Let $B_t$ be a standard Brownian motion. For some non-integer $\alpha>0$, what is $$ \mathbb{E}\left[\left(\int_0^x e^{B_t} dt \right)^{\alpha} \right]? $$
Notes For the integer $\alpha$, I can solve this problem easily by Fubini Theorem as follows: $$ \mathbb{E}\left[\int_0^x \cdots \int_0^x e^{B_{t_1,}+\cdots+B_{t_\alpha}} dt_1 \cdots dt_{\alpha} \right] = \int_0^x \cdots \int_0^x \mathbb{E}\left[e^{B_{t_1,}+\cdots +B_{t_\alpha}}\right] dt_1 \cdots dt_{\alpha}. $$ The term $\mathbb{E}\left[e^{B_{t_1,}+\cdots +B_{t_\alpha}}\right]$ can be easily obtained using the moment generating function of Gaussian.
This question is derived from the other question here. It would be generalized to what is $\mathbb{E}\left[\left(\int_0^x X_t dt \right)^{\alpha} \right]$ for some stochastic process $X_t$?
Thanks.