$t \mapsto \int_{[0, t]} ((1+y)(1+t-y))^{-p} \in \mathscr{O} (t^{-p})$

Question

Show that for $p > 1$

$$ t \mapsto \int_{[0, t]} ((1+y)(1+t-y))^{-p} \in \mathscr{O} (t^{-p}) \quad\text{for} \; t \rightarrow \infty. $$

The best I achieved so far is that it is in $\mathscr{O} (t^{1-p})$.