  Geodesics of both lightrays and timelike particles with nonzero mass are
deflected in a gravitational field. In this work we apply the perturbative
method developed in Ref. \cite{Jia:2020dap} to compute the deflection angle of
both null and timelike rays in the weak field limit for four spacetimes. We
obtained the deflection angles for the Bardeen spacetime to the eleventh order
of $m/b$ where $m$ is the ADM mass and $b$ is the impact parameter, and for the
Hayward, Janis-Newman-Winicour and Einstein-Born-Infeld spacetimes to the
ninth, seventh and eleventh order respectively. The effect of the impact
parameter $b$, velocity $v$ and spacetime parameters on the deflection angle
are analyzed in each of the four spacetimes. It is found that in general, the
perturbative deflection angle depends on and only on the asymptotic behavior of
the metric functions, and in an order-correlated way. Moreover, it is shown
that although these deflection angles are calculated in the large $b/m$ limit,
their minimal valid $b$ can be as small as a few $m$'s as long as the order is
high enough. At these impact parameters, the deflection angle itself is also
found large. As velocity decreases, the deflection angle in all spacetime
studied increases. For a given $b$, if the spacetime parameters allows a
critical velocity $v_c$, then the perturbative deflection angle will deviate
from its true value as $v$ decreases to $v_c$. It is also found that if the
variation of spacetime parameters can only change the spacetime qualitatively
at small but not large radius, then these spacetime parameter will not cause a
qualitative change of the deflection angle, although its value is still
quantitatively affected. The application and possible extension of the work are
discussed.
