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How to evaluate/simplify integration with 4 parts?

How might I evaluate the following indefinite integral?

$$int k , x^a ,(1-x)^b ,(x-y)^c , (1-x+y)^d ,,dx$$

The aim is to get a function of $y$ once I put in my limits, but $k$, $a$, $b$, $c$, and $d$ are all constants. Any tips would be great.

For background, I am trying to evaluate $int f_{Z,W}(z,w)dw$ from this difference between two random Beta-distributed variables question.

Would I have to use integration by parts? How would this be done in practice? Wouldn’t it get very messy?

$$t = x^a; frac{dt}{dx} = frac{x^{a+1}}{a+1}$$

$$u = (1-x)^b; frac{du}{dx} = -frac{(1 – x)^{1 + b}}{1 + b}$$

$$v = (x-y)^c; frac{dv}{dx} = frac{(x – y)^{1 + c}}{1 + c}$$

$$w = (1-x+y)^d; frac{dw}{dx} = -frac{(1 – x + y)^{1 + d}}{1 + d}$$

I noticed that Wolfram (please don’t judge me) couldn’t do it within its time limit, so I guess this is more complicated than I had anticipated. But maybe there’s a ‘trick’ or rule that can simplify my problem to make it more digestible for either a human or machine to evaluate?

Mathematics Asked on November 21, 2021

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