  The ground-state phases of mixed diamond chains with ($S, \tau^{(1)},
\tau^{(2)})=(1/2,1/2,1)$, where $S$ is the magnitude of vertex spins, and
$\tau^{(1)}$ and $\tau^{(2)}$ are those of apical spins, are investigated. The
two apical spins in each unit cell are coupled by an exchange coupling
$\lambda$. The vertex spins are coupled with the top and bottom apical spins by
exchange couplings $1+\delta$ and $1-\delta$, respectively. Although this model
has an infinite number of local conservation laws for $\delta=0$, they are lost
for finite $\delta$. The ground-state phase diagram is determined using the
numerical exact diagonalization and DMRG method in addition to the analytical
approximations in various limiting cases. The phase diagram consists of a
nonmagnetic phase and several kinds of ferrimagnetic phases. We find two
different ferrimagnetic phases without spontaneous translational symmetry
breakdown. It is also found that the quantized ferrimagnetic phases with large
spatial periodicities present for $\delta=0$ are easily destroyed by small
$\delta$ and replaced by a partial ferrimagnetic phase. The nonmagnetic phase
is considered to be a gapless Tomonaga-Luttinger liquid phase based on the
recently extended Lieb-Schultz-Mattis theorem to the site-reflection invariant
spin chains and numerical diagonalization results.
