In my research appeared the following integral:
$$\int e^{-a x^2 } \frac {b x^2}{c + k x} dx$$
Where $a$, $b$, $c$ and $k$ are constants.
I know that the result of this indefinite integral is no conventional function and that such an integral can be solved numerically.
I tried it in all sites for online calculation of integrals with no results.
Has anyone seen this integral before? Is there any special function, like erf or gamma, that was defined to express this integral?
If not possible, perhaps there is some general expression for the integral defined from $0$ to $\infty$
PS: The integral above arose from the product of a function with the Gaussian probability function
If you make $c+k x=t$ and expand, you end with $$I=\frac{b c^2 }{k^3 }\int\frac{ e^{-\frac{a (t-c)^2}{k^2}}}{ t}\,dt-\frac{2 b c }{k^3}\int e^{-\frac{a (t-c)^2}{k^2}}\,dt+\frac{b }{k^3}\int t e^{-\frac{a (t-c)^2}{k^2}}\,dt$$ The problem is with the first integral (the second and third do not make any problem).