OK, to atone for my rash comment, I recalled that I needed something related in:

@article {MR2356429,
AUTHOR = {Rivin, Igor},
TITLE = {Surface area and other measures of ellipsoids},
JOURNAL = {Adv. in Appl. Math.},
FJOURNAL = {Advances in Applied Mathematics},
VOLUME = {39},
YEAR = {2007},
NUMBER = {4},
PAGES = {409--427},
ISSN = {0196-8858},
MRCLASS = {52A38 (28A75 33C65 60F99)},
MRNUMBER = {2356429 (2008k:52012)},
DOI = {10.1016/j.aam.2006.08.009},
URL = {http://dx.doi.org/10.1016/j.aam.2006.08.009},
}

(see page 414). And found it in the book by Mathai and Provost:

@book {MR1192786,
AUTHOR = {Mathai, A. M. and Provost, Serge B.},
TITLE = {Quadratic forms in random variables},
SERIES = {Statistics: Textbooks and Monographs},
VOLUME = {126},
NOTE = {Theory and applications},
PUBLISHER = {Marcel Dekker Inc.},
ADDRESS = {New York},
YEAR = {1992},
PAGES = {xxii+367},
ISBN = {0-8247-8691-2},
MRCLASS = {62H10 (62E15)},
MRNUMBER = {1192786 (94g:62110)},
MRREVIEWER = {Donald R. Jensen},
}

It should be in the vicinity of page 62. The argument uses Laplace transforms, and is not trivial.