Prove that $X_{t} = \int_{0}^{t} e^{W_{s}^2} \,dW_{s}$ is a martingale

Question

Let $W_{t}$ be a Brownian motion, how can I prove that the process $X_{t} = \int_{0}^{t} e^{W_{s}^2} \,dW_{s}$ is a martingale or not? I've found that this process can be a martingale when t $\in[0,1/4]$ and a local martingale when t $\in[1/4,\infty]$, since $E[e^{2W_{t}^2}] \propto \frac{1}{\sqrt{1-4t}}$, but I don't know how to prove $E[X_{t}|F_{s}] = X_{s}$