3.2249 \(\int \frac{1}{\left (1+\sqrt{x}\right ) \sqrt{x}} \, dx\)

Optimal. Leaf size=10 \[ 2 \log \left (\sqrt{x}+1\right ) \]

[Out]

2*Log[1 + Sqrt[x]]

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Rubi [A]  time = 0.0107821, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ 2 \log \left (\sqrt{x}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/((1 + Sqrt[x])*Sqrt[x]),x]

[Out]

2*Log[1 + Sqrt[x]]

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Rubi in Sympy [A]  time = 1.70002, size = 8, normalized size = 0.8 \[ 2 \log{\left (\sqrt{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(sqrt(x) + 1)

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Mathematica [A]  time = 0.00251891, size = 10, normalized size = 1. \[ 2 \log \left (\sqrt{x}+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 + Sqrt[x])*Sqrt[x]),x]

[Out]

2*Log[1 + Sqrt[x]]

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Maple [A]  time = 0.003, size = 9, normalized size = 0.9 \[ 2\,\ln \left ( 1+\sqrt{x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(1/2)/(1+x^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.4368, size = 11, normalized size = 1.1 \[ 2 \, \log \left (\sqrt{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(sqrt(x) + 1)

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Fricas [A]  time = 0.238383, size = 11, normalized size = 1.1 \[ 2 \, \log \left (\sqrt{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(sqrt(x) + 1)

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Sympy [A]  time = 0.347851, size = 8, normalized size = 0.8 \[ 2 \log{\left (\sqrt{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(sqrt(x) + 1)

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GIAC/XCAS [A]  time = 0.234345, size = 11, normalized size = 1.1 \[ 2 \,{\rm ln}\left (\sqrt{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*ln(sqrt(x) + 1)