The given expression can be rearranged as $ \frac{{(\sqrt {15} + 5) + (\sqrt {35} + \sqrt {21} )}}{{\sqrt 3 + 2\sqrt 5 + \sqrt 7 }}$
$ = \frac{{\sqrt 5 (\sqrt 3 + \sqrt 5 ) + \sqrt 7 (\sqrt 5 + \sqrt 3 )}}{{\sqrt 3 + 2\sqrt 5 + \sqrt 7 }}$
$ = \frac{{(\sqrt 3 + \sqrt 5 )(\sqrt 5 + \sqrt 7 )}}{{\sqrt 3 + 2\sqrt 5 + \sqrt 7 }}= \frac{{(\sqrt 3 + \sqrt 5 )(\sqrt 5 + \sqrt 7 )}}{{(\sqrt 3 + \sqrt 5 ) + (\sqrt 5 + \sqrt 7 )}}$
$ = \frac{1}{{\frac{{(\sqrt 3 + \sqrt 5 )}}{{(\sqrt 3 + \sqrt 5 )(\sqrt 5 + \sqrt 7 )}} + \frac{{(\sqrt 5 + \sqrt 7 )}}{{(\sqrt 3 + \sqrt 5 )(\sqrt 5 + \sqrt 7 )}}}}$
$ = \frac{1}{{\frac{1}{{(\sqrt 5 + \sqrt 7 )}} + \frac{1}{{(\sqrt 3 + \sqrt 5 )}}}}$
$ = \frac{1}{{\frac{{(\sqrt 7 - \sqrt 5 )}}{{(\sqrt 7 + \sqrt 5 )(\sqrt 7 - \sqrt 5 )}} + \frac{{(\sqrt 5 - \sqrt 3 )}}{{(\sqrt 5 + \sqrt 3 )(\sqrt 5 - \sqrt 3 )}}}}$
$ = \frac{1}{{\frac{{(\sqrt 7 - \sqrt 5 )}}{2} + \frac{{(\sqrt 5 - \sqrt 3 )}}{2}}}$
$ = \frac{2}{{(\sqrt 7 - \sqrt 3 )}} = \frac{{2(\sqrt 7 + \sqrt 3 )}}{{(\sqrt 7 - \sqrt 3 )(\sqrt 7 + \sqrt 3 )}} = \frac{{\sqrt 7 + \sqrt 3}}{2}$
