{{begin-eqn}} {{eqn | o =        | r = \int \frac {x^2 \ \mathrm d x} {\left({a x^2 + b x + c}\right)^2}       | c =  }} {{eqn | r = \frac 1 a \int \frac {a x^2 \ \mathrm d x} {\left({a x^2 + b x + c}\right)^2}       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 a \int \frac {\left({a x^2 + b x + c - b x - c}\right) \ \mathrm d x} {\left({a x^2 + b x + c}\right)^2}       | c =  }} {{eqn | r = \frac 1 a \int \frac {\left({a x^2 + b x + c }\right) \ \mathrm d x} {\left({a x^2 + b x + c}\right)^2} - \frac 1 a \int \frac {\left({b x + c}\right) \ \mathrm d x} {\left({a x^2 + b x + c}\right)^2}       | c = Linear Combination of Integrals }} {{eqn | r = \frac 1 a \int \frac {\left({a x^2 + b x + c }\right) \ \mathrm d x} {\left({a x^2 + b x + c}\right)^2} - \frac b a \int \frac {x \ \mathrm d x} {\left({a x^2 + b x + c}\right)^2} - \frac c a \int \frac {\mathrm d x} {\left({a x^2 + b x + c}\right)^2}       | c = Linear Combination of Integrals }} {{eqn | r = \frac 1 a \int \frac {\mathrm d x} {a x^2 + b x + c} - \frac b a \int \frac {x \ \mathrm d x} {\left({a x^2 + b x + c}\right)^2} - \frac c a \int \frac {\mathrm d x} {\left({a x^2 + b x + c}\right)^2}       | c = simplification }} {{eqn | r = - \frac b a \left({\frac {- \left({b x + 2 c}\right)} {\left({4 a c - b^2}\right) \left({a x^2 + b x + c}\right)} - \frac b {4 a c - b^2} \int \frac {\mathrm d x} {a x^2 + b x + c} }\right)       | c = Primitive of $\dfrac x {\left({a x^2 + b x + c}\right)^2}$ }} {{eqn | o =       | ro= -       | r = \frac c a \int \frac {\mathrm d x} {\left({a x^2 + b x + c}\right)^2} + \frac 1 a \int \frac {\mathrm d x} {a x^2 + b x + c}       | c =  }} {{eqn | r = \frac {\left({b^2 x + 2 b c}\right)} {a \left({4 a c - b^2}\right) \left({a x^2 + b x + c}\right)} + \frac {b^2} {a \left({4 a c - b^2}\right)} \int \frac {\mathrm d x} {a x^2 + b x + c}       | c = multiplying out }} {{eqn | o =       | ro= -       | r = \frac c a \int \frac {\mathrm d x} {\left({a x^2 + b x + c}\right)^2} + \frac 1 a \int \frac {\mathrm d x} {a x^2 + b x + c}       | c =  }} {{eqn | r = \frac {\left({b^2 x + 2 b c}\right)} {a \left({4 a c - b^2}\right) \left({a x^2 + b x + c}\right)} + \frac {b^2 + 4 a c - b^2} {a \left({4 a c - b^2}\right)} \int \frac {\mathrm d x} {a x^2 + b x + c}       | c = collecting terms in $\displaystyle \int \frac {\mathrm d x} {a x^2 + b x + c}$ }} {{eqn | o =       | ro= -       | r = \frac c a \int \frac {\mathrm d x} {\left({a x^2 + b x + c}\right)^2}       | c =  }} {{eqn | r = \frac {\left({b^2 x + 2 b c}\right)} {a \left({4 a c - b^2}\right) \left({a x^2 + b x + c}\right)} + \frac {4 c} {4 a c - b^2} \int \frac {\mathrm d x} {a x^2 + b x + c}       | c = simplifying }} {{eqn | o =       | ro= -       | r = \frac c a \int \frac {\mathrm d x} {\left({a x^2 + b x + c}\right)^2}       | c =  }} {{eqn | r = \frac {\left({b^2 x + 2 b c}\right)} {a \left({4 a c - b^2}\right) \left({a x^2 + b x + c}\right)} + \frac {4 c} {4 a c - b^2} \int \frac {\mathrm d x} {a x^2 + b x + c}       | c =  }} {{eqn | o =       | ro= -       | r = \frac c a \left({\frac {2 a x + b} {\left({4 a c - b^2}\right) \left({a x^2 + b x + c}\right)} + \frac {2 a} {4 a c - b^2} \int \frac {\mathrm d x} {a x^2 + b x + c} }\right)       | c = Primitive of $\dfrac 1 {\left({a x^2 + b x + c}\right)^2}$ }} {{eqn | r = \frac {\left({b^2 x + 2 b c}\right)} {a \left({4 a c - b^2}\right) \left({a x^2 + b x + c}\right)} + \frac {4 c} {4 a c - b^2} \int \frac {\mathrm d x} {a x^2 + b x + c}       | c =  }} {{eqn | o =       | ro= -       | r = \frac {2 a c x + b c} {a \left({4 a c - b^2}\right) \left({a x^2 + b x + c}\right)} - \frac {2 c } {4 a c - b^2} \int \frac {\mathrm d x} {a x^2 + b x + c}       | c = multiplying out }} {{eqn | r = \frac {\left({b^2 - 2 a c}\right) x + b c} {a \left({4 a c - b^2}\right) \left({a x^2 + b x + c}\right)} + \frac {2 c} {4 a c - b^2} \int \frac {\mathrm d x} {a x^2 + b x + c}       | c = simplifying }} {{end-eqn}} {{qed}}