Indefinite Integrals with $ax^2 + bx + c$

$ax^2 + bx + c$
$\int \dfrac{dx}{ax^2 + bx + c} = \begin{cases} \dfrac{2}{\sqrt{4ac-b^2}} \arctan \dfrac{2ax+b}{\sqrt{4ac-b^2}} \\ \dfrac{1}{\sqrt{b^2-4ac}} \ln\left(\dfrac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}}\right) \end{cases}$
$\int \dfrac{xdx}{ax^2 + bx + c} = \dfrac{1}{2a} \ln\left(ax^2+bx+c\right) - \dfrac{b}{2a}\int \dfrac{dx}{ax^2 + bx + c}$
$\int \dfrac{x^2dx}{ax^2 + bx + c} = \dfrac{x}{a} - \dfrac{b}{2a^2}\ln\left(ax^2+bx+c\right) + \dfrac{b^2-2ac}{2a^2} \dfrac{b}{2a}\int \dfrac{dx}{ax^2 + bx + c}$
$\int \dfrac{x^{m}dx}{ax^2 + bx + c} = \dfrac{x^{m-1}}{\left(m-1\right)a} - \dfrac{c}{a}\int \dfrac{x^{m-2}dx}{ax^2 + bx + c} - \dfrac{b}{a}\int \dfrac{x{m-1}dx}{ax^2 + bx + c}$
$\int \dfrac{dx}{x\left(ax^2 + bx + c\right)} = \dfrac{1}{2c} \ln\left(\dfrac{x^2}{ax^2+bx+c}\right) - \dfrac{b}{2c} \int \dfrac{dx}{ax^2+bx+c}$
$\int \dfrac{dx}{x^2\left(ax^2 + bx + c\right)} = \dfrac{b}{2c^2} \ln\left(\dfrac{ax^2+bx+c}{x^2}\right) - \dfrac{1}{cx} + \dfrac{b^2-2ac}{2c^2} \int \dfrac{dx}{ax^2+bx+c}$
$\int \dfrac{dx}{x^n\left(ax^2 + bx + c\right)} = - \dfrac{1}{\left(n-1\right)cx^{n-1}} - \dfrac{b}{c}\int \dfrac{dx}{x^{n-1}\left(ax^2+bx+c\right)} - \dfrac{a}{c}\int \dfrac{dx}{x^{n-2}\left(ax^2+bx+c\right)}$
$\int \dfrac{dx}{\left(ax^2+bx+c \right)^2} = \dfrac{2ax+b}{\left(4ac-b^2 \right) \left(ax^2+bx+c \right)} + \dfrac{2a}{4ac-b^2} \int \dfrac{dx}{ax^2+bx+c}$
$\int \dfrac{xdx}{\left(ax^2+bx+c^2 \right)^2} = - \dfrac{bx+2c}{\left(4ac-b^2\right)\left(ax^2+bx+c \right)} - \dfrac{b}{4ac-b^2} \int \dfrac{dx}{ax^2+bx+c}$
$\int \dfrac{x^2 dx}{\left(ax^2+bx+c \right)^2} = \dfrac{\left(b^2-2ac\right)x+bc}{a\left(4ac-b^2\right)\left(ax^2+bx+c\right)} + \dfrac{2c}{4ac-b^2} \int \dfrac{dx}{ax^2+bx+c}$
$\int \dfrac{x^{m}dx}{\left(ax^2+bx+c\right)^n} = - \dfrac{x^{m-1}}{\left(2n-m-1\right)a\left(ax^2+bx+c\right)^{n-1}} + \dfrac{\left(m-1\right)c}{\left(2n-m-1\right)a} \int \dfrac{x^{m-2}dx}{\left(ax^2+bx+c\right)^{n}}$
$- \dfrac{\left(n-m\right)b}{\left(2n-m-1\right)a} \int \dfrac{x^{m-1}dx}{\left(ax^2+bx+c\right)^n}$
