Theorem: Skewness of Beta Distribution

Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname {Beta}$ denotes the Beta distribution. 
Then the skewness $\gamma_1$ of $X$ is given by: 
:$\gamma_1 = \dfrac {2 \paren {\beta - \alpha} \sqrt {\alpha + \beta + 1} } {\paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }$

Proof:
From Skewness in terms of Non-Central Moments: 
:$\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$ 
where: 
:$\mu$ is the expectation of $X$. 
:$\sigma$ is the standard deviation of $X$.
We have, by Expectation of Beta Distribution:
:$\expect X = \dfrac {\alpha} {\alpha + \beta}$
By Variance of Beta Distribution:
:$\var X = \sigma^2 = \dfrac {\alpha \beta} {\paren {\alpha + \beta}^2 \paren {\alpha + \beta + 1} }$
so:
:$\sigma = \dfrac {\sqrt {\alpha \beta} } {\paren {\alpha + \beta} \paren {\sqrt {\alpha + \beta + 1 } } }$
From Raw Moment of Beta Distribution, we have:
{{begin-eqn}}
{{eqn	| l = \expect {X^3}
	    | r = \prod_{r \mathop = 0}^2 \frac {\alpha + r} {\alpha + \beta + r}
}}
{{eqn	| r = \paren {\dfrac {\alpha} {\alpha + \beta} } \paren {\dfrac {\alpha + 1} {\alpha + \beta + 1} } \paren {\dfrac {\alpha + 2} {\alpha + \beta + 2} }
}}
{{end-eqn}}
So..
{{begin-eqn}}
{{eqn | l = \gamma_1
      | r = \frac {\paren {\dfrac {\alpha} {\alpha + \beta} } \paren {\dfrac {\alpha + 1} {\alpha + \beta + 1} } \paren {\dfrac {\alpha + 2} {\alpha + \beta + 2} } - 3 \paren {\dfrac {\alpha} {\alpha + \beta} } \paren {\dfrac {\alpha \beta} {\paren {\alpha + \beta}^2 \paren {\alpha + \beta + 1} } } - \paren {\dfrac {\alpha} {\alpha + \beta} }^3} {\paren {\dfrac {\sqrt {\alpha \beta} } {\paren {\alpha + \beta} \paren {\sqrt {\alpha + \beta + 1 } } } }^3 }
      | c = 
}}
{{eqn | r = \frac {\paren {\dfrac {\alpha \paren {\alpha + 1} \paren {\alpha + 2} \paren {\alpha + \beta}^2 - 3 \alpha^2 \beta \paren {\alpha + \beta + 2} - \alpha^3 \paren {\alpha + \beta + 1} \paren {\alpha + \beta + 2} } {\paren {\alpha + \beta}^3 \paren {\alpha + \beta + 1} \paren {\alpha + \beta + 2} } } } {\paren {\dfrac {\paren {\alpha \beta} \sqrt {\alpha \beta} } {\paren {\alpha + \beta}^3 \paren {\paren {\alpha + \beta + 1} \sqrt {\alpha + \beta + 1 } } } } }
      | c = Expanding terms. Addition of Fractions
}}
{{eqn | r = \dfrac {\paren { \paren {\alpha + 1} \paren {\alpha + 2} \paren {\alpha + \beta}^2 - 3 \alpha \beta \paren {\alpha + \beta + 2} - \alpha^2 \paren {\alpha + \beta + 1} \paren {\alpha + \beta + 2} } \paren {\sqrt {\alpha + \beta + 1 } } } { \beta \paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }
      | c = Canceling $\alpha$, $\paren {\alpha + \beta}^3$ and $\paren {\alpha + \beta + 1}$ from numerator and denominator
}}
{{eqn | r = \dfrac {\paren { \paren {\alpha + 1} \paren {\alpha + 2} \paren {\alpha + \beta}^2 - 3 \alpha \beta \paren {\alpha + \beta + 2} - \alpha^2 \paren {\paren {\alpha + \beta }^2 + 3 \paren {\alpha + \beta } + 2 } } \paren {\sqrt {\alpha + \beta + 1 } } } { \beta \paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }
      | c = Grouping the $\paren {\alpha + \beta}$ term.
}}
{{eqn | r = \dfrac {\paren { \paren {\alpha + \beta}^2 \paren {\alpha^2 + 3 \alpha + 2 - \alpha^2 } - 3 \alpha \beta \paren {\alpha + \beta + 2} - \alpha^2 \paren {3 \paren {\alpha + \beta } + 2 } } \paren {\sqrt {\alpha + \beta + 1 } } } { \beta \paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }
      | c = Grouping $\paren {\alpha + \beta}^2$
}}
{{eqn | r = \dfrac {\paren { \paren {\alpha + \beta}^2 \paren {3 \alpha + 2 } - 3 \alpha \beta \paren {\paren {\alpha + \beta } + 2} - \alpha^2 \paren {3 \paren {\alpha + \beta } + 2 } } \paren {\sqrt {\alpha + \beta + 1 } } } { \beta \paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }
      | c = $\alpha^2$ cancels.
}}
{{eqn | r = \dfrac {\paren { \paren {\alpha + \beta} \paren {\paren {\alpha + \beta} \paren {3 \alpha + 2 } - 3 \alpha \beta - 3 \alpha^2 } - 6 \alpha \beta - 2\alpha^2 } \paren {\sqrt {\alpha + \beta + 1 } } } { \beta \paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }
      | c = Factoring out $\paren {\alpha + \beta}$ again
}}
{{eqn | r = \dfrac {\paren { \paren {\alpha + \beta} \paren {\paren {3 \alpha^2 + 2 \alpha + 3 \alpha \beta + 2 \beta } - 3 \alpha \beta - 3 \alpha^2 } - 6 \alpha \beta - 2\alpha^2 } \paren {\sqrt {\alpha + \beta + 1 } } } { \beta \paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }
      | c = Expanding the product
}}
{{eqn | r = \dfrac {\paren { \paren {\alpha + \beta} \paren {2 \alpha + 2 \beta } - 6 \alpha \beta - 2\alpha^2 } \paren {\sqrt {\alpha + \beta + 1 } } } { \beta \paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }
      | c = Canceling $3 \alpha^2$ and $3 \alpha \beta$
}}
{{eqn | r = \dfrac {\paren { \paren {2 \alpha^2 + 2 \alpha \beta + 2 \alpha \beta + 2\beta^2} - 6 \alpha \beta - 2\alpha^2 } \paren {\sqrt {\alpha + \beta + 1 } } } { \beta \paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }
      | c = Expanding the product
}}
{{eqn | r = \dfrac {\paren {2\beta^2 - 2 \alpha \beta } \paren {\sqrt {\alpha + \beta + 1 } } } { \beta \paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }
      | c = Canceling $2 \alpha^2$ and grouping $\alpha \beta$ terms
}}
{{eqn | r = \frac {2 \paren {\beta - \alpha} \sqrt {\alpha + \beta + 1} } {\paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }
      | c = Canceling $\beta$ from numerator and denominator
}}
{{end-eqn}}
{{qed}}