Question about Square Integrable Semimartingale

Question

Let $B = (B_t)_{t \in [0,T]}$ be a Brownian motion and $\alpha = (\alpha_t)_{t \in [0,T]} $ be progressively measurable. Let $$ X = \int_0^\cdot \alpha_t dt + \int_0^\cdot \alpha_tdB_t. $$ If $\alpha$ satisfies $$ E\left[ \int_0^T |\alpha_t|^2 dt \right] < \infty, $$ then $X$ is bounded in $L^2.$ Is the reverse implication also true?