Step 1: Express in terms of sine and cosine.
The given expression is rewritten using sine and cosine functions:
\[
\sqrt{\tan x} = \frac{\sin^{1/2} x}{\cos^{1/2} x}, \quad \sqrt{\cot x} = \frac{\cos^{1/2} x}{\sin^{1/2} x}.
\]
The integral becomes:
\[
I = \int \left( \frac{\sin^{1/2} x}{\cos^{1/2} x} + \frac{\cos^{1/2} x}{\sin^{1/2} x} \right) dx.
\]
Step 2: Rewrite in a simplified form.
Factoring and simplifying yields:
\[
I = \int \frac{\sin x + \cos x}{\sqrt{\sin x \cos x}} dx.
\]
Using the identity:
\[
\sin x + \cos x = \sqrt{2} \sin \left( x + \frac{\pi}{4} \right),
\]
the expression is rewritten as:
\[
I = \int \frac{\sqrt{2} \sin \left( x + \frac{\pi}{4} \right)}{\sqrt{\sin x \cos x}} dx.
\]
Step 3: Use trigonometric substitution.
Using the identity:
\[
\sin x \cos x = \frac{1}{2} \sin 2x,
\]
we obtain:
\[
\sqrt{\sin x \cos x} = \sqrt{\frac{1}{2} \sin 2x} = \frac{\sqrt{\sin 2x}}{\sqrt{2}}.
\]
The integral simplifies to:
\[
I = \int \frac{\sqrt{2} \sin \left( x + \frac{\pi}{4} \right)}{\frac{\sqrt{\sin 2x}}{\sqrt{2}}} dx.
\]
\[
= \int \frac{2\sin \left( x + \frac{\pi}{4} \right)}{\sqrt{\sin 2x}} dx.
\]
Step 4: Substituting \( t = \sin 2x \).
Let:
\[
t = \sin 2x, \quad \frac{dt}{dx} = 2\cos 2x.
\]
Rewriting in terms of \( t \), simplifying, and integrating gives the final result after substituting back \( t = \sin 2x \).
\[
I = \int \frac{2\sin (x + \frac{\pi}{4})}{\sqrt{t}} dx.
\]