3.2248 \(\int \frac{\sqrt{x}}{1+\sqrt{x}} \, dx\)

Optimal. Leaf size=19 \[ x-2 \sqrt{x}+2 \log \left (\sqrt{x}+1\right ) \]

[Out]

-2*Sqrt[x] + x + 2*Log[1 + Sqrt[x]]

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Rubi [A]  time = 0.028977, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ x-2 \sqrt{x}+2 \log \left (\sqrt{x}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(1 + Sqrt[x]),x]

[Out]

-2*Sqrt[x] + x + 2*Log[1 + Sqrt[x]]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - 2 \sqrt{x} + 2 \log{\left (\sqrt{x} + 1 \right )} + 2 \int ^{\sqrt{x}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(x) + 2*log(sqrt(x) + 1) + 2*Integral(x, (x, sqrt(x)))

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Mathematica [A]  time = 0.00637374, size = 19, normalized size = 1. \[ x-2 \sqrt{x}+2 \log \left (\sqrt{x}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

-2*Sqrt[x] + x + 2*Log[1 + Sqrt[x]]

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Maple [A]  time = 0.003, size = 16, normalized size = 0.8 \[ x+2\,\ln \left ( 1+\sqrt{x} \right ) -2\,\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(1+x^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.44349, size = 30, normalized size = 1.58 \[{\left (\sqrt{x} + 1\right )}^{2} - 4 \, \sqrt{x} + 2 \, \log \left (\sqrt{x} + 1\right ) - 4 \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.235673, size = 20, normalized size = 1.05 \[ x - 2 \, \sqrt{x} + 2 \, \log \left (\sqrt{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - 2*sqrt(x) + 2*log(sqrt(x) + 1)

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Sympy [A]  time = 0.328301, size = 17, normalized size = 0.89 \[ - 2 \sqrt{x} + x + 2 \log{\left (\sqrt{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(x) + x + 2*log(sqrt(x) + 1)

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GIAC/XCAS [A]  time = 0.257443, size = 20, normalized size = 1.05 \[ x - 2 \, \sqrt{x} + 2 \,{\rm ln}\left (\sqrt{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - 2*sqrt(x) + 2*ln(sqrt(x) + 1)