3.881 \(\int \frac{x \sqrt [4]{1+x}}{\sqrt [4]{1-x}} \, dx\)

Optimal. Leaf size=213 \[ -\frac{1}{2} (1-x)^{3/4} (x+1)^{5/4}-\frac{1}{4} (1-x)^{3/4} \sqrt [4]{x+1}-\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt{2}}+\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt{2}}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{4 \sqrt{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{4 \sqrt{2}} \]

[Out]

-((1 - x)^(3/4)*(1 + x)^(1/4))/4 - ((1 - x)^(3/4)*(1 + x)^(5/4))/2 + ArcTan[1 -
(Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/(4*Sqrt[2]) - ArcTan[1 + (Sqrt[2]*(1 - x)
^(1/4))/(1 + x)^(1/4)]/(4*Sqrt[2]) - Log[1 + Sqrt[1 - x]/Sqrt[1 + x] - (Sqrt[2]*
(1 - x)^(1/4))/(1 + x)^(1/4)]/(8*Sqrt[2]) + Log[1 + Sqrt[1 - x]/Sqrt[1 + x] + (S
qrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/(8*Sqrt[2])

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Rubi [A]  time = 0.219272, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{1}{2} (1-x)^{3/4} (x+1)^{5/4}-\frac{1}{4} (1-x)^{3/4} \sqrt [4]{x+1}-\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt{2}}+\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt{2}}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{4 \sqrt{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(x*(1 + x)^(1/4))/(1 - x)^(1/4),x]

[Out]

-((1 - x)^(3/4)*(1 + x)^(1/4))/4 - ((1 - x)^(3/4)*(1 + x)^(5/4))/2 + ArcTan[1 -
(Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/(4*Sqrt[2]) - ArcTan[1 + (Sqrt[2]*(1 - x)
^(1/4))/(1 + x)^(1/4)]/(4*Sqrt[2]) - Log[1 + Sqrt[1 - x]/Sqrt[1 + x] - (Sqrt[2]*
(1 - x)^(1/4))/(1 + x)^(1/4)]/(8*Sqrt[2]) + Log[1 + Sqrt[1 - x]/Sqrt[1 + x] + (S
qrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/(8*Sqrt[2])

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Rubi in Sympy [A]  time = 21.633, size = 172, normalized size = 0.81 \[ - \frac{\left (- x + 1\right )^{\frac{3}{4}} \left (x + 1\right )^{\frac{5}{4}}}{2} - \frac{\left (- x + 1\right )^{\frac{3}{4}} \sqrt [4]{x + 1}}{4} - \frac{\sqrt{2} \log{\left (- \frac{\sqrt{2} \sqrt [4]{- x + 1}}{\sqrt [4]{x + 1}} + \frac{\sqrt{- x + 1}}{\sqrt{x + 1}} + 1 \right )}}{16} + \frac{\sqrt{2} \log{\left (\frac{\sqrt{2} \sqrt [4]{- x + 1}}{\sqrt [4]{x + 1}} + \frac{\sqrt{- x + 1}}{\sqrt{x + 1}} + 1 \right )}}{16} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- x + 1}}{\sqrt [4]{x + 1}} - 1 \right )}}{8} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- x + 1}}{\sqrt [4]{x + 1}} + 1 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(1+x)**(1/4)/(1-x)**(1/4),x)

[Out]

-(-x + 1)**(3/4)*(x + 1)**(5/4)/2 - (-x + 1)**(3/4)*(x + 1)**(1/4)/4 - sqrt(2)*l
og(-sqrt(2)*(-x + 1)**(1/4)/(x + 1)**(1/4) + sqrt(-x + 1)/sqrt(x + 1) + 1)/16 +
sqrt(2)*log(sqrt(2)*(-x + 1)**(1/4)/(x + 1)**(1/4) + sqrt(-x + 1)/sqrt(x + 1) +
1)/16 - sqrt(2)*atan(sqrt(2)*(-x + 1)**(1/4)/(x + 1)**(1/4) - 1)/8 - sqrt(2)*ata
n(sqrt(2)*(-x + 1)**(1/4)/(x + 1)**(1/4) + 1)/8

