Primitive of Reciprocal of x squared by a x squared plus b x plus c/Partial Fraction Expansion

Lemma for Primitive of $\dfrac 1 {x^2 \paren {a x^2 + b x + c} }$

 * $\dfrac 1 {x^2 \paren {a x^2 + b x + c} } \equiv \dfrac {-b} {c^2 x} + \dfrac 1 {c x^2} + \dfrac {a b x + b^2 - a c} {c^2 \paren {a x^2 + b x + c} }$

Proof
Setting $x = 0$ in $(1)$:

Equating coefficients of $x$ in $(1)$:

Equating coefficients of $x^3$ in $(1)$:

Equating coefficients of $x^2$ in $(1)$:

Summarising:

Hence the result.