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One of my friends gave me to solve the following integration problem:

$\int \frac{\sqrt x}{1+\sqrt[4]{x-b}} dx=?$ where $b$ is any arbitrary constant.

I tried to solve it by putting $x-b=z^4$ but it only complicated the problem. Can someone point me in the right direction?Thanks in advance for your time.

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maple12 cannot find a closed form for this. –  coffeemath Feb 28 '13 at 15:06
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What have you done to your friend for him or her to want such revenge! –  L. F. Feb 28 '13 at 15:26
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1 Answer 1

Mathematica gives the frightening result below. AppellF1[$a$,$b_1$,Subscript[$b_2$],$c$,$x$,$y$] is the Appell hypergeometric function of two variables $F_1$($a$;$b_1$,$b_2$;$c$;$x$,$y$).

$\frac{1}{210 x^{3/2} (-b+x)^{3/4}}\left(-280 b x^2+280 x^3+210 b x^2 (-b+x)^{1/4}-210 x^3 (-b+x)^{1/4}-840 b x \sqrt{-b+x}-336 b^2 x \sqrt{-b+x}+840 x^2 \sqrt{-b+x}+168 b x^2 \sqrt{-b+x}+168 x^3 \sqrt{-b+x}-420 x^2 (-b+x)^{3/4}-280 (3+2 b) \left(1-\frac{b}{x}\right)^{3/4} x^2 \text{AppellF1}\left[\frac{1}{4},\frac{3}{4},1,\frac{5}{4},\frac{b}{x},\frac{1+b}{x}\right]+56 \left(-5+5 b+4 b^2\right) \left(1-\frac{b}{x}\right)^{1/4} x \sqrt{-b+x} \text{AppellF1}\left[\frac{3}{4},\frac{1}{4},1,\frac{7}{4},\frac{b}{x},\frac{1+b}{x}\right]+112 b \left(1-\frac{b}{x}\right)^{3/4} x \text{AppellF1}\left[\frac{5}{4},\frac{3}{4},1,\frac{9}{4},\frac{b}{x},\frac{1+b}{x}\right]+112 b^2 \left(1-\frac{b}{x}\right)^{3/4} x \text{AppellF1}\left[\frac{5}{4},\frac{3}{4},1,\frac{9}{4},\frac{b}{x},\frac{1+b}{x}\right]-240 b \left(1-\frac{b}{x}\right)^{1/4} \sqrt{-b+x} \text{AppellF1}\left[\frac{7}{4},\frac{1}{4},1,\frac{11}{4},\frac{b}{x},\frac{1+b}{x}\right]-336 b^2 \left(1-\frac{b}{x}\right)^{1/4} \sqrt{-b+x} \text{AppellF1}\left[\frac{7}{4},\frac{1}{4},1,\frac{11}{4},\frac{b}{x},\frac{1+b}{x}\right]-96 b^3 \left(1-\frac{b}{x}\right)^{1/4} \sqrt{-b+x} \text{AppellF1}\left[\frac{7}{4},\frac{1}{4},1,\frac{11}{4},\frac{b}{x},\frac{1+b}{x}\right]+420 \sqrt{1+b} x^{3/2} (-b+x)^{3/4} \text{ArcTanh}\left[\frac{\sqrt{1+b}}{\sqrt{x}}\right]-420 \sqrt{1-\frac{b}{x}} x^2 (-b+x)^{1/4} \text{Log}\left[1+\sqrt{1-\frac{b}{x}}\right]-210 b \sqrt{1-\frac{b}{x}} x^2 (-b+x)^{1/4} \text{Log}\left[1+\sqrt{1-\frac{b}{x}}\right]+210 \sqrt{1+b} \sqrt{1-\frac{b}{x}} x^2 (-b+x)^{1/4} \text{Log}\left[\sqrt{1+b}+\sqrt{1-\frac{b}{x}}-\frac{b}{\sqrt{x}}\right]+210 \sqrt{1+b} \sqrt{1-\frac{b}{x}} x^2 (-b+x)^{1/4} \text{Log}\left[\sqrt{1+b}+\sqrt{1-\frac{b}{x}}+\frac{b}{\sqrt{x}}\right]-210 \sqrt{1+b} \sqrt{1-\frac{b}{x}} x^2 (-b+x)^{1/4} \text{Log}\left[\sqrt{1+b}-\frac{1+b}{\sqrt{x}}\right]-210 \sqrt{1+b} \sqrt{1-\frac{b}{x}} x^2 (-b+x)^{1/4} \text{Log}\left[\sqrt{1+b}+\frac{1+b}{\sqrt{x}}\right]-210 \sqrt{1-\frac{b}{x}} x^2 (-b+x)^{1/4} \text{Log}[x]-105 b \sqrt{1-\frac{b}{x}} x^2 (-b+x)^{1/4} \text{Log}[x]\right)$

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