I was trying to find the asymptotic expansion for $$\int_0^1 \sqrt{t(1-t)}(t+a)^{-x} \; \mathrm{d}t,$$ for $a>0$ as $x \rightarrow \infty$. I have already tried re-writing $$(t+a)^{-x}=\exp(- x\log(t+a)).$$ However, by using Laplace's method for asymptotic expansions I get that $f(t) = \sqrt{t(1-t)}$ vanishes everywhere.
Any ideas are appreciated/welcome. Thank you.