Integrate with rule

$\begin{aligned} I & = \int_1^5 \frac {dx}{\sqrt x(1+x)^\frac 52} & \small \color{#3D99F6} \text{Let }u = \sqrt x \implies du = \frac 1{2\sqrt x} dx \\ & = 2 \int_1^{\sqrt 5} \frac {du}{(1+u^2)^\frac 52} & \small \color{#3D99F6} \text{Let }u = \tan \theta \implies du = \sec^2 \theta \ d\theta \\ & = 2 \int_\frac \pi 4^{\tan^{-1} \sqrt 5} \frac {d\theta}{\sec^3 \theta} \\ & = 2 \int_\frac \pi 4^{\tan^{-1} \sqrt 5} \cos^3 \theta \ d\theta \\ & = 2 \int_{\frac 1{\sqrt 2}}^{\sqrt{\frac 56}} \left(1-\sin^2 \theta\right) d\sin \theta \\ & = 2 \left[\sin \theta - \frac {\sin^3 \theta}3 \right]_{\frac 1{\sqrt 2}}^{\sqrt{\frac 56}} \\ & = \frac {13\sqrt 5}{9\sqrt 6} - \frac 5{6\sqrt 2} \\ & \approx \boxed{0.140} \end{aligned}$