I have found in the wikipedia page following generalisation of Doob's so-called $L^p$ inequality, for general nonnegative submartinagles $X_s$:
$$E[\sup_{0 \le s \le T} X_s] \le \frac{e(1 + E[X_T\log^+(X_T)])}{e -1}$$
But I couldn't find a proof of this on the internet, other than some vague hint. Here $\log^+$ indicates the positive part of the logarithm.
I have tried the following approach through Doob's maximal inequality, which has led me to nothing:
$$\int \sup_{0 \le s \le T} X_s dP = \int P(\sup_{0 \le s \le T} \ge t) dt \le \int ( \int X_T \frac{1_{\{X_T \ge t\}}}{t}dP )dt $$
But now if I change integration order I get $+\infty,$ which doesn't seem right. How to proceed?