{{begin-eqn}} {{eqn | r = \int \frac {\d x} {x \paren {a x^2 + b x + c}^{n + \frac 1 2} }       | o =        | c =  }} {{eqn | r = \int \frac {c \rd x} {c x \paren {a x^2 + b x + c}^{n + \frac 1 2} }       | c = multiplying top and bottom by $c$ }} {{eqn | r = \frac 1 c \int \frac {\paren {a x^2 + b x + c - a x^2 - b x} \rd x} {c x \paren {a x^2 + b x + c}^{n + \frac 1 2} }       | c =  }} {{eqn | r = \frac 1 c \int \frac {\paren {a x^2 + b x + c} \rd x} {x \paren {a x^2 + b x + c}^{n + \frac 1 2} } - \frac a c \int \frac {x^2 \rd x} {x \paren {a x^2 + b x + c}^{n + \frac 1 2} } - \frac b c \int \frac {x \rd x} {x \paren {a x^2 + b x + c}^{n + \frac 1 2} }       | c = Linear Combination of Integrals }} {{eqn | n = 1       | r = \frac 1 c \int \frac {\d x} {x \paren {a x^2 + b x + c}^{n - \frac 1 2} } - \frac a c \int \frac {x \rd x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} } - \frac b c \int \frac {\d x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} }       | c = simplifying }} {{end-eqn}} Now take: {{begin-eqn}} {{eqn | r = \int \frac {x \rd x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} }       | o =        | c =  }} {{eqn | r = \int \frac {2 a x \rd x} {2 a \paren {a x^2 + b x + c}^{n + \frac 1 2} }       | c = multiplying top and bottom by $2 a$ }} {{eqn | r = \int \frac {\paren {2 a x + b - b} \rd x} {2 a \paren {a x^2 + b x + c}^{n + \frac 1 2} }       | c =  }} {{eqn | n = 2       | r = \frac 1 {2 a} \int \frac {\paren {2 a x + b} \rd x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} } - \frac b {2 a} \int \frac {\d x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} }       | c = Linear Combination of Integrals }} {{end-eqn}} Let: {{begin-eqn}} {{eqn | l = z       | r = a x^2 + b x + c       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d z} {\d x}        | r = 2 a x + b       | c = Derivative of Power }} {{eqn | ll= \leadsto       | l = \int \frac {\paren {2 a x + b} \rd x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} }       | r = \int z^{-n - \frac 1 2} \rd z       | c = Integration by Substitution }} {{eqn | r = \frac {z^{-n + \frac 1 2} } {-n + \frac 1 2}       | c = Primitive of Power }} {{eqn | n = 3       | r = \frac {-2} {\paren {2 n - 1} \paren {a x^2 + b x + c}^{n - \frac 1 2} }       | c = simplifying }} {{end-eqn}} Thus: {{begin-eqn}} {{eqn | r = \int \frac {x \rd x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} }       | o =        | c =  }} {{eqn | r = \frac 1 {2 a} \int \frac {\paren {2 a x + b} \rd x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} } - \frac b {2 a} \int \frac {\d x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} }       | c = from $(2)$ }} {{eqn | r = \frac 1 {2 a} \paren {\frac {-2} {\paren {2 n - 1} \paren {a x^2 + b x + c}^{n - \frac 1 2} } } - \frac b {2 a} \int \frac {\d x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} }       | c = from $(3)$ }} {{eqn | r = \frac {-1} {a \paren {2 n - 1} \paren {a x^2 + b x + c}^{n - \frac 1 2} } - \frac b {2 a} \int \frac {\d x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} }       | c = simplifying }} {{end-eqn}} So: {{begin-eqn}} {{eqn | r = \int \frac {\d x} {x \paren {a x^2 + b x + c}^{n + \frac 1 2} }       | o =        | c =  }} {{eqn | r = \frac 1 c \int \frac {\d x} {x \paren {a x^2 + b x + c}^{n - \frac 1 2} } - \frac a c \int \frac {x \rd x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} } - \frac b c \int \frac {\d x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} }       | c = from $(1)$ }} {{eqn | r = \frac 1 c \int \frac {\d x} {x \paren {a x^2 + b x + c}^{n - \frac 1 2} }       | c =  }} {{eqn | o =        | ro= -       | r = \frac a c \paren {\frac {-1} {a \paren {2 n - 1} \paren {a x^2 + b x + c}^{n - \frac 1 2} } - \frac b {2 a} \int \frac {\d x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} } }       | c = from $(3)$ }} {{eqn | o =        | ro= -       | r = \frac b c \int \frac {\d x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} }       | c =  }} {{eqn | r = \frac 1 c \int \frac {\d x} {x \paren {a x^2 + b x + c}^{n - \frac 1 2} }       | c =  }} {{eqn | o =        | ro= +       | r = \frac 1 {\paren {2 n - 1} c \paren {a x^2 + b x + c}^{n - \frac 1 2} } + \frac b {2 c} \int \frac {\d x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} }       | c = simplifying }} {{eqn | o =        | ro= -       | r = \frac b c \int \frac {\d x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} }       | c =  }} {{eqn | r = \frac 1 {\paren {2 n - 1} c \paren {a x^2 + b x + c}^{n - \frac 1 2} } + \frac 1 c \int \frac {\d x} {x \paren {a x^2 + b x + c}^{n - \frac 1 2} } - \frac b {2 c} \int \frac {\d x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} }       | c = gathering terms }} {{end-eqn}} {{qed}}