If $X$ is integrable then $\mathbb{E}[\sqrt{a_0 + a_1X^2}]$ is finite

Question

Let $X$ be an integrable random variable with possibly no moment of order 2.

Let $a_0$ and $a_1$ be two positive scalars. I want to prove that $\mathbb{E}[\sqrt{a_0 + a_1X^2}] < \infty$

I tried to use Jensen or $\sqrt{x} < 1 + x$, but to do that I have to assume that $\mathbb{E}[X^2] < \infty$ which is not the case.