I am trying to find the following integral:
$\int\limits_0^\infty {x{e^{ - (a{x^2} + b)}}{I_0}\left( {\sqrt {c{x^2} + dx + k} } \right)dx} $
Where, $x,a,b,c,d,k \in \mathbb{R}$ and $I_0(.)$ is the modified bessel function of the first kind.
I know there is a basic rule for the, for $k>0$:
$\int\limits_0^\infty {{x^{2k - 1}}{e^{ - (a{x^2})}}{I_0}\left( {bx} \right)dx} = \frac{{\Gamma \left( k \right)}}{{2{a^k}}}{L_{ - k}}\left( {\frac{{{b^2}}}{{4a}}} \right)$
Where, $L$ is Laguerre polynomial.
I am having a problem because of the argument inside $I_0(.)$ which is ${\sqrt {c{x^2} + dx + k} }$. Can anyone suggest to me how to do the integration?