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

Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^2 \rd x} {\paren {a x^2 + b x + c}^2} = \frac {\paren {b^2 - 2 a c} x + b c} {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {2 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}$

Proof:
{{begin-eqn}}
{{eqn | o = 
      | r = \int \frac {x^2 \rd x} {\paren {a x^2 + b x + c}^2}
      | c = 
}}
{{eqn | r = \frac 1 a \int \frac {a x^2 \rd x} {\paren {a x^2 + b x + c}^2}
      | c = Primitive of Constant Multiple of Function
}}
{{eqn | r = \frac 1 a \int \frac {\paren {a x^2 + b x + c - b x - c} \rd x} {\paren {a x^2 + b x + c}^2}
      | c = 
}}
{{eqn | r = \frac 1 a \int \frac {\paren {a x^2 + b x + c} \rd x} {\paren {a x^2 + b x + c}^2} - \frac 1 a \int \frac {\paren {b x + c} \rd x} {\paren {a x^2 + b x + c}^2}
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac 1 a \int \frac {\paren {a x^2 + b x + c} \rd x} {\paren {a x^2 + b x + c}^2} - \frac b a \int \frac {x \rd x} {\paren {a x^2 + b x + c}^2} - \frac c a \int \frac {\d x} {\paren {a x^2 + b x + c}^2}
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac 1 a \int \frac {\d x} {a x^2 + b x + c} - \frac b a \int \frac {x \rd x} {\paren {a x^2 + b x + c}^2} - \frac c a \int \frac {\d x} {\paren {a x^2 + b x + c}^2}
      | c = simplification
}}
{{eqn | r = - \frac b a \paren {\frac {- \paren {b x + 2 c} } {\paren {4 a c - b^2} \paren {a x^2 + b x + c} } - \frac b {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c} }
      | c = Primitive of $\dfrac x {\paren {a x^2 + b x + c}^2}$
}}
{{eqn | o =
      | ro= -
      | r = \frac c a \int \frac {\d x} {\paren {a x^2 + b x + c}^2} + \frac 1 a \int \frac {\d x} {a x^2 + b x + c}
      | c = 
}}
{{eqn | r = \frac {\paren {b^2 x + 2 b c} } {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {b^2} {a \paren {4 a c - b^2} } \int \frac {\d x} {a x^2 + b x + c}
      | c = multiplying out
}}
{{eqn | o =
      | ro= -
      | r = \frac c a \int \frac {\d x} {\paren {a x^2 + b x + c}^2} + \frac 1 a \int \frac {\d x} {a x^2 + b x + c}
      | c = 
}}
{{eqn | r = \frac {\paren {b^2 x + 2 b c} } {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {b^2 + 4 a c - b^2} {a \paren {4 a c - b^2} } \int \frac {\d x} {a x^2 + b x + c}
      | c = collecting terms in $\ds \int \frac {\d x} {a x^2 + b x + c}$
}}
{{eqn | o =
      | ro= -
      | r = \frac c a \int \frac {\d x} {\paren {a x^2 + b x + c}^2}
      | c = 
}}
{{eqn | r = \frac {\paren {b^2 x + 2 b c} } {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {4 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}
      | c = simplifying
}}
{{eqn | o =
      | ro= -
      | r = \frac c a \int \frac {\d x} {\paren {a x^2 + b x + c}^2}
      | c = 
}}
{{eqn | r = \frac {\paren {b^2 x + 2 b c} } {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {4 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}
      | c = 
}}
{{eqn | o =
      | ro= -
      | r = \frac c a \paren {\frac {2 a x + b} {\paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {2 a} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c} }
      | c = Primitive of $\dfrac 1 {\paren {a x^2 + b x + c}^2}$
}}
{{eqn | r = \frac {\paren {b^2 x + 2 b c} } {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {4 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}
      | c = 
}}
{{eqn | o =
      | ro= -
      | r = \frac {2 a c x + b c} {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } - \frac {2 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}
      | c = multiplying out
}}
{{eqn | r = \frac {\paren {b^2 - 2 a c} x + b c} {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {2 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}