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

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Theorem

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

Then:

$\ds \int \frac {\d x} {x^2 \paren {\sqrt {a x^2 + b x + c} }^3} = -\frac {a x^2 + 2 b x + c} {c^2 x \sqrt {a x^2 + b x + c} } + \frac {b^2 - 2 a c} {2 c^2} \int \frac {\d x} {\paren {\sqrt {a x^2 + b x + c} }^3} - \frac {3 b} {2 c^2} \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }$


Proof

\(\ds \) \(\) \(\ds \int \frac {\d x} {x^2 \paren {\sqrt {a x^2 + b x + c} }^3}\)
\(\ds \) \(=\) \(\ds \int \frac {c \rd x} {c x^2 \paren {\sqrt {a x^2 + b x + c} }^3}\) multiplying top and bottom by $c$
\(\ds \) \(=\) \(\ds \int \frac {\paren {a x^2 + b x + c - a x^2 - b x} \rd x} {c x^2 \paren {\sqrt {a x^2 + b x + c} }^3}\)
\(\ds \) \(=\) \(\ds \frac 1 c \int \frac {\paren {a x^2 + b x + c} \rd x} {x^2 \paren {\sqrt {a x^2 + b x + c} }^3} - \frac a c \int \frac {x^2 \rd x} {x^2 \paren {\sqrt {a x^2 + b x + c} }^3} - \frac b c \int \frac {x \rd x} {x^2 \paren {\sqrt {a x^2 + b x + c} }^3}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac 1 c \int \frac {\d x} {x^2 \sqrt {a x^2 + b x + c} } - \frac a c \int \frac {\d x} {\paren {\sqrt {a x^2 + b x + c} }^3} - \frac b c \int \frac {\d x} {x \paren {\sqrt {a x^2 + b x + c} }^3}\) simplifying
\(\ds \) \(=\) \(\ds \frac 1 c \paren {-\frac {\sqrt {a x^2 + b x + c} } {c x} - \frac b {2 c} \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } } - \frac a c \int \frac {\d x} {\paren {\sqrt {a x^2 + b x + c} }^3}\) Primitive of $\dfrac 1 {x^2 \sqrt {a x^2 + b x + c} }$
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac b c \paren {\frac 1 {c \sqrt {a x^2 + b x + c} } + \frac 1 c \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } - \frac b {2 c} \int \frac {\d x} {\paren {\sqrt {a x^2 + b x + c} }^3} }\) Primitive of $\dfrac 1 {x \paren {\sqrt {a x^2 + b x + c} }^3}$
\(\ds \) \(=\) \(\ds -\frac {a x^2 + b x + c} {c^2 x \sqrt {a x^2 + b x + c} } - \frac b {2 c^2} \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } - \frac {2 a c} {2 c^2} \int \frac {\d x} {\paren {\sqrt {a x^2 + b x + c} }^3}\) arranging common denominators for like terms
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac {b x} {c^2 x \sqrt {a x^2 + b x + c} } - \frac b {c^2} \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } + \frac {b^2} {2 c^2} \int \frac {\d x} {\paren {\sqrt {a x^2 + b x + c} }^3}\)
\(\ds \) \(=\) \(\ds -\frac {a x^2 + 2 b x + c} {c^2 x \sqrt {a x^2 + b x + c} } + \frac {b^2 - 2 a c} {2 c^2} \int \frac {\d x} {\paren {\sqrt {a x^2 + b x + c} }^3} - \frac {3 b} {2 c^2} \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }\) assembling final form

$\blacksquare$


Sources

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