Primitive of Odd Power of x over Power of a x squared plus b x plus c

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Theorem

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

Then:

$\ds \int \frac {x^{2 n - 1} \rd x} {\paren {a x^2 + b x + c}^n} = \frac 1 a \int \frac {x^{2 n - 3} \rd x} {\paren {a x^2 + b x + c}^{n - 1} } - \frac c a \int \frac {x^{2 n - 3} \rd x} {\paren {a x^2 + b x + c}^n} - \frac b a \int \frac {x^{2 n - 2} \rd x} {\paren {a x^2 + b x + c}^n}$


Proof

\(\ds \) \(\) \(\ds \int \frac {x^{2 n - 1} \rd x} {\paren {a x^2 + b x + c}^n}\)
\(\ds \) \(=\) \(\ds \frac 1 a \int \frac {x^{2 n - 3} a x^2 \rd x} {\paren {a x^2 + b x + c}^n}\) Primitive of Constant Multiple of Function
\(\ds \) \(=\) \(\ds \frac 1 a \int \frac {x^{2 n - 3} \paren {a x^2 + b x + c - b x - c} \rd x} {\paren {a x^2 + b x + c}^n}\)
\(\ds \) \(=\) \(\ds \frac 1 a \int \frac {x^{2 n - 3} \rd x} {\paren {a x^2 + b x + c}^{n - 1} } - \frac c a \int \frac {x^{2 n - 3} \rd x} {\paren {a x^2 + b x + c}^n} - \frac b a \int \frac {x^{2 n - 2} \rd x} {\paren {a x^2 + b x + c}^n}\) Linear Combination of Primitives

$\blacksquare$


Sources