Primitive of Reciprocal of Root of a x squared plus b x plus c

Theorem

$a$ greater than $0$

Let $a \in \R_{> 0}$.

Then for $x \in \R$ such that $a x^2 + b x + c > 0$:

$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \begin {cases}

\dfrac 1 {\sqrt a} \ln \size {2 \sqrt a \sqrt {a x^2 + b x + c} + 2 a x + b} + C & : b^2 - 4 a c > 0 \\ \dfrac 1 {\sqrt a} \map \arsinh {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C & : b^2 - 4 a c < 0 \\ \dfrac 1 {\sqrt a} \ln \size {2 a x + b} + C & : b^2 - 4 a c = 0 \end {cases}$ where $\arsinh$ denotes the area hyperbolic sine function.

$a$ less than $0$

Let $a \in \R_{< 0}$.

Then for $x \in \R$ such that $a x^2 + b x + c > 0$:

$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \dfrac {-1} {\sqrt {-a} } \map \arcsin {\dfrac {2 a x + b} {\sqrt {\size {b^2 - 4 a c} } } } + C$

given that $b^2 \ne 4 a c$.

$a$ equal to Zero

When $a = 0$ we have the following result:

$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \frac {2 \sqrt {b x + c} } b + C$

when $a = 0$.

Examples

Primitive of $\dfrac 1 {\sqrt {2 + 4 x - 3 x^2} }$

$\ds \int \dfrac {\d x} {\sqrt {2 + 4 x - 3 x^2} } = \dfrac 1 {\sqrt 3} \map \arcsin {\dfrac {3 x - 2} {10} } + C$

Primitive of $\dfrac 1 {\sqrt {x^2 + 4 x + 5} }$

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

Primitive of $\dfrac 1 {\sqrt {2 x - x^2} }$

$\ds \int \dfrac {\d x} {\sqrt {2 x - x^2} } = \map \arcsin {x - 1} + C$

Primitive of $\dfrac 1 {\sqrt {2 x + x^2} }$

$\ds \int \dfrac {\d x} {2 x + x^2} = $

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {a x^2 + b x + c}$: $14.280$