Is there a method available to obtain a closed-form solution to an integral in the following form:
$$I(x)=\int_{\mathcal{A}} \mathrm{d}x f(x)\left(\frac{\mathrm{d}^n}{\mathrm{d}x^n}g(x)\right)$$
in terms of $f(x)$, $g(x)$, $\frac{\mathrm{d}^m}{\mathrm{d}x^m}g(x)$, $\int_{\mathcal{A}}f(x)\mathrm{d}x$, $\int_{\mathcal{A}}g(x)\mathrm{d}x$, with $\mathcal{A}$ a (possibly infinite) segment on a real number line?
If there is such a method, is it extendable to multiple dimensions?
The specific problem I am trying to solve involves Hermite polynomials and has the following form:
$$\int_{-\infty}^{\infty}\mathrm{d}x_1\int_{-\infty}^{\infty}\mathrm{d}x_2\int_{-\infty}^{\infty}\mathrm{d}x_3e^{-f(x_1,x_2,x_3)}e^{x_1^2}\left(\frac{\mathrm{d}^{n_1}}{\mathrm{d}x_1^{n_1}}e^{-x_1^2}\right)e^{x_2^2}\left(\frac{\mathrm{d}^{n_2}}{\mathrm{d}x_2^{n_2}}e^{-x_2^2}\right)e^{x_3^2}\left(\frac{\mathrm{d}^{n_3}}{\mathrm{d}x_3^{n_3}}e^{-x_3^2}\right),$$
where $f(x_1,x_2,x_3)\geq0$ is a positive quadratic polynomial.
I found a very messy solution using standard methods (integration of Gaussian functions). My solution has an ugly summation, and now I am wondering if a "prettier" solution exists.