****** Note: In response to the comment below, I've modified the problem ******
I'm trying to evaluate the following indefinite integral
$$ \Large\int x^{-\frac{3}{2}} e^{-\frac{K}{x}\Big[(x-a)^2+(x-b)^2+(x-c)^2\Bigr]} $$
where $ K,a,b >0$ and $c\ge 0$. For the curious, this models the transient dispersion of a pollutant with a time dependent mass source ($x$ is really a time variable)
Wolfram Alpha is unable to integrate this particular form
Fair enough. I'm willing to do some legwork, so I recast the integral as
$$ \Large\int x^{-\frac{3}{2}} e^{-\Big[Ax+B+\frac{C}{x}\Bigr]} =e^{-B}\Large\int x^{-\frac{3}{2}} e^{-\Big[Ax+\frac{C}{x}\Bigr]} $$
where $$ A=3K\\ B=-2K(a+b+c)\\ C=K(a^2+b^2+c^2) $$
Note that $A>0$, $C>0$ and critically, $B<0$ given the bounds on $K,a,b$ and $c$. For this form Wolfram yielded
which however is potentially problematic in terms of evaluation due to the resultant $e^{2\sqrt{AC}}$ term, given that $AC>0$. Looking at the original form of the integral, I was not expecting a term with a potentially large positive exponential argument, not to mention that the integral itself is now multiplied by another exponential term with a potentially large positive argument ($e^{-B}$ given $B<0$)
The alternate forms of the integral that Wolfram Alpha suggests do not help either
Now it turns out that for my particular case of interest, the magnitudes of $A$, $B$, and $C$ are not excessive (see the plot below), but I'm concerned about the general applicability of the expression. Is there a more robust form that can be constructed?
I did a sanity check of the integral with a quick-and-dirty trapezoidal rule and the two curves in the plot are graphically indistinguishable.