$\int \dfrac{x^{2n-1}dx}{\left(ax^2+bx+c\right)^{n}} = \dfrac{1}{a} \int \dfrac{x^{2n-3}dx}{\left(ax^2+bx+c\right)^{n-1}} - \dfrac{c}{a} \int \dfrac{x^{2n-3}dx}{\left(ax^2+bx+c\right)^{n}} - \dfrac{b}{a} \int \dfrac{x^{2n-2}dx}{\left(ax^2+bx+c\right)^n}$
$\int \dfrac{dx}{x\left(ax^2+bx+c\right)^2} = \dfrac{1}{2c\left(ax^2+bx+c\right)} - \dfrac{b}{2c} \int \dfrac{dx}{\left(ax^2+bx+c\right)^2} + \dfrac{1}{c} \int \dfrac{dx}{x\left(ac^2+bx+c\right)}$
$\int \dfrac{dx}{x^2\left(ax^2+bx+c\right)^2} = -\dfrac{1}{cx\left(ax^2+bx+c\right)} - \dfrac{3a}{c} \int \dfrac{dx}{\left(ax^2+bx+c\right)^2} - \dfrac{2b}{c} \int \dfrac{dx}{x\left(ac^2+bx+c\right)^2}$
$\int \dfrac{dx}{x^{m}\left(ax^2+bx+c\right)^{n}} = -\dfrac{1}{\left(m-1\right)cx^{m-1}\left(ax^2+bx+c\right)^{n-1}} - \dfrac{\left(m+2n-3\right)a}{\left(m-1\right)c} \int \dfrac{dx}{x^{m-2}\left(ax^2+bx+c\right)^{n}}$
$+ \dfrac{\left(m+n-2\right)b}{\left(m-1\right)c} \int \dfrac{dx}{x^{m-1}\left(ac^2+bx+c\right)^{n}}$
$\sqrt{ax^2 + bx + c}$
$\int \dfrac{dx}{\sqrt{ax^2+bx+c}} = \begin{cases} \dfrac{1}{\sqrt{a}} \ln\left(2\sqrt{a}\sqrt{ax^2+bx+c}+ax+b\right)\\ -\dfrac{1}{\sqrt{-a}} \arcsin\left(\dfrac{2ax+b}{\sqrt{b^2-4ac}}\right)\quad ou\quad \dfrac{1}{\sqrt{a}} argsh\left(\dfrac{2ax+b}{\sqrt{4ac-b^2}}\right) \end{cases}$
$\int \dfrac{xdx}{\sqrt{ax^2+bx+c}} = \dfrac{\sqrt{ax^2+bx+c}}{a} - \dfrac{b}{2a} \int \dfrac{dx}{\sqrt{ax^2+bx+c}}$
$\int \dfrac{x^2dx}{\sqrt{ax^2+bx+c}} = \dfrac{2ax-3b}{4a^2}\sqrt{ax^2+bx+c}+\dfrac{3b^2-4ac}{8a^2}\int\dfrac{dx}{\sqrt{ax^2+bx+c}}$
$\int \dfrac{dx}{x\sqrt{ax^2+bx+c}} = \begin{cases} -\dfrac{1}{\sqrt{c}}\ln\left(\dfrac{2\sqrt{c}\sqrt{ax^2+bx+c}+bx+2c}{x}\right)\\ -\dfrac{1}{\sqrt{-c}} \arcsin\left(\dfrac{bx+2c}{\left|x\right\vert\sqrt{b^2-4ac}}\right)\quad ou\quad -\dfrac{1}{\sqrt{c}}argsh\left(\dfrac{bx+2c}{\left|x\right\vert\sqrt{4ac-b^2}}\right)\end{cases}$
$\int \dfrac{dx}{x^2\sqrt{ax^2+bx+c}} = -\dfrac{\sqrt{ax^2+bx+c}}{cx} - \dfrac{b}{2c}\int\dfrac{dx}{x\sqrt{ax^2+bx+c}}$
$\int \sqrt{ax^2+bx+c} dx = \dfrac{\left(2ax+b\right)\sqrt{ax^2+bx+c}}{4a} + \dfrac{4ac-b^2}{8a}\int\dfrac{dx}{\sqrt{ax^2+bx+c}}$
$\int x\sqrt{ax^2+bx+c} dx = \dfrac{\left(ax^2+bx+c\right)^{3/2}}{3a} - \dfrac{b\left(2ax+b\right)}{8a^2}\sqrt{ax^2+bx+c} - \dfrac{b\left(4ac-b^2\right)}{16a^2} \int \dfrac{dx}{\sqrt{ax^2+bx+c}}$
