}\,\int \!f \left ( t \right ) \,{\rm d}t+11664\, \left ( \int \!f \left ( t \right ) \,{\rm d}t \right ) ^{2} \right ) ^{2} }}}}\,{\rm d}t+{\it \_C3} \right \} , \left \{ y \left ( t \right ) ={ \frac {1}{4\, \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{3}} \left ( 4\,x \left ( t \right ) \left ( {\frac {\rm d}{ {\rm d}t}}x \left ( t \right ) \right ) ^{3}+ \left ( {\frac {{\rm d}^{2} }{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) f \left ( t \right ) - \left ( {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) \right ) {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) -\sqrt {-16\,f \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{5}+ \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) ^{2} \left ( f \left ( t \right ) \right ) ^{2}-2\, \left ( { \frac {\rm d}{{\rm d}t}}f \left ( t \right ) \right ) \left ( {\frac { {\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) f \left ( t \right ) {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) \right ) ^{2} \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{2}} \right ) },y \left ( t \right ) ={\frac {1}{4\, \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{3}} \left ( 4\,x \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{3}+ \left ( { \frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) f \left ( t \right ) - \left ( {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) \right ) {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +\sqrt { -16\,f \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{5}+ \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) ^{2} \left ( f \left ( t \right ) \right ) ^{2 }-2\, \left ( {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) \right ) \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) f \left ( t \right ) {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) \right ) ^{2} \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) ^{2}} \right ) } \right \} , \left \{ z \left ( t \right ) ={ \frac { \left ( x \left ( t \right ) \right ) ^{2}{\frac {\rm d}{{\rm d}t }}x \left ( t \right ) -x \left ( t \right ) \left ( {\frac {\rm d}{ {\rm d}t}}x \left ( t \right ) \right ) y \left ( t \right ) -f \left ( t \right ) }{x \left ( t \right ) {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) - \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) y \left ( t \right ) }} \right \} ] \right \} \]