Theorem: Primitive of Power of x over Power of a x squared plus b x plus c

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

Proof:
With a view to expressing the problem 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}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = \frac {-\paren {2 a x + b} n} {\paren {a x^2 + b x + c}^{n + 1} }
      | c = Derivative of Power and Chain Rule for Derivatives
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = x^m
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = \frac {x^{m + 1} } {m + 1}
      | c = Primitive of Power
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {\paren {a x^2 + b x + c}^n}
      | r = \int \frac 1 {\paren {a x^2 + b x + c}^n} x^m \rd x
      | c = 
}}
{{eqn | r = \frac 1 {\paren {a x^2 + b x + c}^n} \frac {x^{m + 1} } {m + 1} - \int \frac {x^{m + 1} } {m + 1} \frac {-\paren {2 a x + b} n} {\paren {a x^2 + b x + c}^{n + 1} } \rd x
      | c = Integration by Parts
}}
{{eqn | n = 1
      | r = \frac {x^{m + 1} } {\paren {m + 1} \paren {a x^2 + b x + c}^n} + \frac n {m + 1} \int \frac {\paren {2 a x + b} x^{m + 1} } {\paren {a x^2 + b x + c}^{n + 1} } \rd x
      | c = simplifying
}}
{{end-eqn}}
Meanwhile:
{{begin-eqn}}
{{eqn | o = 
      | r = \int \frac {x^m \rd x} {\paren {a x^2 + b x + c}^n}
      | c = 
}}
{{eqn | r = \frac 1 a \int \frac {x^{m - 2} a x^2 \rd x} {\paren {a x^2 + b x + c}^n}
      | c = Primitive of Constant Multiple of Function
}}
{{eqn | r = \frac 1 a \int \frac {x^{m - 2} \paren {a x^2 + b x + c - b x - c} \rd x} {\paren {a x^2 + b x + c}^n}
      | c = 
}}
{{eqn | r = \frac 1 a \int \frac {x^{m - 2} \rd x} {\paren {a x^2 + b x + c}^{n - 1} }
      | c = Linear Combination of Primitives
}}
{{eqn | o =
      | ro= -
      | r = \frac b a \int \frac {x^{m - 1} \rd x} {\paren {a x^2 + b x + c}^n} - \frac c a \int \frac {x^{m - 2} \rd x} {\paren {a x^2 + b x + c}^n}
      | c = and simplification
}}
{{eqn | r = \frac 1 a \paren {\frac {x^{m - 1} } {\paren {m - 1} \paren {a x^2 + b x + c}^{n - 1} } + \frac {n - 1} {m - 1} \int \frac {\paren {2 a x + b} x^{m - 1} } {\paren {a x^2 + b x + c}^n} \rd x}
      | c = from $(1)$
}}
{{eqn | o =
      | ro= -
      | r = \frac b a \int \frac {x^{m - 1} \rd x} {\paren {a x^2 + b x + c}^n} - \frac c a \int \frac {x^{m - 2} \rd x} {\paren {a x^2 + b x + c}^n}
      | c = and simplification
}}
{{eqn | r = \frac {x^{m - 1} } {a \paren {m - 1} \paren {a x^2 + b x + c}^{n - 1} } + \frac {2 \paren {n - 1} } {m - 1} \int \frac {x^m} {\paren {a x^2 + b x + c}^n} \rd x
      | c = multiplying out
}}
{{eqn | o =
      | ro= +
      | r = \frac {b \paren {n - 1} } {a \paren {m - 1} } \int \frac {x^{m - 1} } {\paren {a x^2 + b x + c}^n} \rd x
      | c = 
}}
{{eqn | o =
      | ro= -
      | r = \frac b a \int \frac {x^{m - 1} \rd x} {\paren {a x^2 + b x + c}^n} - \frac c a \int \frac {x^{m - 2} \rd x} {\paren {a x^2 + b x + c}^n}
      | c = 
}}
{{eqn | ll= \leadsto
      | o = 
      | r = \paren {1 - \frac {2 \paren {n - 1} } {m - 1} } \int \frac {x^m \rd x} {\paren {a x^2 + b x + c}^n}
      | c = rearranging
}}
{{eqn | r = \frac {x^{m - 1} } {a \paren {m - 1} \paren {a x^2 + b x + c}^{n - 1} }
      | c = 
}}
{{eqn | o =
      | ro= +
      | r = \paren {\frac {b \paren {n - 1} } {a \paren {m - 1} } - \frac b a} \int \frac {x^{m - 1} } {\paren {a x^2 + b x + c}^n} \rd x
      | c = 
}}
{{eqn | o =
      | ro= -
      | r = \frac c a \int \frac {x^{m - 2} \rd x} {\paren {a x^2 + b x + c}^n}
      | c = 
}}
{{eqn | ll= \leadsto
      | o = 
      | r = \int \frac {x^m \rd x} {\paren {a x^2 + b x + c}^n}
      | c = 
}}
{{eqn | r = \frac {x^{m - 1} } {\paren {2 n - m - 1} a \paren {a x^2 + b x + c}^{n - 1} }
      | c = 
}}
{{eqn | o =
      | ro= +
      | r = \frac {\paren {m - 1} c} {\paren {2 n - m - 1} a} \int \frac {x^{m - 2} \rd x} {\paren {a x^2 + b x + c}^n}
      | c = 
}}
{{eqn | o =
      | ro= -
      | r = \frac {\paren {n - m} b} {\paren {2 n - m - 1} a} \int \frac {x^{m - 1} \rd x} {\paren {a x^2 + b x + c}^n}
      | c = simplification
}}
{{end-eqn}}
{{qed}}