Integral of exponential and algebraic function I would like to know, if it exists, a closed form for:
$$\int_0^{+\infty}\,\frac{k}{ak^4+bk^2+c}\,\,\exp(-wk^2)\,\,dk$$
with $a,b,c,w>0$.
Thanks!
 A: As an alternate approach, you may simply express in terms of a Laplace transform; to pick up where @Harry Peter left off:
$$I(w)=\int_0^{\infty} dk \, k \frac{e^{-w k^2}}{a k^4+b k^2+c}  = \frac12 \int_0^{\infty} dx \frac{e^{-w x}}{a x^2+b x+c}$$
Let the roots of the denominator of the integrand be $x_+$ and $x_-$.  Then
$$I(w) = \frac{1}{2 a} \frac{1}{x_+-x_-} \left [\int_0^{\infty}dx \frac{e^{-w x}}{x-x_+} - \int_0^{\infty}dx \frac{e^{-w x}}{x-x_-} \right ]$$
if these integrals exist. Now,
$$x_{\pm} = \frac{-b \pm \sqrt{b^2-4 a c}}{2 a}$$
and because $a$, $b$, and $c$ are all positive, then the roots $x_{\pm}$ are either negative or complex, so the integrals will exist.  If, however, the roots are equal, a slightly different approach will be required.
Each of these integrals maybe expressed in terms of either an incomplete gamma function, or as Harry notes, an exponential integral:
$$I(w) = \frac{1}{2 \sqrt{b^2-4 a c}} \left [e^{w x_+} E_1\left (-w x_+\right) - e^{w x_-} E_1\left (-w x_-\right)  \right ]$$
For real, distinct, negative roots, the above expression is no problem.  For complex roots ($b^2-4 a c \lt 0$), however, care must be taken to respect the branch cut along the negative real axis that is part of the definition of $E_1$.  
For double roots ($b^2-4 a c=0$), we recognize that
$$I(w) = \frac{1}{a} \int_0^{\infty}dx \frac{e^{-w x}}{(x-x_+)^2} $$
where $x_+ = -b/(2 a) \lt 0$.  In this case, we use a generalized exponential integral, again, related to the incomplete gamma function, and is
$$I(w) = \frac{e^{w x_+}}{a} E_2(-w x_+)$$
A: You can evaluate the integral by using partial fraction and the following identity

$$\int_{0}^{\infty}\frac{e^{-wx^2}}{\alpha x+\beta}dx= \frac{1}{2\pi \beta \sqrt{w} }\,
G^{3, 2}_{2, 3}\left({\frac {{\beta}^{2}w}{{\alpha}^{2}}}\, \Big\vert\,^{\frac{1}{2}, 1}_{1, \frac{1}{2}, \frac{1}{2}}\right),$$

where $G^{3, 2}_{2, 3}$ is the Meijer G-function.
