Can anybody help me in solving this integration?

Question

i've tried all the existing methods to solve it but failed. if anybody can suggest any way to solve this that would be a great help to me. $$\int\frac{r(r^p-1)^q}{\left(1-\frac{r-1}{r_{\rm{max}}-1}\right)^q}dr$$

Answer 1

Note that your integral can be written as: $$(r_{max}-1)^q\int r\left(\frac{r^p - 1}{r_{max} - r}\right)^qdr$$ Which, if $q$ is a known integer, this can be expressed as sum of hypergeometric functions multiplied by powers of $r$, by expanding the numerator.

I doubt however that you can derive a generalized formula.

Answer 2

Assuming that $p$ and $q$ are not dependent on $r$, we have $$\int\frac{r(r^p-1)^q}{\left(1-\frac{r-1}{r_{\rm{max}}-1}\right)^q}dr$$ $$=\int r\left(\frac{r^p-1}{1-\frac{r-1}{r_{\rm{max}}-1}}\right)^q dr$$ $$=\int r\left(\frac{(r^p-1)(r_{\rm{max}}-1)}{r_{\rm{max}}-r}\right)^q dr$$ $$=(r_{\rm{max}}-1)^q\int r\left(\frac{r^p-1}{r_{\rm{max}}-r}\right)^q dr$$ Note that this integral does not have a solution in terms of elementary mathematical functions.