$\int x^2\sqrt{ax^2+bx+c} dx = \dfrac{6ax-5b}{24a^2} \left(ax^2+bx+c\right)^{3/2} + \dfrac{5b^2-4ac}{16a^2}\int\sqrt{ax^2+bx+c} dx$
$\int \dfrac{\sqrt{ax^2+bx+c}}{x} dx = \sqrt{ax^2+bx+c} + \dfrac{b}{2} \int\dfrac{dx}{\sqrt{ax^2+bx+c}} + c\int\dfrac{dx}{x\sqrt{ax^2+bx+c}}$
$\int \dfrac{\sqrt{ax^2+bx+c}}{x^2} dx = -\dfrac{\sqrt{ax^2+bx+c}}{x} + a\int\dfrac{dx}{\sqrt{ax^2+bx+c}} + \dfrac{b}{2}\int\dfrac{dx}{x\sqrt{ax^2+bx+c}}$
$\int \dfrac{dx}{\left(ax^2+bx+c\right)^{3/2}} = \dfrac{2\left(2ax+b\right)}{\left(4ac-b^2\right)\sqrt{ax^2+bx+c}}$
$\int \dfrac{xdx}{\left(ax^2+bx+c\right)^{3/2}} = \dfrac{2\left(2bx+2c\right)}{\left(b^2-4ac\right)\sqrt{ax^2+bx+c}}$
$\int \dfrac{x^2dx}{\left(ax^2+bx+c\right)^{3/2}} = \dfrac{\left(2b^2-4ac\right)x+2bc}{a\left(4ac-b^2\right)\sqrt{ax^2+bx+c}}+\dfrac{1}{a}\int\dfrac{dx}{\sqrt{ax^2+bx+c}}$
$\int\dfrac{dx}{x\left(ax^2+bx+c\right)^{3/2}} = \dfrac{1}{c\sqrt{ax^2+bx+c}} + \dfrac{1}{c}\int\dfrac{dx}{x\sqrt{ax^2+x+c}} - \dfrac{b}{2c}\int\dfrac{dx}{\left(ax^2+bx+c\right)^{3/2}}$
$\int\dfrac{dx}{x^2\left(ax^2+bx+c\right)^{3/2}} = -\dfrac{ax^2+2bx+c}{c^2x\sqrt{ax^2+bx+c}} + \dfrac{v^2-2ac}{2c^2}\int\dfrac{dx}{\left(ax^2+x+c\right)^{3/2}} - \dfrac{3b}{2c^2}\int\dfrac{dx}{x\sqrt{ax^2+bx+c}}$
$\int\left(ax^2+bx+c\right)^{n+1/2} dx = \dfrac{\left(2ax+b\right)\left(ax^2+bx+c\right)^{n+1/2}}{4a\left(n+1\right)} + \dfrac{\left(2n+1\right)\left(4ac-b^2\right)}{8a\left(n+1\right)}\int\left(ax^2+bx+c\right)^{n-1/2}dx$
$\int x\left(ax^2+bx+c\right)^{n+1/2}dx = \dfrac{\left(ax^2+bx+c\right)^{n+3/2}}{a\left(2n+3\right)} - \dfrac{b}{2a}\int \left((ax^2+bx+c\right)^{n+1/2} dx$
$\int \dfrac{dx}{\left(ax^2+bx+c\right)^{n+1/2}} = \dfrac{2\left(2ax+b\right)}{\left(2n-1\right)\left(4ac-b^2\right)\left(ax^2+bx+c\right)^{n-1/2}} + \dfrac{8a\left(n-1\right)}{\left(2n-1\right)\left(4ac-b^2\right)} \int \dfrac{dx}{\left(ax^2+bx+c\right)^{n-1/2}}$
$\int \dfrac{dx}{x\left(ax^2+bx+c\right)^{n+1/2}} = \dfrac{1}{\left(2n-1\right)c\left(ax^2+bx+c\right)^{n-1/2}} + \dfrac{1}{c} \int \dfrac{dx}{x\left(ax^2+bx+c\right)^{n-1/2}} - \dfrac{b}{2c} \int \dfrac{dx}{\left(ax^2+bx+c\right)^{n+1/2}}$

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