I am trying to solve the following integral involving Meijer-g function
$\int_0^{\infty} \left(\varrho x + \nu\right)^{p+\frac{1}{2}-\frac{k}{2}} G_{1,3}^{3,0}\left(\begin{matrix} \frac{\kappa^2}{2}\\ \frac{m}{2}-\frac{1}{2},n ,\frac{\kappa^2}{2}-1\end{matrix} , Ax\right) \, dx $
Should $\nu$ be zero, I could solve it very easily, but I don't know how to solve it when $\nu$ is not zero.
So far I have tried to do using substitution and by parts.
Plugin in this expression into mathematica:
Integrate[((rx + v)^(p + 1/2 + k/2)) MeijerG[{{}, {kappa^2/2}}, {{m/2 - 1/2, n, kappa^2/2 - 1}, {}}, A*x], {x, 0, [Infinity]}]
I get the following answer:
$ v^{\frac{1}{2} (k+2 p+1)} \left(\frac{2 A^{\frac{1}{2} (-k-2 p-3)} \Gamma \left(\frac{1}{2} (k+m+2 p+2)\right) \Gamma \left(\frac{k}{2}+n+p+\frac{3}{2}\right) \left(\frac{r}{v}\right)^{\frac{1}{2} (k+2 p+1)} \, _2F_3\left(-\frac{k}{2}-p-\frac{1}{2},-\frac{\kappa^2}{2}-p-\frac{k}{2}-\frac{1}{2};-\frac{\kappa^2}{2}-p-\frac{k}{2}+\frac{1}{2},-\frac{k}{2}-p-\frac{m}{2},-\frac{k}{2}-n-p-\frac{1}{2};-\frac{A v}{r}\right)}{k+\kappa^2+2 p+1}-\frac{2 A^{\frac{m-1}{2}} \Gamma \left(\frac{m+1}{2}\right) \Gamma \left(-\frac{m}{2}+n+\frac{1}{2}\right) \left(\frac{r}{v}\right)^{-\frac{m}{2}-\frac{1}{2}} \Gamma \left(-\frac{k}{2}-p-\frac{m}{2}-1\right) \, _2F_3\left(\frac{m}{2}+\frac{1}{2},-\frac{\kappa^2}{2}+\frac{m}{2}+\frac{1}{2};-\frac{\kappa^2}{2}+\frac{m}{2}+\frac{3}{2},\frac{m}{2}-n+\frac{1}{2},\frac{k}{2}+\frac{m}{2}+p+2;-\frac{A v}{r}\right)}{\left(-\kappa^2+m+1\right) \Gamma \left(-\frac{k}{2}-p-\frac{1}{2}\right)}+\frac{2 A^n \Gamma (n+1) \Gamma \left(\frac{1}{2} (m-2 n-1)\right) \left(\frac{r}{v}\right)^{-n-1} \Gamma \left(-\frac{k}{2}-n-p-\frac{3}{2}\right) \, _2F_3\left(n+1,-\frac{\kappa^2}{2}+n+1;-\frac{\kappa^2}{2}+n+2,-\frac{m}{2}+n+\frac{3}{2},\frac{k}{2}+n+p+\frac{5}{2};-\frac{A v}{r}\right)}{\left(\kappa^2-2 n-2\right) \Gamma \left(-\frac{k}{2}-p-\frac{1}{2}\right)}+\frac{A^{\frac{\kappa^2}{2}-1} \Gamma \left(\frac{\kappa^2}{2}\right) \Gamma \left(\frac{1}{2} \left(-\kappa^2+m+1\right)\right) \Gamma \left(-\frac{\kappa^2}{2}+n+1\right) \left(\frac{r}{v}\right)^{-\frac{\kappa^2}{2}} \Gamma \left(\frac{1}{2} \left(-\kappa^2-k-2 p-1\right)\right)}{\Gamma \left(-\frac{k}{2}-p-\frac{1}{2}\right)}\right)$
Is there a way to solve it in a simpler form?
