I need to bound the expectation of a nonnegative random variable that satisfies a Poisson-type tail bound:

$\mathbb{P}( X \geq t ) \leq \min( d \cdot (\frac{a}{t} )^{t}, \ 1)$ for $t > 0$ 

where $a > 0$ and $d \geq 3$.  The answer appears to be

$\mathbb{E} X \leq {\rm const} \cdot \max( a,\ \frac{\log d}{\log \log d} )$

Is there a simple way to do this?  The references I checked helpfully told me that the calculation is "standard".  My approach is too awful to set in print.