Problem
How can I integrate $$\int_{0}^{1} x^a (1-x)^b (1-2x)^c \, \mathrm dx $$ where $a,b,c$ are some constants.
Solution Attempt
I've tried using Beta Function and integrate it by parts, however, it didn't work.
2026-02-22 20:40:38.1771792838
Integrating $\int_{0}^{1} x^a (1-x)^b (1-2x)^c \, \mathrm dx $
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This is not a Beta function in the general case.
As stated in the comments, it's the Hypergeometric function.
The Euler integral for this function has the following form:
$${_2 F_1} (\alpha,\beta;\gamma;z)=\frac{1}{B(\beta,\gamma-\beta)} \int_0^1 t^{\beta-1} (1-t)^{\gamma-\beta-1} (1- zt)^{-\alpha} dt$$
Where Beta function still appears.
Now apply this definition for your case.
Note that $\alpha$ and $\beta$ are interchangeable in this function, so you get two different integral definitions, which of course give the same values.
For the other ways to express the function and for its properties, see for example https://en.wikipedia.org/wiki/Hypergeometric_function