Simplify: $\;$ $\dfrac{a^2 + 10a + 25 + 2 \sqrt{5} \left(\sqrt{a^3} + 5 \sqrt{a}\right)}{\left(a^2 - 25\right) \left[\left(\sqrt{a^3} - \sqrt{125}\right) \left(a + \sqrt{5a} + 5\right)^{-1}\right]^{-1}}$
$\dfrac{a^2 + 10a + 25 + 2 \sqrt{5} \left(\sqrt{a^3} + 5 \sqrt{a}\right)}{\left(a^2 - 25\right) \left[\left(\sqrt{a^3} - \sqrt{125}\right) \left(a + \sqrt{5a} + 5\right)^{-1}\right]^{-1}}$
$= \dfrac{\left(a + 5\right)^2 + 2 \sqrt{5 a} \left(a + 5\right)}{\left(a + 5\right) \left(a - 5\right) \left[\dfrac{\left(\sqrt{a}\right)^3 - \left(\sqrt{5}\right)^3}{a + \sqrt{5a} + 5}\right]^{-1}}$
$= \dfrac{\left(a + 5\right) \left(a + 5 + 2 \sqrt{5a}\right)}{\dfrac{\left(a + 5\right) \left(a - 5\right) \left(a + \sqrt{5a} + 5\right)}{\left(\sqrt{a}\right)^3 - \left(\sqrt{5}\right)^3}}$
$= \dfrac{\left(a + 2 \sqrt{5a} + 5\right) \left[\left(\sqrt{a}\right)^3 - \left(\sqrt{5}\right)^3\right]}{\left(a - 5\right) \left(a + \sqrt{5a} + 5\right)}$
$= \dfrac{\left(\sqrt{a} + \sqrt{5}\right)^2 \left(\sqrt{a} - \sqrt{5}\right) \left(a + \sqrt{5a} + 5\right)}{\left[\left(\sqrt{a}\right)^2 - \left(\sqrt{5}\right)^2\right] \left(a + \sqrt{5a} + 5\right)}$
$= \dfrac{\left(\sqrt{a} + \sqrt{5}\right)^2 \left(\sqrt{a} - \sqrt{5}\right)}{\left(\sqrt{a} + \sqrt{5}\right) \left(\sqrt{a} - \sqrt{5}\right)}$
$= \sqrt{a} + \sqrt{5}$