# Symbolic expectations and double integrals

I'd like to compute the following expectation ($U$ and $V$ are independent and normally distributed or Gaussian) $a_{k,p}=E(|U|^p|U+\sqrt{k-1}V|^p)$

Is there a way to directly compute those expectations in Sage ?

If I write it down, I obtain a double integral which I already tried to compute with maxima like this (ok here, you can only see the starting point with the inner integral with respect to $v$ first)

```
load(distrib);
n(x):=pdf_normal(x, 0, 1);
inner_integral: integrate(abs(u+sqrt(k-1)*v)^p*n(v), v, minf, inf);
```

but to no avail.

NB: The absolute central moment $a_{1,p/2}$ can be easily obtained with maxima with

```
ratsimp(integrate(abs(x)^p*n(x), x, minf, inf));
```

but no 'simple' expression as for the aforementioned double integral.

Any hint?

Since Sage uses Maxima for all of its symbolic integration, I find it unlikely we'll do better. We do have numerical routines of various types, but I don't think this is what you are looking for.

@kcrisman I thought that sometimes Sage could use sympy or some other CAS for symbolic integration. But then, I'm stuck here then.