Primitive of x squared over square of a x squared plus b x plus c

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Theorem

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

Then:

$\ds \int \frac {x^2 \rd x} {\paren {a x^2 + b x + c}^2} = \frac {\paren {b^2 - 2 a c} x + b c} {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {2 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}$


Proof

\(\ds \) \(\) \(\ds \int \frac {x^2 \rd x} {\paren {a x^2 + b x + c}^2}\)
\(\ds \) \(=\) \(\ds \frac 1 a \int \frac {a x^2 \rd x} {\paren {a x^2 + b x + c}^2}\) Primitive of Constant Multiple of Function
\(\ds \) \(=\) \(\ds \frac 1 a \int \frac {\paren {a x^2 + b x + c - b x - c} \rd x} {\paren {a x^2 + b x + c}^2}\)
\(\ds \) \(=\) \(\ds \frac 1 a \int \frac {\paren {a x^2 + b x + c} \rd x} {\paren {a x^2 + b x + c}^2} - \frac 1 a \int \frac {\paren {b x + c} \rd x} {\paren {a x^2 + b x + c}^2}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac 1 a \int \frac {\paren {a x^2 + b x + c} \rd x} {\paren {a x^2 + b x + c}^2} - \frac b a \int \frac {x \rd x} {\paren {a x^2 + b x + c}^2} - \frac c a \int \frac {\d x} {\paren {a x^2 + b x + c}^2}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac 1 a \int \frac {\d x} {a x^2 + b x + c} - \frac b a \int \frac {x \rd x} {\paren {a x^2 + b x + c}^2} - \frac c a \int \frac {\d x} {\paren {a x^2 + b x + c}^2}\) simplification
\(\ds \) \(=\) \(\ds - \frac b a \paren {\frac {- \paren {b x + 2 c} } {\paren {4 a c - b^2} \paren {a x^2 + b x + c} } - \frac b {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c} }\) Primitive of $\dfrac x {\paren {a x^2 + b x + c}^2}$
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac c a \int \frac {\d x} {\paren {a x^2 + b x + c}^2} + \frac 1 a \int \frac {\d x} {a x^2 + b x + c}\)
\(\ds \) \(=\) \(\ds \frac {\paren {b^2 x + 2 b c} } {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {b^2} {a \paren {4 a c - b^2} } \int \frac {\d x} {a x^2 + b x + c}\) multiplying out
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac c a \int \frac {\d x} {\paren {a x^2 + b x + c}^2} + \frac 1 a \int \frac {\d x} {a x^2 + b x + c}\)
\(\ds \) \(=\) \(\ds \frac {\paren {b^2 x + 2 b c} } {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {b^2 + 4 a c - b^2} {a \paren {4 a c - b^2} } \int \frac {\d x} {a x^2 + b x + c}\) collecting terms in $\ds \int \frac {\d x} {a x^2 + b x + c}$
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac c a \int \frac {\d x} {\paren {a x^2 + b x + c}^2}\)
\(\ds \) \(=\) \(\ds \frac {\paren {b^2 x + 2 b c} } {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {4 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}\) simplifying
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac c a \int \frac {\d x} {\paren {a x^2 + b x + c}^2}\)
\(\ds \) \(=\) \(\ds \frac {\paren {b^2 x + 2 b c} } {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {4 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac c a \paren {\frac {2 a x + b} {\paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {2 a} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c} }\) Primitive of $\dfrac 1 {\paren {a x^2 + b x + c}^2}$
\(\ds \) \(=\) \(\ds \frac {\paren {b^2 x + 2 b c} } {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {4 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac {2 a c x + b c} {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } - \frac {2 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}\) multiplying out
\(\ds \) \(=\) \(\ds \frac {\paren {b^2 - 2 a c} x + b c} {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {2 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}\) simplifying

$\blacksquare$


Sources

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