3.887 \(\int \frac{\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^5} \, dx\)

Optimal. Leaf size=137 \[ -\frac{(1-x)^{3/4} \sqrt [4]{x+1}}{4 x^4}-\frac{7 (1-x)^{3/4} \sqrt [4]{x+1}}{24 x^3}-\frac{29 (1-x)^{3/4} \sqrt [4]{x+1}}{96 x^2}-\frac{83 (1-x)^{3/4} \sqrt [4]{x+1}}{192 x}-\frac{11}{64} \tan ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac{11}{64} \tanh ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right ) \]

[Out]

-((1 - x)^(3/4)*(1 + x)^(1/4))/(4*x^4) - (7*(1 - x)^(3/4)*(1 + x)^(1/4))/(24*x^3
) - (29*(1 - x)^(3/4)*(1 + x)^(1/4))/(96*x^2) - (83*(1 - x)^(3/4)*(1 + x)^(1/4))
/(192*x) - (11*ArcTan[(1 + x)^(1/4)/(1 - x)^(1/4)])/64 - (11*ArcTanh[(1 + x)^(1/
4)/(1 - x)^(1/4)])/64

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Rubi [A]  time = 0.212428, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{(1-x)^{3/4} \sqrt [4]{x+1}}{4 x^4}-\frac{7 (1-x)^{3/4} \sqrt [4]{x+1}}{24 x^3}-\frac{29 (1-x)^{3/4} \sqrt [4]{x+1}}{96 x^2}-\frac{83 (1-x)^{3/4} \sqrt [4]{x+1}}{192 x}-\frac{11}{64} \tan ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac{11}{64} \tanh ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)^(1/4)/((1 - x)^(1/4)*x^5),x]

[Out]

-((1 - x)^(3/4)*(1 + x)^(1/4))/(4*x^4) - (7*(1 - x)^(3/4)*(1 + x)^(1/4))/(24*x^3
) - (29*(1 - x)^(3/4)*(1 + x)^(1/4))/(96*x^2) - (83*(1 - x)^(3/4)*(1 + x)^(1/4))
/(192*x) - (11*ArcTan[(1 + x)^(1/4)/(1 - x)^(1/4)])/64 - (11*ArcTanh[(1 + x)^(1/
4)/(1 - x)^(1/4)])/64

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Rubi in Sympy [A]  time = 16.232, size = 116, normalized size = 0.85 \[ - \frac{11 \operatorname{atan}{\left (\frac{\sqrt [4]{x + 1}}{\sqrt [4]{- x + 1}} \right )}}{64} - \frac{11 \operatorname{atanh}{\left (\frac{\sqrt [4]{x + 1}}{\sqrt [4]{- x + 1}} \right )}}{64} - \frac{83 \left (- x + 1\right )^{\frac{3}{4}} \sqrt [4]{x + 1}}{192 x} - \frac{29 \left (- x + 1\right )^{\frac{3}{4}} \sqrt [4]{x + 1}}{96 x^{2}} - \frac{7 \left (- x + 1\right )^{\frac{3}{4}} \sqrt [4]{x + 1}}{24 x^{3}} - \frac{\left (- x + 1\right )^{\frac{3}{4}} \sqrt [4]{x + 1}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)**(1/4)/(1-x)**(1/4)/x**5,x)

[Out]

-11*atan((x + 1)**(1/4)/(-x + 1)**(1/4))/64 - 11*atanh((x + 1)**(1/4)/(-x + 1)**
(1/4))/64 - 83*(-x + 1)**(3/4)*(x + 1)**(1/4)/(192*x) - 29*(-x + 1)**(3/4)*(x +
1)**(1/4)/(96*x**2) - 7*(-x + 1)**(3/4)*(x + 1)**(1/4)/(24*x**3) - (-x + 1)**(3/
4)*(x + 1)**(1/4)/(4*x**4)

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Mathematica [C]  time = 0.160648, size = 124, normalized size = 0.91 \[ \frac{-\frac{132 x F_1\left (1;\frac{1}{4},\frac{3}{4};2;\frac{1}{x},-\frac{1}{x}\right )}{8 x F_1\left (1;\frac{1}{4},\frac{3}{4};2;\frac{1}{x},-\frac{1}{x}\right )-3 F_1\left (2;\frac{1}{4},\frac{7}{4};3;\frac{1}{x},-\frac{1}{x}\right )+F_1\left (2;\frac{5}{4},\frac{3}{4};3;\frac{1}{x},-\frac{1}{x}\right )}-\frac{48}{x^4}-\frac{56}{x^3}-\frac{10}{x^2}+83 x-\frac{27}{x}+58}{192 \sqrt [4]{1-x} (x+1)^{3/4}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(1 + x)^(1/4)/((1 - x)^(1/4)*x^5),x]

[Out]

(58 - 48/x^4 - 56/x^3 - 10/x^2 - 27/x + 83*x - (132*x*AppellF1[1, 1/4, 3/4, 2, x
^(-1), -x^(-1)])/(8*x*AppellF1[1, 1/4, 3/4, 2, x^(-1), -x^(-1)] - 3*AppellF1[2,
1/4, 7/4, 3, x^(-1), -x^(-1)] + AppellF1[2, 5/4, 3/4, 3, x^(-1), -x^(-1)]))/(192
*(1 - x)^(1/4)*(1 + x)^(3/4))

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5}}\sqrt [4]{1+x}{\frac{1}{\sqrt [4]{1-x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^(1/4)/(1-x)^(1/4)/x^5,x)

[Out]

int((1+x)^(1/4)/(1-x)^(1/4)/x^5,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x + 1\right )}^{\frac{1}{4}}}{x^{5}{\left (-x + 1\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(1/4)/(x^5*(-x + 1)^(1/4)),x, algorithm="maxima")

[Out]

integrate((x + 1)^(1/4)/(x^5*(-x + 1)^(1/4)), x)

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Fricas [A]  time = 0.22764, size = 158, normalized size = 1.15 \[ \frac{66 \, x^{4} \arctan \left (\frac{{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}}}{x - 1}\right ) + 33 \, x^{4} \log \left (\frac{x +{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - 1}{x - 1}\right ) - 33 \, x^{4} \log \left (-\frac{x -{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - 1}{x - 1}\right ) - 2 \,{\left (83 \, x^{3} + 58 \, x^{2} + 56 \, x + 48\right )}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}}}{384 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(1/4)/(x^5*(-x + 1)^(1/4)),x, algorithm="fricas")

[Out]

1/384*(66*x^4*arctan((x + 1)^(1/4)*(-x + 1)^(3/4)/(x - 1)) + 33*x^4*log((x + (x
+ 1)^(1/4)*(-x + 1)^(3/4) - 1)/(x - 1)) - 33*x^4*log(-(x - (x + 1)^(1/4)*(-x + 1
)^(3/4) - 1)/(x - 1)) - 2*(83*x^3 + 58*x^2 + 56*x + 48)*(x + 1)^(1/4)*(-x + 1)^(
3/4))/x^4

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)**(1/4)/(1-x)**(1/4)/x**5,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x + 1\right )}^{\frac{1}{4}}}{x^{5}{\left (-x + 1\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(1/4)/(x^5*(-x + 1)^(1/4)),x, algorithm="giac")

[Out]

integrate((x + 1)^(1/4)/(x^5*(-x + 1)^(1/4)), x)