$L = \{ww_rx, \text{ where } w,x \in \{a,b\}^*\}$ is definitely regular $w$ can always be considered to be empty string $(\epsilon).$ So, this just becomes $(a+b)^*$ language.
What about the following one?
Whether $L=\{ww_rx, \text{ where } w,x \in \{a,b\}^+\}$ regular?