Integral of $$$x^{n} \left(1 - x\right)$$$ with respect to $$$x$$$

Find $$$\int x^{n} \left(1 - x\right)\, dx$$$.

This integral does not have a closed form:

$${\color{red}{\int{x^{n} \left(1 - x\right) d x}}} = {\color{red}{\frac{x^{n + 1} {{}_{2}F_{1}\left(\begin{matrix} -1, n + 1 \\ n + 2 \end{matrix}\middle| {x} \right)}}{n + 1}}}$$

Therefore,

$$\int{x^{n} \left(1 - x\right) d x} = \frac{x^{n + 1} {{}_{2}F_{1}\left(\begin{matrix} -1, n + 1 \\ n + 2 \end{matrix}\middle| {x} \right)}}{n + 1}$$

Simplify:

$$\int{x^{n} \left(1 - x\right) d x} = \frac{x^{n + 1} \left(n - x \left(n + 1\right) + 2\right)}{\left(n + 1\right) \left(n + 2\right)}$$

Add the constant of integration:

$$\int{x^{n} \left(1 - x\right) d x} = \frac{x^{n + 1} \left(n - x \left(n + 1\right) + 2\right)}{\left(n + 1\right) \left(n + 2\right)}+C$$

$$$\int x^{n} \left(1 - x\right)\, dx = \frac{x^{n + 1} \left(n - x \left(n + 1\right) + 2\right)}{\left(n + 1\right) \left(n + 2\right)} + C$$$A