$
\begin{aligned}
& \text { For } x < 1, I=\int \frac{x-x^2}{\sqrt{1-x}} d x=\int \frac{x(1-x)}{\sqrt{1-x}} d x \\
& =\int x \sqrt{1-x} d x
\end{aligned}
$
Let $1-x=t^2$
$
\begin{array}{lc}
\Rightarrow & d x=-2 t d t \\
\text { So, } & I=\int\left(1-t^2\right) t(-2 t) d t=2 \int\left(t^4-t^2\right) d t \\
\Rightarrow & I=2\left[\frac{t^5}{5}-\frac{t^3}{3}\right]+c \\
\Rightarrow & I=\frac{2}{15} t^3\left(3 t^2-5\right)+c \\
\Rightarrow & I=\frac{2}{15}(1-x)^{3 / 2}[3(1-x)-5]+c \\
\Rightarrow & I=\frac{-2}{15}(1-x)^{3 / 2}(3 x+2)+c .
\end{array}
$