  Tripartite mutual information (TMI) is an efficient observable to quantify
the ability of scrambler for unitary time-evolution operator with quenched
many-body Hamiltonian. In this paper, we give numerical demonstrations of the
TMI in disorder-free (translational invariant) spin models with 3-body and
4-body multiple-spin interactions. The dynamical behavior of the TMI of these
models does {\it not} exhibit linear light-cone for sufficiently strong
interactions. In early-time evolution, the TMI displays distinct negative
increase behavior fitted by a logarithmic-like function. This is in contrast to
the conventional linear light-cone behavior present in the XXZ model and its
near integrable vicinity. The late-time evolution of the TMI in finite-size
systems is also numerically investigated. The multiple-spin interactions make
the system nearly integrable and weakly suppress the spread of information and
scrambling. The observation of the late-time value of the TMI indicates that
the scrambling nature of the system changes by interactions and this change can
be characterized by a phase-transition-like behavior of the TMI, reflecting the
system's integrability and breaking of eigenstate thermalization hypothesis.
