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

Optimal. Leaf size=62 \[ 2 \sqrt{x}+\frac{4}{3} \log \left (\sqrt [4]{x}+1\right )-\frac{2}{3} \log \left (\sqrt{x}-\sqrt [4]{x}+1\right )+\frac{4 \tan ^{-1}\left (\frac{1-2 \sqrt [4]{x}}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0755541, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692 \[ 2 \sqrt{x}+\frac{4}{3} \log \left (\sqrt [4]{x}+1\right )-\frac{2}{3} \log \left (\sqrt{x}-\sqrt [4]{x}+1\right )+\frac{4 \tan ^{-1}\left (\frac{1-2 \sqrt [4]{x}}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.799, size = 61, normalized size = 0.98 \[ 2 \sqrt{x} + \frac{4 \log{\left (\sqrt [4]{x} + 1 \right )}}{3} - \frac{2 \log{\left (- \sqrt [4]{x} + \sqrt{x} + 1 \right )}}{3} - \frac{4 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [4]{x}}{3} - \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x) + 4*log(x**(1/4) + 1)/3 - 2*log(-x**(1/4) + sqrt(x) + 1)/3 - 4*sqrt(3)
*atan(sqrt(3)*(2*x**(1/4)/3 - 1/3))/3

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Mathematica [A]  time = 0.02472, size = 62, normalized size = 1. \[ 2 \sqrt{x}+\frac{4}{3} \log \left (\sqrt [4]{x}+1\right )-\frac{2}{3} \log \left (\sqrt{x}-\sqrt [4]{x}+1\right )-\frac{4 \tan ^{-1}\left (\frac{2 \sqrt [4]{x}-1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 46, normalized size = 0.7 \[ 2\,\sqrt{x}+{\frac{4}{3}\ln \left ( 1+\sqrt [4]{x} \right ) }-{\frac{2}{3}\ln \left ( 1-\sqrt [4]{x}+\sqrt{x} \right ) }-{\frac{4\,\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,\sqrt [4]{x}-1 \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1/x^(1/4)+x^(1/2)),x)

[Out]

2*x^(1/2)+4/3*ln(1+x^(1/4))-2/3*ln(1-x^(1/4)+x^(1/2))-4/3*3^(1/2)*arctan(1/3*(2*
x^(1/4)-1)*3^(1/2))

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Maxima [A]  time = 1.53031, size = 61, normalized size = 0.98 \[ -\frac{4}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{4}} - 1\right )}\right ) + 2 \, \sqrt{x} - \frac{2}{3} \, \log \left (\sqrt{x} - x^{\frac{1}{4}} + 1\right ) + \frac{4}{3} \, \log \left (x^{\frac{1}{4}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.221387, size = 77, normalized size = 1.24 \[ -\frac{2}{9} \, \sqrt{3}{\left (\sqrt{3} \log \left (\sqrt{x} - x^{\frac{1}{4}} + 1\right ) - 2 \, \sqrt{3} \log \left (x^{\frac{1}{4}} + 1\right ) - 3 \, \sqrt{3} \sqrt{x} + 6 \, \arctan \left (\frac{2}{3} \, \sqrt{3} x^{\frac{1}{4}} - \frac{1}{3} \, \sqrt{3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*sqrt(3)*(sqrt(3)*log(sqrt(x) - x^(1/4) + 1) - 2*sqrt(3)*log(x^(1/4) + 1) -
3*sqrt(3)*sqrt(x) + 6*arctan(2/3*sqrt(3)*x^(1/4) - 1/3*sqrt(3)))

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Sympy [A]  time = 1.33415, size = 68, normalized size = 1.1 \[ 2 \sqrt{x} + \frac{4 \log{\left (\sqrt [4]{x} + 1 \right )}}{3} - \frac{2 \log{\left (- 4 \sqrt [4]{x} + 4 \sqrt{x} + 4 \right )}}{3} - \frac{4 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} \sqrt [4]{x}}{3} - \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x) + 4*log(x**(1/4) + 1)/3 - 2*log(-4*x**(1/4) + 4*sqrt(x) + 4)/3 - 4*sqr
t(3)*atan(2*sqrt(3)*x**(1/4)/3 - sqrt(3)/3)/3

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GIAC/XCAS [A]  time = 0.213663, size = 61, normalized size = 0.98 \[ -\frac{4}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{4}} - 1\right )}\right ) + 2 \, \sqrt{x} - \frac{2}{3} \,{\rm ln}\left (\sqrt{x} - x^{\frac{1}{4}} + 1\right ) + \frac{4}{3} \,{\rm ln}\left (x^{\frac{1}{4}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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