MHB Indefinite integral in division form

[Elina_Gilbert]

I have the following integration -

$$\int \frac{2}{x - b \frac{x^{m - n + 1}}{(-x + 1)^m}} \, dx $$

To solve this I did the following -
$$\int \frac{1 - b \frac{x^{m - n}}{(-x + 1)^m}+1 + b \frac{x^{m - n}}{(-x + 1)^m}}{x(1 - b \frac{x^{m - n}}{(-x + 1)^m})} \, dx $$

Which gives me -

$$log(x) + C+ \int \frac{1 + b \frac{x^{m - n}}{(-x + 1)^m}}{x(1 - b \frac{x^{m - n}}{(-x + 1)^m})} \, dx $$

No matter what substitution I do, I couldn't solve the integral -

$$\int \frac{1 + b \frac{x^{m - n}}{(-x + 1)^m}}{x(1 - b \frac{x^{m - n}}{(-x + 1)^m})} \, dx $$

Can anyone please suggest what I did wrong? Please suggest me another method to solve this?

 

[Prove It]

May I ask what context this integral comes from?