Moment Estimate for Simple Linear SDE

Question

In a paper I am reading, it is claimed that the SDE $$ dX_t = b X_t dt + (\sigma X_t + \beta_t) dB_t , \quad t \in [0,T],$$ satisfies $$ E\left[ \sup_{t \in [0,T]} |X_t|^p \right] < \infty, \forall p \in [2,\infty), $$ where $b, \sigma \in \mathbb{R}$ are constants and $\beta$ is a process satisfying $$ E \left[ \int_0^T | \beta_t |^2 dt \right] < \infty . $$ Is this true?