Help me solve the indefinite integral

Question

So, I ran across this in my workbook and have no idea where to start. I can't recognize any basic form or any substitution which I can make. $$I=\int\frac{dx}{\sqrt[n]{(x-a)^{n+1}(x-b)^{n-1}}}$$

Answer 1

If $a = b$, then the integrand is $(x - a)^{-2}$, and $\int (x - a)^{-2}\, dx = -(x - a)^{-1} + C$. Suppose $a \neq b$ and let $x = a + (b - a)u$. Then $x - a = (b - a)u$, $x - b = (b - a)(u - 1)$, and $dx = (b - a)\, du$. So

\begin{align}\int \frac{dx}{\sqrt[n]{(x - a)^{n+1}(x - b)^{n-1}}} &= \int \frac{(b - a)\, du}{\sqrt[n]{[(b - a)u]^{n+1}[(b - a)(u - 1)]^{n-1}}}\\ & = (b - a)^{-1} \int u^{-\frac{n+1}{n}}(u - 1)^{-\frac{n-1}{n}}\, du\\ & = (b - a)^{-1} \int \left(\frac{u-1}{u}\right)^{\frac{1}{n} - 1} u^{-2}\, du\\ & = n(b - a)^{-1}\left(\frac{u-1}{u}\right)^{1/n} + C\\ & = n(b - a)^{-1} \left(\frac{x - b}{x - a}\right)^{1/n} + C. \end{align}