Answer
$$ 2 \left(\frac{1}{3}x^{3/2}+\frac{1}{2}x+\sqrt{x}+\ln|\sqrt{x}-1|\right)+C$$
Work Step by Step
Given $$\int \frac{x}{\sqrt{x}-1} d x $$
Let $$ u^2 =x\ \ \ \ \ \ 2udu=dx$$
Then
\begin{align*}
\int \frac{x}{\sqrt{x}-1} d x&=2\int \frac{u^3 du}{u-1}\\
&=2\int \left(u^2+u+1+\frac{1}{u-1}\right)du\\
&=2 \left(\frac{1}{3}u^3+\frac{1}{2}u^2+u+\ln|u-1|\right)+C\\
&= 2 \left(\frac{1}{3}x^{3/2}+\frac{1}{2}x+\sqrt{x}+\ln|\sqrt{x}-1|\right)+C
\end{align*}
