I'm looking to the result of the following integral: $$\int_0^{\infty} f_G(g)\exp(-tg) dg $$ where: $f_G(g) = \frac{ckg^{(c-1)}}{(1+g^c)^{k+1}}$ is a PDF not defined in 0 but with $\int_0^\infty f_G(g) = 1$, and $c$, $k$, $t$ are variables.

Knowing that for both the functions: $f_G(g)$ and $\exp(-tg)$ the integral exist, I don't understand with I am not able to compute integral of the product of the two functions using Mathematica with the following expression:

Integrate[c*k*(x^(c - 1)/(1 + x^c)^(k + 1))*Exp[-t*x], {x, 0, Infinity}].

I do not want to compute the integral in a numeric way, I'm just looking for an expression that simplify the integral, considering that this is just an inner integral. Thus, I can proceed numerically with the computation of the others external integrals.

Any suggestion on how I can obtain a simplified expression using Mathematica or any other tool or simplified assumption?


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