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

I=∫(dx/((px+q)(√(ax^2 +bx+c))))

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

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

I=∫(dx/((px+q)(√(ax^2 +bx+c)))) u=(1/(px+q)) ⇒x=(1/(up))−(q/p)⇒du=−(p/((px+q)^2 ))dx=−(u^2 p)dx I=−∫(u/( (√(a((1/(up))−(q/p))^2 +b((1/(up))−(q/p))+c))))∙(du/((u^2 p))) =∓∫(du/( (√(a(1−uq)^2 +b(up−u^2 pq)+cu^2 p^2 )))) a(1−2uq+u^2 q^2 )+b(up−u^2 pq)+cu^2 p^2 =(aq^2 −bpq+cp^2 )u^2 +(−2aq+bp)u+(a) I=±∫(du/( (√((aq^2 −b+cp^2 )u^2 +(−2aq+bp)u+(a)))))

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

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