probability - Concentration inequality to bound expectation

Let $X$ be a non-negative r.v. so that

$$P(X \geq t) \leq C \exp\bigg\{\frac{-t^2/2}{\sigma^2 + bt}\bigg\}$$
for positive $\sigma, b$ and $C\geq 1$. Show that

$$E[X] \leq 2\sigma (\sqrt{\pi} + \sqrt{\log C}) + 4b(1 + \log C)$$

I know how to usually get concentration bound or bound the deviation from the mean, but no idea how to bound this expectation here. Any help would be really appreciated.