Without calculating the integral, prove that this is true

Question

How to prove that this is true without calculating the integral?

$$\int \frac{\sqrt{1 + \sqrt{x}}}{\sqrt{x}}\,dx = \frac{4}{3}\bigl(1 + x^{\frac{1}{2}}\bigr)^{\frac{3}{2}} + C$$

Answer 1

It is absolutely not difficult to prove it's true even with calculating the integral, because $$\left(1+\sqrt{x}\right)'=\frac{1}{2}\frac{1}{\sqrt{x}}$$ thus $$\int \frac{\sqrt{1+\sqrt{x}}}{\sqrt{x}}dx=2\int \left(\sqrt{1+\sqrt{x}} \right)d\left(1+\sqrt{x}\right)=...$$ and substituting $y=1+\sqrt{x}$ $$...=2\int \sqrt{y} dy=\frac{4}{3}y^{\frac{3}{2}}+C$$