I've been trying to find an asymptotic expansion of the following series

$$C(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$$

and

$$L(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n(n!{\sqrt{n}}) }$$

around $+\infty$, in the from

$$\exp(x^2)\Big(1+\frac{a_1}{x}+\frac{a_2}{x^2} + .. +\frac{a_k}{x^k}\Big) + O\Big(\frac{\exp(x^2)}{x^{k+1}}\Big)$$

where $x$ is a positive real number. As far as I progressed, I obtained only

$$C(x) = \exp(x^2) + \frac{\exp(x^2)}{x} + O\Big(\frac{\exp(x^2)}{x}\Big).$$

I tried to use ideas from https://math.stackexchange.com/questions/484367/upper-bound-for-an-infinite-series-with-a-square-root?rq=1, https://math.stackexchange.com/questions/115410/whats-the-sum-of-sum-limits-k-1-infty-fractkkk, https://math.stackexchange.com/questions/378024/infinite-series-involving-sqrtn?noredirect=1&lq=1, but I was unable to make them work in my case.

Any suggestions would be greatly appreciated!

(If someone has a solid culture in this kind of things, is there are any specific names for $C(x) $ and $L(x) $ ?).

PS:

This question was asked on the math.SE but was closed as duplicate of https://math.stackexchange.com/questions/2117742/lim-x-rightarrow-infty-sqrtxe-x-left-sum-k%ef%bc%9d1-infty-fracxk/2123100#2123100. However, the latter question provides only the first term of the asymptotic expansion and does not address sufficiently the problem considered here.