I am trying to solve the following equations

$\int^\infty_0 \exp\left(-\frac{\alpha}{x+1}\right)\exp(-c x) x^{\frac{n-1}{2}} I_{n-1}\left(\sqrt{\beta x}\right)\ \mathrm{d}x $


$\int^\infty_0 \exp\left(-\frac{\alpha}{x}\right)\exp(-cx)(x-1)^{\frac{n-1}{2}} I_{n-1}\left(\sqrt{\beta(x-1)}\right) \mathrm{d}x $

I tried using Mellin convolution, but I always struggle with the inversion part. Any other hints or clues are welcomed.


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