I am trying to integrate this:

$$\int_0^\infty \log(1+x^r)x^{a-1}e^{-\beta x}I_v(kx) \ \mathrm dx$$ where $r, a, \beta, v, k$ are arbitrary constants, $v$ is the order of the modified bessel function of the first kind and $r\leq 1$.
I found a possible way to work it out using Meijer G-function, which is by transforming the logarithm, the exponential and the Bessel function to their equivalent Meijer G-function form, which could be integrated in mathematica.
However, I don't know how to deal with Meijer G-function in Mathematica. please help me with this.


You may find examples at Wolfram directly.

Alpha may return an answer as here for the numerical evaluation of :


Note that the terms between $G$ and '$($' in $G^{m,n}_{p,q}$ allow to count the elements at the second level and will appear only 'implicitly', the term between '$($' and '$|$' is the $z$ from $z^{-s}$ ($3$ here) and must be put alone at the right of the function.
In the required syntax for MeijerG the first parameters will be written $\{\{a_l\},\{a_r\}\}$ corresponding to the high part at the right of the representation (with $a_l$ constituted of $n$ parameters separated by ',' ).
The second parameter will be $\{\{b_l\},\{b_r\}\}$ for the low part with $b_l$ constituted of $m$ parameters to give something like :

MeijerG[{{}, {2}}, {{1/2, 3/2}, {}}, 3]

Alpha will also often produce 'Time-out'...


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