\[ \left \{f(t) (a x(t)+b y(t))+x'(t)=g(t),f(t) (c x(t)+d y(t))+y'(t)=h(t)\right \} \] ✗ Mathematica : cpu = 0.0055809 (sec), leaf count = 0 , could not solve
DSolve[{f[t]*(a*x[t] + b*y[t]) + Derivative[1][x][t] == g[t], f[t]*(c*x[t] + d*y[t]) + Derivative[1][y][t] == h[t]}, {x[t], y[t]}, t]
✓ Maple : cpu = 1.095 (sec), leaf count = 2606
\[ \left \{ \left \{ x \left ( t \right ) ={{\rm e}^{\int \!-{\frac {f \left ( t \right ) }{2\,a+2\,d} \left ( \tan \left ( {\frac {-1+ \left ( a+d \right ) \int \!f \left ( t \right ) \,{\rm d}t}{2\, \left ( a+d \right ) ^{2}}\sqrt {-{a}^{4}-4\,{a}^{2}bc+2\,{a}^{2}{d}^{2}-8\,abcd-4\,bc{d}^{2}-{d}^{4}}} \right ) \sqrt {-{a}^{4}-4\,{a}^{2}bc+2\,{a}^{2}{d}^{2}-8\,abcd-4\,bc{d}^{2}-{d}^{4}}+ \left ( a+d \right ) ^{2} \right ) }\,{\rm d}t}}{\it \_C2}+{{\rm e}^{\int \!-{\frac {f \left ( t \right ) }{2\,a+2\,d} \left ( \tan \left ( {\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2\,a+2\,d}\sqrt {-{a}^{4}-4\,{a}^{2}bc+2\,{a}^{2}{d}^{2}-8\,abcd-4\,bc{d}^{2}-{d}^{4}}} \right ) \sqrt {-{a}^{4}-4\,{a}^{2}bc+2\,{a}^{2}{d}^{2}-8\,abcd-4\,bc{d}^{2}-{d}^{4}}+ \left ( a+d \right ) ^{2} \right ) }\,{\rm d}t}}{\it \_C1}-2\,{\frac {a+d}{\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }} \left ( \int \!-{\frac {dg \left ( t \right ) \left ( f \left ( t \right ) \right ) ^{2}-bh \left ( t \right ) \left ( f \left ( t \right ) \right ) ^{2}+ \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) \right ) f \left ( t \right ) -g \left ( t \right ) {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) }{ \left ( f \left ( t \right ) \right ) ^{2}}{{\rm e}^{{\frac {1}{2\,a+2\,d} \left ( \sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }\int \!f \left ( t \right ) \tan \left ( {\frac {\int \!f \left ( t \right ) \,{\rm d}t\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }}{2\,a+2\,d}} \right ) \,{\rm d}t+\int \!f \left ( t \right ) \,{\rm d}t \left ( a+d \right ) ^{2} \right ) }}} \left ( -\tan \left ( {\frac {\int \!f \left ( t \right ) \,{\rm d}t\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }}{2\,a+2\,d}} \right ) +\tan \left ( 1/2\,{\frac { \left ( -1+ \left ( a+d \right ) \int \!f \left ( t \right ) \,{\rm d}t \right ) \sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }}{ \left ( a+d \right ) ^{2}}} \right ) \right ) ^{-1}}\,{\rm d}t{{\rm e}^{{\frac {1}{2\,a+2\,d} \left ( \sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }\int \!f \left ( t \right ) \tan \left ( 1/2\,{\frac { \left ( -1+ \left ( a+d \right ) \int \!f \left ( t \right ) \,{\rm d}t \right ) \sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }}{ \left ( a+d \right ) ^{2}}} \right ) \,{\rm d}t+\int \!f \left ( t \right ) \,{\rm d}t \left ( a+d \right ) ^{2} \right ) }}}-\int \!-{\frac {dg \left ( t \right ) \left ( f \left ( t \right ) \right ) ^{2}-bh \left ( t \right ) \left ( f \left ( t \right ) \right ) ^{2}+ \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) \right ) f \left ( t \right ) -g \left ( t \right ) {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) }{ \left ( f \left ( t \right ) \right ) ^{2}}{{\rm e}^{{\frac {1}{2\,a+2\,d} \left ( \sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }\int \!f \left ( t \right ) \tan \left ( 1/2\,{\frac { \left ( -1+ \left ( a+d \right ) \int \!f \left ( t \right ) \,{\rm d}t \right ) \sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }}{ \left ( a+d \right ) ^{2}}} \right ) \,{\rm d}t+\int \!f \left ( t \right ) \,{\rm d}t \left ( a+d \right ) ^{2} \right ) }}} \left ( -\tan \left ( {\frac {\int \!f \left ( t \right ) \,{\rm d}t\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }}{2\,a+2\,d}} \right ) +\tan \left ( 1/2\,{\frac { \left ( -1+ \left ( a+d \right ) \int \!f \left ( t \right ) \,{\rm d}t \right ) \sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }}{ \left ( a+d \right ) ^{2}}} \right ) \right ) ^{-1}}\,{\rm d}t{{\rm e}^{{\frac {1}{2\,a+2\,d} \left ( \sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }\int \!f \left ( t \right ) \tan \left ( {\frac {\int \!f \left ( t \right ) \,{\rm d}t\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }}{2\,a+2\,d}} \right ) \,{\rm d}t+\int \!f \left ( t \right ) \,{\rm d}t \left ( a+d \right ) ^{2} \right ) }}} \right ) {{\rm e}^{{\frac {1}{2\,a+2\,d} \left ( -\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }\int \!f \left ( t \right ) \tan \left ( 1/2\,{\frac { \left ( -1+ \left ( a+d \right ) \int \!f \left ( t \right ) \,{\rm d}t \right ) \sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }}{ \left ( a+d \right ) ^{2}}} \right ) \,{\rm d}t-\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }\int \!f \left ( t \right ) \tan \left ( {\frac {\int \!f \left ( t \right ) \,{\rm d}t\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }}{2\,a+2\,d}} \right ) \,{\rm d}t-2\,\int \!f \left ( t \right ) \,{\rm d}t \left ( a+d \right ) ^{2} \right ) }}}},y \left ( t \right ) =-{\frac {1}{b \left ( a+d \right ) f \left ( t \right ) } \left ( {{\rm e}^{{\frac {1}{2\,a+2\,d} \left ( \sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }\int \!f \left ( t \right ) \tan \left ( {\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2\,a+2\,d}\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }} \right ) \,{\rm d}t+\int \!f \left ( t \right ) \,{\rm d}t \left ( a+d \right ) ^{2} \right ) }}}{{\rm e}^{{\frac {1}{2\,a+2\,d} \left ( -\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }\int \!f \left ( t \right ) \tan \left ( {\frac {-1+ \left ( a+d \right ) \int \!f \left ( t \right ) \,{\rm d}t}{2\, \left ( a+d \right ) ^{2}}\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }} \right ) \,{\rm d}t-\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }\int \!f \left ( t \right ) \tan \left ( {\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2\,a+2\,d}\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }} \right ) \,{\rm d}t-2\,\int \!f \left ( t \right ) \,{\rm d}t \left ( a+d \right ) ^{2} \right ) }}}f \left ( t \right ) \left ( a+d \right ) \left ( {a}^{2}-\tan \left ( {\frac {-1+ \left ( a+d \right ) \int \!f \left ( t \right ) \,{\rm d}t}{2\, \left ( a+d \right ) ^{2}}\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }} \right ) \sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }-{d}^{2} \right ) \int \!-{\frac {dg \left ( t \right ) \left ( f \left ( t \right ) \right ) ^{2}-bh \left ( t \right ) \left ( f \left ( t \right ) \right ) ^{2}+ \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) \right ) f \left ( t \right ) -g \left ( t \right ) {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) }{ \left ( f \left ( t \right ) \right ) ^{2}}{{\rm e}^{{\frac {1}{2\,a+2\,d} \left ( \sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }\int \!f \left ( t \right ) \tan \left ( {\frac {-1+ \left ( a+d \right ) \int \!f \left ( t \right ) \,{\rm d}t}{2\, \left ( a+d \right ) ^{2}}\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }} \right ) \,{\rm d}t+\int \!f \left ( t \right ) \,{\rm d}t \left ( a+d \right ) ^{2} \right ) }}} \left ( -\tan \left ( {\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2\,a+2\,d}\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }} \right ) +\tan \left ( {\frac {-1+ \left ( a+d \right ) \int \!f \left ( t \right ) \,{\rm d}t}{2\, \left ( a+d \right ) ^{2}}\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }} \right ) \right ) ^{-1}}\,{\rm d}t-{{\rm e}^{{\frac {1}{2\,a+2\,d} \left ( \sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }\int \!f \left ( t \right ) \tan \left ( {\frac {-1+ \left ( a+d \right ) \int \!f \left ( t \right ) \,{\rm d}t}{2\, \left ( a+d \right ) ^{2}}\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }} \right ) \,{\rm d}t+\int \!f \left ( t \right ) \,{\rm d}t \left ( a+d \right ) ^{2} \right ) }}}{{\rm e}^{{\frac {1}{2\,a+2\,d} \left ( -\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }\int \!f \left ( t \right ) \tan \left ( {\frac {-1+ \left ( a+d \right ) \int \!f \left ( t \right ) \,{\rm d}t}{2\, \left ( a+d \right ) ^{2}}\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }} \right ) \,{\rm d}t-\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }\int \!f \left ( t \right ) \tan \left ( {\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2\,a+2\,d}\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }} \right ) \,{\rm d}t-2\,\int \!f \left ( t \right ) \,{\rm d}t \left ( a+d \right ) ^{2} \right ) }}}f \left ( t \right ) \left ( a+d \right ) \left ( {a}^{2}-\tan \left ( {\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2\,a+2\,d}\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }} \right ) \sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }-{d}^{2} \right ) \int \!-{\frac {dg \left ( t \right ) \left ( f \left ( t \right ) \right ) ^{2}-bh \left ( t \right ) \left ( f \left ( t \right ) \right ) ^{2}+ \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) \right ) f \left ( t \right ) -g \left ( t \right ) {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) }{ \left ( f \left ( t \right ) \right ) ^{2}}{{\rm e}^{{\frac {1}{2\,a+2\,d} \left ( \sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }\int \!f \left ( t \right ) \tan \left ( {\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2\,a+2\,d}\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }} \right ) \,{\rm d}t+\int \!f \left ( t \right ) \,{\rm d}t \left ( a+d \right ) ^{2} \right ) }}} \left ( -\tan \left ( {\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2\,a+2\,d}\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }} \right ) +\tan \left ( {\frac {-1+ \left ( a+d \right ) \int \!f \left ( t \right ) \,{\rm d}t}{2\, \left ( a+d \right ) ^{2}}\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }} \right ) \right ) ^{-1}}\,{\rm d}t+{\frac {1}{2}\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) } \left ( {\it \_C2}\,f \left ( t \right ) \left ( {a}^{2}-\tan \left ( {\frac {-1+ \left ( a+d \right ) \int \!f \left ( t \right ) \,{\rm d}t}{2\, \left ( a+d \right ) ^{2}}\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }} \right ) \sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }-{d}^{2} \right ) {{\rm e}^{\int \!-{\frac {f \left ( t \right ) }{2\,a+2\,d} \left ( \tan \left ( {\frac {-1+ \left ( a+d \right ) \int \!f \left ( t \right ) \,{\rm d}t}{2\, \left ( a+d \right ) ^{2}}\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }} \right ) \sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }+ \left ( a+d \right ) ^{2} \right ) }\,{\rm d}t}}+{\it \_C1}\,f \left ( t \right ) \left ( {a}^{2}-\tan \left ( {\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2\,a+2\,d}\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }} \right ) \sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }-{d}^{2} \right ) {{\rm e}^{\int \!-{\frac {f \left ( t \right ) }{2\,a+2\,d} \left ( \tan \left ( {\frac {\int \!f \left ( t \right ) \,{\rm d}t}{2\,a+2\,d}\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }} \right ) \sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }+ \left ( a+d \right ) ^{2} \right ) }\,{\rm d}t}}-2\,g \left ( t \right ) \left ( a+d \right ) \right ) } \right ) {\frac {1}{\sqrt {- \left ( a+d \right ) ^{2} \left ( {a}^{2}-2\,ad+4\,bc+{d}^{2} \right ) }}}} \right \} \right \} \]