Problem. I want to solve the following: $$ \mathbb{E}\left[\left(\int_0^x t^{\beta -1} e^{-\gamma t + (1-t)B_t} \, dt \right)^{1/\beta } \right], $$ where $B_t$ is a standard Brownian motion, $0<\beta<1$, and $0<\gamma<\infty$.
Here what I have tried. For integer power $1/\beta$, I can obtain the expectation simply. Let $1/\beta=1$, then by the Fubini Theorem, we have $$ \mathbb{E}\left[\int_0^x t^{\beta -1} e^{-\gamma t + (1-t)B_t}dt \right] = \int_0^x t^{\beta -1} e^{-\gamma t} \mathbb{E} \left[e^{(1-t)B_t}\right]dt \\ = \int_0^x t^{\beta -1} e^{-\gamma t} e^{\frac{1}{2}t(1-t)^2}dt, $$ where the last equality is due to the moment generating function of $B_t$ is $e^{t(1-t)^2/2}$. The final integral may be related to the incomplete gamma function, but not exactly. Similarly, I can obtain the expectation with other integers.
However, when the power is non-integer, I have no idea how to solve it.
Please let me know how to solve this.