Integral of the form $\int_{0}^{\infty}e^{-ax}e^{-x^{4}}dx$

Question

I'm searching a closed form for the integral of the form :

$$ \int_{0}^{\infty}e^{-ax}e^{-x^\frac{\alpha}{2}}dx $$

especially for $\alpha=8$

After several attempts, I find that the general form can be written on the basis of hypergeometric functions. what do you think ?