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$\ds{{\cal J}\pars{\alpha, \beta, c} \equiv \int_{0}^{\infty}{1 \over c + x}\,
{\beta^{\alpha} \over \Gamma\pars{\alpha}}\,x^{\alpha - 1}\expo{-\beta x}\,\dd x
= {\beta^{\alpha} \over \Gamma\pars{\alpha}}\,{\cal K}\pars{\alpha, \beta, c}}\etiqueta{1}$
\begin{align}
&\mbox{where}\\
{\cal K}\pars{\alpha, \beta, c}
& \equiv \int_{0}^{\infty}{x^{\alpha - 1}\expo{-\beta x} \over c + x}\,\dd x
=
\beta^{1 - \alpha}\int_{0}^{\infty}{\pars{\beta x}^{\alpha - 1}\expo{-\beta x} \over \beta x + \beta c}\,\dd\pars{\beta x}
\\[3mm]&=
\beta^{1 - \alpha}\int_{0}^{\infty}{x^{-\pars{1 - \alpha}}\expo{-x} \over x + \beta c}\,\dd x
=
\beta^{1 - \alpha}\,
{\Gamma\pars{1 - \bracks{1 - \alpha}}\Gamma\pars{1 - \alpha,\beta c}
\over
\expo{-\beta c}\pars{\beta c}^{\alpha}}
\\&\mbox{See G&R}\ {\bf 8.353}.3\,,\quad 7^{\underline{\rm a}}\ \mbox{ed. Page 900}. \\&\mbox{The result is valid whenever}\ \Re\alpha >0\ \mbox{and}\ \beta, c > 0.
\\[3mm]
{\cal K}\pars{a,b,c} &= \beta^{1 - 2\alpha}c^{-\alpha}\expo{\beta c}
\Gamma\pars{\alpha}\Gamma\pars{1 - \alpha,\beta c}
\end{align}
Sustituyendo este resultado en $\pars{1}$, encontramos:
$$
\color{#0000ff}{\large%
\int_{0}^{\infty}{x^{\alpha - 1}\expo{-\beta x} \over c + x}\,\dd x
=
\beta^{1 - \alpha}c^{-\alpha}\expo{\beta c}\Gamma\pars{1 - \alpha,\beta c}}
$$
Aviso que es un poco diferente de la OP propuesta de respuesta.
