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?