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Mathematica [C]  time = 0.0450334, size = 51, normalized size = 0.24 \[ \frac{1}{4} \sqrt [4]{x+1} \left (2^{3/4} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{x+1}{2}\right )-(1-x)^{3/4} (2 x+3)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(x*(1 + x)^(1/4))/(1 - x)^(1/4),x]

[Out]

((1 + x)^(1/4)*(-((1 - x)^(3/4)*(3 + 2*x)) + 2^(3/4)*Hypergeometric2F1[1/4, 1/4,
5/4, (1 + x)/2]))/4

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \[ \int{x\sqrt [4]{1+x}{\frac{1}{\sqrt [4]{1-x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(1+x)^(1/4)/(1-x)^(1/4),x)

[Out]

int(x*(1+x)^(1/4)/(1-x)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x + 1\right )}^{\frac{1}{4}} x}{{\left (-x + 1\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(1/4)*x/(-x + 1)^(1/4),x, algorithm="maxima")

[Out]

integrate((x + 1)^(1/4)*x/(-x + 1)^(1/4), x)

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Fricas [A]  time = 0.239405, size = 371, normalized size = 1.74 \[ -\frac{1}{4} \,{\left (2 \, x + 3\right )}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{x - 1}{\sqrt{2}{\left (x - 1\right )} \sqrt{\frac{\sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + x - \sqrt{x + 1} \sqrt{-x + 1} - 1}{x - 1}} + \sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + x - 1}\right ) + \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{x - 1}{\sqrt{2}{\left (x - 1\right )} \sqrt{-\frac{\sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - x + \sqrt{x + 1} \sqrt{-x + 1} + 1}{x - 1}} + \sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - x + 1}\right ) - \frac{1}{16} \, \sqrt{2} \log \left (\frac{2 \,{\left (\sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + x - \sqrt{x + 1} \sqrt{-x + 1} - 1\right )}}{x - 1}\right ) + \frac{1}{16} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - x + \sqrt{x + 1} \sqrt{-x + 1} + 1\right )}}{x - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(1/4)*x/(-x + 1)^(1/4),x, algorithm="fricas")

[Out]

-1/4*(2*x + 3)*(x + 1)^(1/4)*(-x + 1)^(3/4) + 1/4*sqrt(2)*arctan((x - 1)/(sqrt(2
)*(x - 1)*sqrt((sqrt(2)*(x + 1)^(1/4)*(-x + 1)^(3/4) + x - sqrt(x + 1)*sqrt(-x +
1) - 1)/(x - 1)) + sqrt(2)*(x + 1)^(1/4)*(-x + 1)^(3/4) + x - 1)) + 1/4*sqrt(2)
*arctan((x - 1)/(sqrt(2)*(x - 1)*sqrt(-(sqrt(2)*(x + 1)^(1/4)*(-x + 1)^(3/4) - x
+ sqrt(x + 1)*sqrt(-x + 1) + 1)/(x - 1)) + sqrt(2)*(x + 1)^(1/4)*(-x + 1)^(3/4)
- x + 1)) - 1/16*sqrt(2)*log(2*(sqrt(2)*(x + 1)^(1/4)*(-x + 1)^(3/4) + x - sqrt
(x + 1)*sqrt(-x + 1) - 1)/(x - 1)) + 1/16*sqrt(2)*log(-2*(sqrt(2)*(x + 1)^(1/4)*
(-x + 1)^(3/4) - x + sqrt(x + 1)*sqrt(-x + 1) + 1)/(x - 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x \sqrt [4]{x + 1}}{\sqrt [4]{- x + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(1+x)**(1/4)/(1-x)**(1/4),x)

[Out]

Integral(x*(x + 1)**(1/4)/(-x + 1)**(1/4), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x + 1\right )}^{\frac{1}{4}} x}{{\left (-x + 1\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(1/4)*x/(-x + 1)^(1/4),x, algorithm="giac")

[Out]

integrate((x + 1)^(1/4)*x/(-x + 1)^(1/4), x)