Answer
\[4\sqrt{1+\sqrt{x}}+C\]
Work Step by Step
Let \[I=\int\frac{1}{\sqrt{x+x^{\frac{3}{2}}}}dx\]
\[I=\int\frac{dx}{\sqrt{x(1+\sqrt{x})}}dx\]
\[I=\int\frac{dx}{\sqrt{x}(\sqrt{1+\sqrt{x}}})dx\;\;\;\ldots (1)\]
Substitute $1+\sqrt{x}=t\;\;\;\ldots (2)$
\[\Rightarrow \frac{1}{2\sqrt{x}}dx=dt\]
(1) becomes
\[I=\int\frac{2}{\sqrt{t}}dt=2\int t^{-\frac{1}{2}}dt\]
\[I=2\times 2\sqrt {t}+C\]
Where $C$ is constant of integration
\[I=4\sqrt{t}+C\]
From (2)
\[I=4\sqrt{1+\sqrt{x}}+C\]
Hence \[I=4\sqrt{1+\sqrt{x}}+C\;\;.\]
