Find a random variable which is the function of X and U and has a moment generating function which is a definite integral

Question

I'm self teaching probability and came across this interesting question:

Find a random variable which is the function of X and U and which has the moment generating function: $$\int_{0}^{1} M_X(tu) du $$

Here $U$ is uniformly distributed on $(0,1)$.

Would really appreciate