$\frac{3+\sqrt{5}}{2\sqrt{2}+\sqrt{3}+\sqrt{5}}+\frac{3-\sqrt{5}}{2\sqrt{2}-\sqrt{3}-\sqrt{5}}$$=\frac{\left( 3+\sqrt{5} \right)\left( 2\sqrt{2}-\sqrt{3}-\sqrt{5} \right)}{8-{{\left( \sqrt{3}+\sqrt{5} \right)}^{2}}}+\frac{\left( 3-\sqrt{5} \right)\left( 2\sqrt{2}+\sqrt{3}+\sqrt{5} \right)}{8-{{\left( \sqrt{3}+\sqrt{5} \right)}^{2}}}$
$=\frac{12\sqrt{2}-2\sqrt{15}-10}{-2\sqrt{15}}$$=\frac{5+\sqrt{15}-6\sqrt{2}}{\sqrt{15}}$