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Summary

In mathematics, a quadratic integral is an integral of the form

$\int {\frac {dx}{a+bx+cx^{2}}}.$ It can be evaluated by completing the square in the denominator.

$\int {\frac {dx}{a+bx+cx^{2}}}={\frac {1}{c}}\int {\frac {dx}{\left(x+{\frac {b}{2c}}\right)^{2}+\left({\frac {a}{c}}-{\frac {b^{2}}{4c^{2}}}\right)}}.$ Positive-discriminant case

Assume that the discriminant q = b2 − 4ac is positive. In that case, define u and A by

$u=x+{\frac {b}{2c}},$ and
$-A^{2}={\frac {a}{c}}-{\frac {b^{2}}{4c^{2}}}={\frac {1}{4c^{2}}}\left(4ac-b^{2}\right).$ The quadratic integral can now be written as

$\int {\frac {dx}{a+bx+cx^{2}}}={\frac {1}{c}}\int {\frac {du}{u^{2}-A^{2}}}={\frac {1}{c}}\int {\frac {du}{(u+A)(u-A)}}.$ ${\frac {1}{(u+A)(u-A)}}={\frac {1}{2A}}\left({\frac {1}{u-A}}-{\frac {1}{u+A}}\right)$ allows us to evaluate the integral:
${\frac {1}{c}}\int {\frac {du}{(u+A)(u-A)}}={\frac {1}{2Ac}}\ln \left({\frac {u-A}{u+A}}\right)+{\text{constant}}.$ The final result for the original integral, under the assumption that q > 0, is

$\int {\frac {dx}{a+bx+cx^{2}}}={\frac {1}{\sqrt {q}}}\ln \left({\frac {2cx+b-{\sqrt {q}}}{2cx+b+{\sqrt {q}}}}\right)+{\text{constant}}.$ Negative-discriminant case

In case the discriminant q = b2 − 4ac is negative, the second term in the denominator in

$\int {\frac {dx}{a+bx+cx^{2}}}={\frac {1}{c}}\int {\frac {dx}{\left(x+{\frac {b}{2c}}\right)^{2}+\left({\frac {a}{c}}-{\frac {b^{2}}{4c^{2}}}\right)}}.$ is positive. Then the integral becomes
{\begin{aligned}{\frac {1}{c}}\int {\frac {du}{u^{2}+A^{2}}}&={\frac {1}{cA}}\int {\frac {du/A}{(u/A)^{2}+1}}\\[9pt]&={\frac {1}{cA}}\int {\frac {dw}{w^{2}+1}}\\[9pt]&={\frac {1}{cA}}\arctan(w)+\mathrm {constant} \\[9pt]&={\frac {1}{cA}}\arctan \left({\frac {u}{A}}\right)+{\text{constant}}\\[9pt]&={\frac {1}{c{\sqrt {{\frac {a}{c}}-{\frac {b^{2}}{4c^{2}}}}}}}\arctan \left({\frac {x+{\frac {b}{2c}}}{\sqrt {{\frac {a}{c}}-{\frac {b^{2}}{4c^{2}}}}}}\right)+{\text{constant}}\\[9pt]&={\frac {2}{\sqrt {4ac-b^{2}\,}}}\arctan \left({\frac {2cx+b}{\sqrt {4ac-b^{2}}}}\right)+{\text{constant}}.\end{aligned}} 