Primitive of Reciprocal of Root of a x squared plus b x plus c/Examples/x^2 + 4 x + 5

Examples of Use of Primitive of $\dfrac 1 {\sqrt {a x^2 + b x + c} }$

 * $\ds \int \dfrac {\d x} {\sqrt {x^2 + 4 x + 5} } = \map \ln {x + 2 + \sqrt {x^2 + 4 x + 5} } + C$

Proof
We aim to use Primitive of $\dfrac 1 {\sqrt {a x^2 + b x + c} }$ with:

We note that:

Hence from Primitive of $\dfrac 1 {\sqrt {a x^2 + b x + c} }$:
 * $\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \dfrac 1 {\sqrt a} \map \arsinh {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C$

Substituting for $a$, $b$ and $c$ and simplifying: