  A ferrimagnetic spin model composed of $S=1/2$ spin-dimers and $S=5/2$
spin-chains is studied by combining the bond-operator representation (for
$S=1/2$ spin-dimers) and Holstein-Primakoff transformation (for $S=5/2$ spins).
A finite interaction $J_{\rm DF}$ between the spin-dimer and the spin chain
makes the spin chains ordered antiferromagnetically and the spin dimers
polarized. The effective interaction between the spin chains, mediated by the
spin dimers, is calculated up to the third order. The staggered magnetization
in the spin dimer is shown proportional to $J_{\rm DF}$. It presents an
effective staggered field reacting on the spin chains. The degeneracy of the
triplons is lifted due to the chain magnetization and a mode with longitudinal
polarization is identified. Due to the triplon-magnon interaction, the
hybridized triplon-like excitations show different behaviors near the vanishing
$J_{\rm DF}$. On the other hand, the hybridized magnon-like excitations open a
gap $\Delta_A\sim J_{\rm DF}$. These results consist well with the experiments
on Cu$_{2}$Fe$_{2}$Ge$_{4}$O$_{13}$.
