∫(( dx)/((√(x^3 )) ^3 (√(1 +^4 (√x^3 ))))) = ?


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Question Number 106830 by malwaan last updated on 08/Aug/20

$\int \frac{d x}{\sqrt{x^{3}}^{3} \sqrt{1 +^{4} \sqrt{x^{3}}}} = ?$
Commented by mathmax by abdo last updated on 07/Aug/20

$its seems that \int \frac{tdt}{{(1 + t^{3})}^{\frac{1}{3}}} is not solvable . . .!$
Commented by bobhans last updated on 07/Aug/20

$\int \frac{dx}{\sqrt[4]{x^{2}} \sqrt[3]{1 + \sqrt[4]{x^{3}}}} [let \sqrt[4]{x} = t]$ $\int \frac{{4 t}^{3} dt}{t^{2} \sqrt[3]{1 + t^{3}}} = \int \frac{4 t}{\sqrt[3]{1 + t^{3}}} dt$ $next . . .$
Commented by Her_Majesty last updated on 07/Aug/20

$t h i s e n d s u p w i t h_{2} F_{1} - f u n c t i o n . . .$
Commented by malwaan last updated on 07/Aug/20

$t h i s i s f o r s e c o n d a r y s c h o o l s$ $y o u r - m a j e s t y$
Commented by Sarah85 last updated on 07/Aug/20

$\int \frac{t}{\sqrt[3]{1 + t^{3}}} d t =_{2} F_{1} (\frac{1}{3}; \frac{2}{3}; \frac{5}{3}; - t^{3}) \times \frac{t^{2}}{2}$ $so Her Majesty is right$
Commented by malwaan last updated on 08/Aug/20

$I = \int x^{- \frac{3}{2}} {(1 + x^{\frac{3}{4}})}^{- \frac{1}{3}} d x$ $t = {(\frac{1 + x^{\frac{3}{4}}}{x^{\frac{3}{4}}})}^{\frac{1}{3}}$ $I t h i n k w e w i l l g e t t h e s o l u t i o n$ $w i t h e l e m e n t a r y f u n c t i o n s$
Commented by malwaan last updated on 08/Aug/20

$s o r r y e v e r y b o d y$ $I d i d a m i s t a k e w h e n p o s t i n g$ $t h e q u e s t i o n$
	Answered by malwaan last updated on 08/Aug/20

	$I = \int x^{- \frac{2}{3}} {(1 + x^{\frac{3}{4}})}^{- \frac{1}{3}} d x$ $t = {(\frac{1 + x^{\frac{3}{4}}}{x^{\frac{3}{4}}})}^{\frac{1}{3}} = (1 + x^{- \frac{3}{4}})$ $t^{3} = 1 + x^{- \frac{3}{4}}$ $x^{- \frac{3}{4}} = t^{3} - 1; x^{\frac{3}{4}} = \frac{1}{t^{3} - 1}$ $x = {(t^{3} - 1)}^{- \frac{4}{3}}$ $\Rightarrow d x = - 4 t^{2} {(t^{3} - 1)}^{- \frac{7}{3}} d t$ $\int {[{(t^{3} - 1)}^{- \frac{4}{3}}]}^{- \frac{3}{2}} {[1 + \frac{1}{t^{3} - 1}]}^{- \frac{1}{3}} (- 4 t^{2}) {(t^{3} - 1)}^{- \frac{7}{3}} d t$ $= - 4 \int t^{2} {(t^{3} - 1)}^{2} t^{- 1} {(t^{3} - 1)}^{\frac{1}{3}} {(t^{3} - 1)}^{- \frac{7}{3}} d t$ $= - 4 \int t d t = - 4 \frac{t^{2}}{2} + C$ $= - 2 t^{2} + C$ $= - 2^{3} \sqrt{{(1 + x^{- \frac{3}{4}})}^{2}} + C$
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