Root, Root, Square Root

Algebra Level 3

\[ \dfrac1{2\sqrt1+1\sqrt2} + \dfrac1{3\sqrt2+2\sqrt3} + \dfrac1{4\sqrt3+3\sqrt4} + \cdots \\ \cdots +\dfrac{1}{24\sqrt{23} + 23\sqrt{24}} + \dfrac1{25\sqrt{24} + 24\sqrt{25}} = \, ? \]

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