Evaluate the Integral \(\int x^{km+1}(a+x^{k})^{-m} \mathrm{d}x\)

How to evaluate
\begin{equation}
\int x^{km+1}(a+x^{k})^{-m} \mathrm{d}x
\end{equation}
was part of a question at Mathematics Stack Exchange. Here is my solution.

Let \(y = -\frac{x^{k}}{a}\)
\begin{align}
\int x^{km+1}(a+x^{k})^{-m} \mathrm{d}x &= \frac{1}{k}a^{2/k}(-1)^{m+2/k} \int (1-y)^{-m} y^{m-1+2/k} \mathrm{d}y \\
&= \frac{1}{k}a^{2/k}(-1)^{m+2/k} \mathrm{B}_{y}\left(m+\frac{2}{k},1-m\right) \\
&= \frac{1}{k}a^{2/k}(-1)^{m+2/k} \frac{y^{m+2/k}}{m+2/k} \,{}_{2}\mathrm{F}_{1}\left(m+\frac{2}{k},m;m+1+\frac{2}{k};y\right) \\
&= \frac{1}{km+2} \,\frac{1}{a^{m}} x^{km+2} \,{}_{2}\mathrm{F}_{1}\left(m+\frac{2}{k},m;m+1+\frac{2}{k};-\frac{x^{k}}{a}\right)
\end{align}

Notes:
1.
\begin{equation}
\mathrm{B}_{z}(p,q) = \int_{0}^{z} t^{p-1} (1-t)^{q-1} \mathrm{d}t
\end{equation}
is the incomplete beta function.
2.
\begin{equation}
\mathrm{B}_{z}(p,q) = \frac{z^{p}}{p} \,{}_{2}\mathrm{F}_{1}(p,1-q;p+1;z)
\end{equation}
is the incomplete beta function in terms of Gauss’s hypergeometric function.

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