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Question Number 138656 by ajfour last updated on 16/Apr/21

$$I=\int \frac{dx}{(px+q)\sqrt{{ax}^2+bx+c}}$$

Answered by Ar Brandon last updated on 16/Apr/21

$$\mathcal{I}=\int \frac{\mathrm{dx}}{(\mathrm{px}+\mathrm{q})\sqrt{{\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}}}$$$$\mathrm{u}=\frac{1}{\mathrm{px}+\mathrm{q}}\Rightarrow \mathrm{x}=\frac{1}{\mathrm{up}}-\frac{\mathrm{q}}{\mathrm{p}}\Rightarrow \mathrm{du}=-\frac{\mathrm{p}}{{(\mathrm{px}+\mathrm{q})}^2}\mathrm{dx}=-\left({\mathrm{u}}^2\mathrm{p}\right)\mathrm{dx}$$$$\mathcal{I}=-\int \frac{\mathrm{u}}{\sqrt{\mathrm{a}{(\frac{1}{\mathrm{up}}-\frac{\mathrm{q}}{\mathrm{p}})}^2+\mathrm{b}(\frac{1}{\mathrm{up}}-\frac{\mathrm{q}}{\mathrm{p}})+\mathrm{c}}}\cdot \frac{\mathrm{du}}{\left({\mathrm{u}}^2\mathrm{p}\right)}$$$$=\mp \int \frac{\mathrm{du}}{\sqrt{\mathrm{a}{(1-\mathrm{uq})}^2+\mathrm{b}(\mathrm{up}-{\mathrm{u}}^2\mathrm{pq})+{\mathrm{cu}}^2{\mathrm{p}}^2}}$$$$\mathrm{a}(1-2\mathrm{uq}+{\mathrm{u}}^2{\mathrm{q}}^2)+\mathrm{b}(\mathrm{up}-{\mathrm{u}}^2\mathrm{pq})+{\mathrm{cu}}^2{\mathrm{p}}^2$$$$=({\mathrm{aq}}^2-\mathrm{bpq}+{\mathrm{cp}}^2){\mathrm{u}}^2+(-2\mathrm{aq}+\mathrm{bp})\mathrm{u}+\left(\mathrm{a}\right)$$$$\mathcal{I}=\pm \int \frac{\mathrm{du}}{\sqrt{({\mathrm{aq}}^2-\mathrm{b}+{\mathrm{cp}}^2){\mathrm{u}}^2+(-2\mathrm{aq}+\mathrm{bp})\mathrm{u}+\left(\mathrm{a}\right)}}$$

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