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

Optimal. Leaf size=42 \[ 2 \sqrt{x}-\log \left (x+\sqrt{x}+1\right )-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0632325, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ 2 \sqrt{x}-\log \left (x+\sqrt{x}+1\right )-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.44955, size = 42, normalized size = 1. \[ 2 \sqrt{x} - \log{\left (\sqrt{x} + x + 1 \right )} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt{x}}{3} + \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0169869, size = 42, normalized size = 1. \[ 2 \sqrt{x}-\log \left (x+\sqrt{x}+1\right )-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 34, normalized size = 0.8 \[ -\ln \left ( 1+x+\sqrt{x} \right ) -{\frac{2\,\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+2\,\sqrt{x} \right ) } \right ) }+2\,\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(1+x+x^(1/2)),x)

[Out]

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

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Maxima [A]  time = 0.787762, size = 45, normalized size = 1.07 \[ -\frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \sqrt{x} + 1\right )}\right ) + 2 \, \sqrt{x} - \log \left (x + \sqrt{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.273461, size = 57, normalized size = 1.36 \[ -\frac{1}{3} \, \sqrt{3}{\left (\sqrt{3} \log \left (x + \sqrt{x} + 1\right ) - 2 \, \sqrt{3} \sqrt{x} + 2 \, \arctan \left (\frac{2}{3} \, \sqrt{3} \sqrt{x} + \frac{1}{3} \, \sqrt{3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sqrt(3)*(sqrt(3)*log(x + sqrt(x) + 1) - 2*sqrt(3)*sqrt(x) + 2*arctan(2/3*sq
rt(3)*sqrt(x) + 1/3*sqrt(3)))

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Sympy [A]  time = 1.9015, size = 46, normalized size = 1.1 \[ 2 \sqrt{x} - \log{\left (\sqrt{x} + x + 1 \right )} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} \sqrt{x}}{3} + \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.262999, size = 45, normalized size = 1.07 \[ -\frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \sqrt{x} + 1\right )}\right ) + 2 \, \sqrt{x} -{\rm ln}\left (x + \sqrt{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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