I need to numerically approximate integral functions of the form
$g(x)=\int_0^\infty f(x)x^ae^{-b\log^2{x} -cx^2+dx}$dx
$a,b,c,d>0$
I have tried a Gauss-Laguerre quadrature after a change of variable $-y=dx$ but then appears the logarithm of a negative term. I was thinking also in a Gauss-Hermite quadrature, but it only works in the interval $(-\infty,\infty)$. Any hints? Any specific quadrature rule I can use? Any variable transform?
Yes, if d is positive the substitution y= -dx (I presume you mean the "d times x" in the exponential, not the diffential "dx". It is not a good idea to use "d" as a constant multiplying x in an integral with respect to x!) then x will be negative so that ln(x) does not exist. What reason do you have for making that substitution?