100 Followers Problem

$\begin{aligned} \int \frac {dx}{\sqrt{e^{5x}}\sqrt[4]{(e^{2x}+e^{-2x})^3}} & = - \sqrt[p]{q+e^{-rx}} + C & \small \color{#3D99F6} \text{Differentiate both sides w.r.t. }x \\ \frac 1{\sqrt{e^{5x}}\sqrt[4]{(e^{2x}+e^{-2x})^3}} & = \frac d{dx} \left(- \sqrt[p]{q+e^{-rx}} + C \right) \\ \frac 1{e^{\frac 52x} \left(e^{2x}+e^{-2x}\right)^\frac 34} & = - \frac 1p \left(q+e^{-rx}\right)^{\frac 1p -1}\left(-re^{-rx}\right) \\ \frac 1{e^{4x} \left(1+e^{-4x}\right)^\frac 34} & = \frac r {p e^{rx} \left(q+e^{-rx}\right)^{\frac {p -1}p}} \end{aligned}$

Comparing the constants on both sides we get $p=4$ , $q=1$ and $r=4$ . $\implies p+q+r = 4+1+4 = \boxed{9}$ .