I am wondering if there is a non-elementary function in literature that describes the integration of $e^{f(x)}/f(x)$ with respect to $x$, where $f(x)$ is:
$$\sqrt{A + Bx + Cx^2}$$
or
$$\sqrt{(A-x)^2 + C}$$
Is it possible or will series expansion / Guassian quadrature or other numerical methods be necessary to estimate the integral?
Thank you,
F
$\int\dfrac{e^{f(x)}}{f(x)}dx=\int\dfrac{1}{f(x)}\sum\limits_{n=0}^\infty\dfrac{(f(x))^n}{n!}dx=\int\sum\limits_{n=0}^\infty\dfrac{(f(x))^{n-1}}{n!}dx$
Then find $\int(f(x))^{n-1}~dx$ for any non-negative integral $n$ .
When $f(x)=\sqrt{A+Bx+Cx^2}$ and $f(x)=\sqrt{(A-x)^2+C}$ , $\int(f(x))^{n-1}~dx$ have close-forms for any non-negative integral $n$ .