Integral with the incomplete upper gamma function

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Can anyone help me integrate this?

$$\int_0^1 \frac{1}{x^{1/p}} \left[\frac{1-x^{1/p}}{x^{1/p}} \right]^{m/n-1} \Gamma\;\left(A, \left[\frac{1-x^{1/p}}{x^{1/p}} \right]^{1/n}\right) \,\mathrm{d}x,$$

where $p$, $m$, $n$ and $A$ are real and positive, and $\Gamma(\cdot,\cdot)$ denotes the incomplete upper gamma function.