Integral of modified Bessel function of second kind first order multiply by incomplete gamma function? Is there any possible solution or approximation for that given integral? $$\int_0^\infty {\big(v^{\frac{m}{2}-\frac{1}{4}}\big)}K_1\Bigg[\frac{2\sqrt[4]{v}}{l}\Bigg]\Gamma\left[m,-\frac{a+b v}{c}\right]\text{d}v.$$
 A: Partial Answer:
N.B this can be seen as a Mellin Transform, for this particular function (when $a=0$), the Mellin transform appears to be expressible in terms of the Meijer-G function. 
$$
\int_0^\infty x^{s-1} K_1\left(\frac{2\sqrt[4]{x}}{l}\right)\Gamma\left(m,\frac{bx}{c}\right)\; dx = 
$$
$$
\frac{\left(-\frac{b}{c}\right)^{-s} G_{2,5}^{4,2}\left(-\frac{c}{16 b l^4}|
   \begin{array}{c}
    1-s,-m-s+1 \\
    -\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{3}{4},-s \\
   \end{array}
   \right)}{4 \pi }
$$
with 
$$
s = \frac{m}{2} + \frac{3}{4}
$$
but I understand that this solution may be hard to work with.
A: I suppose that the approach could be the same as for your previous post.
If $a=0$, we have
$$\frac{1}{4 \pi }\left(-\frac{b}{c}\right)^{-\frac{2m+3}{4}}\,\,G_{2,5}^{4,2}\left(-\frac{c}{16\, b\, l^4}|
\begin{array}{c}
 \frac{1-6 m}{4} ,\frac{1-2m}{4}  \\
 -\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{3}{4},-\frac{2m+3}{4} )
\end{array}
\right)$$
Now, write
$$\Gamma\left[m,-\frac{a+b v}{c}\right]=\sum_{p=0}^\infty d_n\, a^n$$ with
$$d_0=\Gamma \left(m,-\frac{b v}{c}\right)\qquad d_1=\frac 1c\left(-\frac{b}{c}\right)^{m-1} e^{\frac{b }{c}v}$$
$$d_{n+2}=\frac{(n+1)  (b v+c (m-n-1))\,d_{n+1}+n\, d_n}{b c (n+1) (n+2) v}$$
The problem is that I am unable to compute even the first integral.
