Theorem: Primitive of Reciprocal of Power of x by Power of a x squared plus b x plus c

Let $a \in \R_{\ne 0}$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^m \paren {a x^2 + b x + c}^n}
      | r = \frac {-1} {\paren {m - 1} c x^{m - 1} \paren {a x^2 + b x + c}^{n - 1} }
}}
{{eqn | o =
      | ro= -
      | r = \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}
}}
{{eqn | o =
      | ro= -
      | r = \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}}

Proof:
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 Primitives
}}
{{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:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
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 Primitives
}}
{{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}}