Primitive of p x + q over Root of a x squared plus 2 b x plus c

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Theorem

$\ds \int \dfrac {p x + q} {\sqrt {a x^2 + 2 b x + c} } \rd x = \dfrac p a \sqrt {a x^2 + 2 b x + c} + \paren {q - \dfrac {p b} a} \int \dfrac {\d x} {\sqrt {a x^2 + 2 b x + c} }$


Proof 1

\(\ds \int \dfrac {p x + q} {\sqrt {a x^2 + 2 b x + c} } \rd x\) \(=\) \(\ds \dfrac p {2 a} \int \dfrac {2 a x + \frac {2 a q} p} {\sqrt {a x^2 + 2 b x + c} } \rd x\)
\(\ds \) \(=\) \(\ds \dfrac p {2 a} \int \dfrac {2 a x + 2 b - 2 b + \frac {2 a q} p} {\sqrt {a x^2 + 2 b x + c} } \rd x\)
\(\ds \) \(=\) \(\ds \dfrac p {2 a} \int \dfrac {2 a x + 2 b} {\sqrt {a x^2 + 2 b x + c} } \rd x + \dfrac p {2 a} \paren {\dfrac {2 a q} p - 2 b} \int \dfrac {\d x} {\sqrt {a x^2 + 2 b x + c} }\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \dfrac p {2 a} \int \dfrac {2 a x + 2 b} {\sqrt {a x^2 + 2 b x + c} } \rd x + \paren {q - \dfrac {p b} a} \int \dfrac {\d x} {\sqrt {a x^2 + 2 b x + c} }\) simplifying
We have from Power Rule for Derivatives that $\map {\dfrac \d {\d x} } {a x^2 + b x + c} = 2 a x + 2 b$, so:
\(\ds \) \(=\) \(\ds \dfrac p {2 a} \paren {2 \sqrt {a x^2 + 2 b x + c} } + \paren {q - \dfrac {p b} a} \int \dfrac {\d x} {\sqrt {a x^2 + 2 b x + c} }\) Square Root of Function under Derivative
\(\ds \) \(=\) \(\ds \dfrac p a \sqrt {a x^2 + 2 b x + c} + \paren {q - \dfrac {p b} a} \int \dfrac {\d x} {\sqrt {a x^2 + 2 b x + c} }\) simplifying

$\blacksquare$


Proof 2

\(\ds \int \dfrac {p x + q} {\sqrt {a x^2 + 2 b x + c} } \rd x\) \(=\) \(\ds p \int \dfrac x {\sqrt {a x^2 + 2 b x + c} } \rd x + q \int \dfrac {\d x} {\sqrt {a x^2 + 2 b x + c} }\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds p \paren {\frac {\sqrt {a x^2 + 2 b x + c} } a - \frac {2 b} {2 a} \int \frac {\d x} {\sqrt {a x^2 + 2 b x + c} } } + q \int \dfrac {\d x} {\sqrt {a x^2 + 2 b x + c} }\) Primitive of $\dfrac x {\sqrt {a x^2 + 2 b x + c} }$
\(\ds \) \(=\) \(\ds \dfrac p a \sqrt {a x^2 + 2 b x + c} + \paren {q - \dfrac {p b} a} \int \dfrac {\d x} {\sqrt {a x^2 + 2 b x + c} }\) simplifying

$\blacksquare$


Examples

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

$\ds \int \dfrac {1 + x} {\sqrt {2 + 4 x - 3 x^2} } \rd x = -\dfrac 1 3 \sqrt {2 + 4 x - 3 x^2} + \dfrac 5 {3 \sqrt 3} \map \arcsin {\dfrac {3 x - 2} {\sqrt {10} } } + C$
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