First: {{begin-eqn}} {{eqn | o =        | r = \int \frac {\d x} {x^m \paren {a x^2 + b x + c}^n}       | c =  }} {{eqn | r = \int \frac {c \rd x} {c x^m \paren {a x^2 + b x + c}^n}       | c = multiplying top and bottom by $c$ }} {{eqn | r = \frac 1 c \int \frac {c \rd x} {x^m \paren {a x^2 + b x + c}^n}       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 c \int \frac {a x^2 + b x + c - a x^2 - b x} {x^m \paren {a x^2 + b x + c}^n} \rd x       | c = adding and subtracting $a x^2 + b x$ }} {{eqn | r = \frac 1 c \int \frac {\paren {a x^2 + b x + c} \rd x} {x^m \paren {a x^2 + b x + c}^n} - \frac a c \int \frac {x^2 \rd x} {x^m \paren {a x^2 + b x + c}^n}       | c = Linear Combination of Integrals }} {{eqn | o =        | ro= -       | r = \frac b c \int \frac {x \rd x} {x^m \paren {a x^2 + b x + c}^n}       | c =  }} {{eqn | n = 1       | r = \frac 1 c \int \frac {\d x} {x^m \paren {a x^2 + b x + c}^{n - 1} } - \frac a c \int \frac {\d x} {x^{m - 2} \paren {a x^2 + b x + c}^n}       | c = simplification }} {{eqn | o =        | ro= -       | r = \frac b c \int \frac {\d x} {x^{m - 1} \paren {a x^2 + b x + c}^n}       | c =  }} {{end-eqn}} Next, with a view to obtaining an expression in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \frac 1 {\paren {a x^2 + b x + c}^{n - 1} }       | c =  }} {{eqn | r = \paren {a x^2 + b x + c}^{-\paren {n - 1} }       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = -\paren {n - 1} \paren {a x^2 + b x + c}^{-\paren {n - 1} - 1} \paren {2 a x + b}       | c = Chain Rule for Derivatives and Derivative of Power }} {{eqn | r = \frac {-\paren {n - 1} \paren {2 a x + b} } {\paren {a x^2 + b x + c}^n}       | c = simplifying }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = \frac 1 {x^m}       | c =  }} {{eqn | r = x^-m       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {x^{-m + 1} } {-m + 1}       | c = Primitive of Power }} {{eqn | r = \frac {-1} {\paren {m - 1} x^{m - 1} }       | c = simplifying }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \frac {\d x} {x^m \paren {a x^2 + b x + c}^{n - 1} }       | r = \int \frac 1 {\paren {a x^2 + b x + c}^{n - 1} } \frac 1 {x^m} \rd x       | c = Note the index is $n - 1$ }} {{eqn | r = \frac 1 {\paren {a x^2 + b x + c}^{n - 1} } \frac {-1} {\paren {m - 1} x^{m - 1} }       | c = Integration by Parts }} {{eqn | o =        | ro= -       | r = \int \frac {-1} {\paren {m - 1} x^{m - 1} } \frac {-\paren {n - 1} \paren {2 a x + b} } {\paren {a x^2 + b x + c}^n} \rd x       | c =  }} {{eqn | n = 2       | r = \frac {-1} {\paren {m - 1} x^{m - 1} \paren {a x^2 + b x + c}^{n - 1} }       | c = simplification }} {{eqn | o =        | ro= -       | r = \frac {2 a \paren {n - 1} } {m - 1} \int \frac {\d x} {x^{m - 2} \paren {a x^2 + b x + c}^n}       | c = and Linear Combination of Integrals }} {{eqn | o =        | ro= -       | r = \frac {b \paren {n - 1} } {m - 1} \int \frac {\d x} {x^{m - 1} \paren {a x^2 + b x + c}^n}       | c =  }} {{end-eqn}} Thus: {{begin-eqn}} {{eqn | o =        | r = \int \frac {\d x} {x^m \paren {a x^2 + b x + c}^n}       | c =  }} {{eqn | r = \frac 1 c \int \frac {\d x} {x^m \paren {a x^2 + b x + c}^{n - 1} } - \frac a c \int \frac {\d x} {x^{m - 2} \paren {a x^2 + b x + c}^n} - \frac b c \int \frac {\d x} {x^{m - 1} \paren {a x^2 + b x + c}^n}       | c = $(1)$ }} {{eqn | r = \frac 1 c \paren {\frac {-1} {\paren {m - 1} x^{m - 1} \paren {a x^2 + b x + c}^{n - 1} } - \frac {2 a \paren {n - 1} } {m - 1} \int \frac {\d x} {x^{m - 2} \paren {a x^2 + b x + c}^n} - \frac {b \paren {n - 1} } {m - 1} \int \frac {\d x} {x^{m - 1} \paren {a x^2 + b x + c}^n} }       | c = $(2)$ }} {{eqn | o =        | ro= -       | r = \frac a c \int \frac {\d x} {x^{m - 2} \paren {a x^2 + b x + c}^n} - \frac b c \int \frac {\d x} {x^{m - 1} \paren {a x^2 + b x + c}^n}       | c =  }} {{eqn | r = \frac {-1} {c \paren {m - 1} x^{m - 1} \paren {a x^2 + b x + c}^{n - 1} } - \frac {2 a \paren {n - 1} } {c \paren {m - 1} } \int \frac {\d x} {x^{m - 2} \paren {a x^2 + b x + c}^n} - \frac {b \paren {n - 1} } {c \paren {m - 1} } \int \frac {\d x} {x^{m - 1} \paren {a x^2 + b x + c}^n}       | c =  }} {{eqn | o =        | ro= -       | r = \frac a c \int \frac {\d x} {x^{m - 2} \paren {a x^2 + b x + c}^n} - \frac b c \int \frac {\d x} {x^{m - 1} \paren {a x^2 + b x + c}^n}       | c =  }} {{eqn | r = \frac {-1} {\paren {m - 1} c x^{m - 1} \paren {a x^2 + b x + c}^{n - 1} } - \frac {\paren {m + 2 n - 3} a} {\paren {m - 1} c} \int \frac {\d x} {x^{m - 2} \paren {a x^2 + b x + c}^n} - \frac {\paren {m - n + 2} b} {\paren {m - 1} c} \int \frac {\d x} {x^{m - 1} \paren {a x^2 + b x + c}^n} }} {{end-eqn}} {{qed}}