  The cosmological observations of cosmic microwave background and large-scale
structure indicate that our universe has a nearly scaling invariant power
spectrum of the primordial perturbation. However, the exact origin for this
primordial spectrum is still unclear. Here, we propose the Weyl scaling
invariant $R^2+R^3$ gravity that gives rise to inflation that is responsible
for the primordial perturbation in the early universe. We develop both analytic
and numerical treatments on inflationary observables, and find this model gives
a distinctive scalar potential that can support two different patterns of
inflation. The first one is similar to that occurs in the pure $R^2$ model, but
with a wide range of tensor-to-scalar ratio $r$ from $\mathcal O(10^{-4})$ to
$\mathcal O(10^{-2})$. The other one is a new situation with not only slow-roll
inflation but also a short stage of oscillation-induced accelerating expansion.
Both patterns of inflation have viable parameter spaces that can be probed by
future experiments on cosmic microwave background and primordial gravitational
waves.
