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

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Theorem

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

Then:

$\ds \int \frac {x^2 \rd x} {\sqrt {a x^2 + b x + c} } = \frac {2 a x - 3 b} {4 a^2} \sqrt {a x^2 + b x + c} + \frac {3 b^2 - 4 a c} {8 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$


Proof

First:

\(\ds z\) \(=\) \(\ds a x^2 + b x + c\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d z} {\d x}\) \(=\) \(\ds 2 a x + b\) Derivative of Power


Then:

\(\ds \) \(\) \(\ds \int \frac {x^2 \rd x} {\sqrt {a x^2 + b x + c} }\)
\(\ds \) \(=\) \(\ds \frac 1 a \int \frac {a x^2 \rd x} {\sqrt {a x^2 + b x + c} }\)
\(\ds \) \(=\) \(\ds \frac 1 a \int \frac {\paren {a x^2 + b x + c - b x - c} \rd x} {\sqrt {a x^2 + b x + c} }\)
\(\ds \) \(=\) \(\ds \frac 1 a \int \frac {\paren {a x^2 + b x + c} \rd x} {\sqrt {a x^2 + b x + c} } - \frac b a \int \frac {x \rd x} {\sqrt {a x^2 + b x + c} } - \frac c a \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac 1 a \int \sqrt {a x^2 + b x + c} \rd x - \frac b a \int \frac {x \rd x} {\sqrt {a x^2 + b x + c} } - \frac c a \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) simplifying
\(\ds \) \(=\) \(\ds \frac 1 a \paren {\frac {\paren {2 a x + b} \sqrt {a x^2 + b x + c} } {4 a} + \frac {4 a c - b^2} {8 a} \int \frac {\d x} {\sqrt {a x^2 + b x + c} } }\) Primitive of $\sqrt {a x^2 + b x + c}$
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac b a \int \frac {x \rd x} {\sqrt {a x^2 + b x + c} } - \frac c a \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\)
\(\ds \) \(=\) \(\ds \frac 1 a \paren {\frac {\paren {2 a x + b} \sqrt {a x^2 + b x + c} } {4 a} + \frac {4 a c - b^2} {8 a} \int \frac {\d x} {\sqrt {a x^2 + b x + c} } }\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac b a \paren {\frac {\sqrt {a x^2 + b x + c} } a - \frac b {2 a} \int \frac {\d x} {\sqrt {a x^2 + b x + c} } } - \frac c a \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) Primitive of $\dfrac x {\sqrt {a x^2 + b x + c} }$
\(\ds \) \(=\) \(\ds \frac {\paren {2 a x + b} \sqrt {a x^2 + b x + c} } {4 a^2} + \frac {4 a c - b^2} {8 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) simplifying
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac {b \sqrt {a x^2 + b x + c} } {a^2} + \frac {b^2} {2 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} } - \frac c a \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\)
\(\ds \) \(=\) \(\ds \frac {\paren {2 a x + b} \sqrt {a x^2 + b x + c} } {4 a^2} + \frac {4 a c - b^2} {8 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) common denominators
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac {4 b \sqrt {a x^2 + b x + c} } {4 a^2} + \frac {4 b^2} {8 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} } - \frac {8 a c} {8 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\)
\(\ds \) \(=\) \(\ds \frac {\paren {2 a x + b - 4 b} \sqrt {a x^2 + b x + c} } {4 a^2} + \frac {4 a c - b^2 + 4 b^2 - 8 a c} {8 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) combining like terms
\(\ds \) \(=\) \(\ds \frac {2 a x - 3 b} {4 a^2} \sqrt {a x^2 + b x + c} + \frac {3 b^2 - 4 a c} {8 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) simplifying

$\blacksquare$


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