How to prove for that a given non-regular language its given super-set is also non-regular?
Since you already know that $L_1$ is nonregular, here is a simple proof.
Suppose that $L$ is regular. Then since $a^*ba^*$ is regular, the language $L \cap a^*ba^*$ should be regular. But $L \cap a^*ba^* = L_1$, which is not regular. Hence $L$ is nonregular.