Taylor series of a rational function

Question

I am facing some complicated integral, which part of it is

$$\frac{z^{M-1}}{(1+(\eta z)^n)^p}$$

I think if I find the taylor series of this part the integral might be solved. So, can someone help me please to find the taylor series of this?

Note: $${\eta, n, p}$$ are real positive numbers, while $${M}$$ is a positive integer.

Sample plots for different values of these parameters is attached.

Answer 1

Hint

Just focuse on the denominator and define $y={(\eta z})^n$. So $$\frac{1}{(1+y)^p} \simeq 1-p y+\frac{1}{2} \left(p^2+p\right) y^2+\frac{1}{6} \left(-p^3-3 p^2-2 p\right) y^3+\frac{1}{24} \left(p^4+6 p^3+11 p^2+6 p\right) y^4+\frac{1}{120} \left(-p^5-10 p^4-35 p^3-50 p^2-24 p\right) y^5+O\left(y^6\right) $$ Replace now $y$ by its definition and multiply by the numerator.

I am sure that you can take from here.