  We use the diagrammatic $T$-matrix approach to analyze the three-body
scattering problem between two identical fermions and a third particle (which
could be a different species of fermion or a boson). We calculate the s-wave
dimer-atom scattering length for all mass ratios, and our results exactly match
the results of Petrov. In particular, we list the exact dimer-atom scattering
lengths for all available two-species Fermi-Fermi and Bose-Fermi mixtures. In
addition, unlike that of the equal-mass particles case where the three-body
scattering $T$-matrix decays monotonically as a function of the outgoing
momentum, we show that, after an initial rapid drop, this function changes sign
and becomes negative at large momenta and then decays slowly to zero when the
mass ratio of the fermions to the third particle is higher than a critical
value (around 6.5). As the mass ratio gets higher, modulations of the
$T$-matrix become more apparent with multiple sign changes, related to the
"fall of a particle to the center" phenomenon and to the emergence of
three-body Efimov bound states.
