Lemmata for Euler's Third Substitution/Lemma 3

Theorem
Let Euler's third substitution be employed in order to evaluate a primitive of the form:
 * $\ds \map R {x, \sqrt {a x^2 + b x + c} } \rd x$

where $R$ is a rational function of $x$ and $\sqrt {a x^2 + b x + c}$.

Thus:
 * $\ds \sqrt {a x^2 + b x + c} = \sqrt {a \paren {x - \alpha} \paren {x - \beta} } = \paren {x - \alpha} t$

Then we have:
 * $\dfrac {\d x} {\d t} = \dfrac {2 t a \paren {\beta - \alpha} } {\paren {a - t^2}^2}$

Also see

 * Definition:Euler's Third Substitution