Primitive of Root of a x + b over x squared

Theorem

 * $\ds \int \frac {\sqrt {a x + b} } {x^2} \rd x = -\frac {\sqrt {a x + b} } x + \frac a 2 \int \frac {\d x} {x \sqrt {a x + b} }$

Proof
From Reduction Formula for Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $a x + b$:


 * $\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \paren {a x + b}^{n - 1} \rd x$

Putting $m = -2$ and $n = \dfrac 1 2$:

Also see

 * Primitive of Reciprocal of $x \sqrt {a x + b}$ for $\ds \int \frac {\d x} {x \sqrt {a x + b} }$