This question has so long preoccupied my mind.Please help me to solve it.

Question: Assume $X_t$ described by the following stochastic differential equation $$dX_t^{\,\alpha}=\alpha X_t^{\,\alpha} dt+dW_t$$ where $W_t$ is a standard wiener process and $\alpha \in R$. How do I compute

$$ E\left[exp\left((\beta-\alpha)\int_{0}^{T}X_t^{\,\alpha}\,dX_t+\frac{\alpha^2}{2}\int_{0}^{T}X_t^{\,2\alpha}\,dt\right)\right]$$ for all $\beta<\frac{1}{T}$

This question has so long preoccupied my mind.Please help me to solve it.

Question: Assume $X_t$ described by the following stochastic differential equation $$dX_t^{\,\alpha}=\alpha X_t^{\,\alpha} dt+dW_t$$ where $W_t$ is a standard wiener process and $\alpha \in R$. How do I compute

$$ E\left[exp\left((\beta-\alpha)\int_{0}^{T}X_t^{\,\alpha}\,dX_t^\alpha+\frac{\alpha^2}{2}\int_{0}^{T}X_t^{\,2\alpha}\,dt\right)\right]$$ for all $\beta<\frac{1}{T}$