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

Optimal. Leaf size=25 \[ 2 \sqrt{x}-4 \sqrt [4]{x}+4 \log \left (\sqrt [4]{x}+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.0240742, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ 2 \sqrt{x}-4 \sqrt [4]{x}+4 \log \left (\sqrt [4]{x}+1\right ) \]

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 = 0.00990667, size = 25, normalized size = 1. \[ 2 \sqrt{x}-4 \sqrt [4]{x}+4 \log \left (\sqrt [4]{x}+1\right ) \]

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 = 0.008, size = 20, normalized size = 0.8 \[ -4\,\sqrt [4]{x}+4\,\ln \left ( 1+\sqrt [4]{x} \right ) +2\,\sqrt{x} \]

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

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Maxima [A]  time = 0.745472, size = 26, normalized size = 1.04 \[ 2 \, \sqrt{x} - 4 \, x^{\frac{1}{4}} + 4 \, \log \left (x^{\frac{1}{4}} + 1\right ) \]

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 = 0.271106, size = 26, normalized size = 1.04 \[ 2 \, \sqrt{x} - 4 \, x^{\frac{1}{4}} + 4 \, \log \left (x^{\frac{1}{4}} + 1\right ) \]

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 = 0.555739, size = 22, normalized size = 0.88 \[ - 4 \sqrt [4]{x} + 2 \sqrt{x} + 4 \log{\left (\sqrt [4]{x} + 1 \right )} \]

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 = 0.280659, size = 26, normalized size = 1.04 \[ 2 \, \sqrt{x} - 4 \, x^{\frac{1}{4}} + 4 \,{\rm ln}\left (x^{\frac{1}{4}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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