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

Optimal. Leaf size=27 \[ 2 \sqrt{x}+4 \sqrt [4]{x}+4 \log \left (1-\sqrt [4]{x}\right ) \]

[Out]

4*x^(1/4) + 2*Sqrt[x] + 4*Log[1 - x^(1/4)]

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Rubi [A]  time = 0.0248985, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ 2 \sqrt{x}+4 \sqrt [4]{x}+4 \log \left (1-\sqrt [4]{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

4*x^(1/4) + 2*Sqrt[x] + 4*Log[1 - x^(1/4)]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 4 \sqrt [4]{x} + 4 \log{\left (- \sqrt [4]{x} + 1 \right )} + 4 \int ^{\sqrt [4]{x}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-x**(1/4)+x**(1/2)),x)

[Out]

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

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Mathematica [A]  time = 0.0128025, size = 27, normalized size = 1. \[ 2 \sqrt{x}+4 \sqrt [4]{x}+4 \log \left (1-\sqrt [4]{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

4*x^(1/4) + 2*Sqrt[x] + 4*Log[1 - x^(1/4)]

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Maple [A]  time = 0.011, size = 20, normalized size = 0.7 \[ 4\,\sqrt [4]{x}+2\,\sqrt{x}+4\,\ln \left ( \sqrt [4]{x}-1 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^(1/4)+x^(1/2)),x)

[Out]

4*x^(1/4)+2*x^(1/2)+4*ln(x^(1/4)-1)

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Maxima [A]  time = 0.733607, size = 26, normalized size = 0.96 \[ 2 \, \sqrt{x} + 4 \, x^{\frac{1}{4}} + 4 \, \log \left (x^{\frac{1}{4}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.270019, size = 26, normalized size = 0.96 \[ 2 \, \sqrt{x} + 4 \, x^{\frac{1}{4}} + 4 \, \log \left (x^{\frac{1}{4}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.580411, size = 22, normalized size = 0.81 \[ 4 \sqrt [4]{x} + 2 \sqrt{x} + 4 \log{\left (\sqrt [4]{x} - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-x**(1/4)+x**(1/2)),x)

[Out]

4*x**(1/4) + 2*sqrt(x) + 4*log(x**(1/4) - 1)

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GIAC/XCAS [A]  time = 0.282616, size = 27, normalized size = 1. \[ 2 \, \sqrt{x} + 4 \, x^{\frac{1}{4}} + 4 \,{\rm ln}\left ({\left | x^{\frac{1}{4}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x) + 4*x^(1/4) + 4*ln(abs(x^(1/4) - 1))