How to find this integral $I=\int_{0}^{1} t^{-1/2}\left(1-\frac{t}{1+4y}\right)^{-1/2-ix}dt$

Question

Question:

Find the closed form of

$$I(x,y)=\int_{0}^{1}t^{-1/2}\left(1-\dfrac{t}{1+4y}\right)^{-1/2-ix}dt$$

where $i^2=-1$

I have used Wolfram Alpha but it can't help me out. How to find it?

Thank you

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I have explained why I can't comment, because in China we can't comment in MSE, (I don't know why?) Now, I can only post questions and I also can answer questions , but I can't comment.

china math110 also said that he can't comment. Here is the proof: How find all positive real $\beta$ such A finite number of $\left|\frac{p}{q}-\sqrt{2}\right|<\frac{\beta}{q^2}$

How prove $f(a_{i})=0$ if $\int_{0}^{1}x^kf(x)dx=0,k=1,2,3,\cdots,n$

Answer 1

If your x and y are independent of t, then the answer is simply an incomplete beta function.