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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]