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

Optimal. Leaf size=32 \[ x-2 \sqrt{x}+\frac{4 \tan ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0500143, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ x-2 \sqrt{x}+\frac{4 \tan ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x/(1 + Sqrt[x] + x),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - 2 \sqrt{x} + \frac{4 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt{x}}{3} + \frac{1}{3}\right ) \right )}}{3} + 2 \int ^{\sqrt{x}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(x) + 4*sqrt(3)*atan(sqrt(3)*(2*sqrt(x)/3 + 1/3))/3 + 2*Integral(x, (x, s
qrt(x)))

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Mathematica [A]  time = 0.0102977, size = 32, normalized size = 1. \[ x-2 \sqrt{x}+\frac{4 \tan ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 26, normalized size = 0.8 \[ x+{\frac{4\,\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+2\,\sqrt{x} \right ) } \right ) }-2\,\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(1+x+x^(1/2)),x)

[Out]

x+4/3*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 = 0.762577, size = 34, normalized size = 1.06 \[ \frac{4}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \sqrt{x} + 1\right )}\right ) + x - 2 \, \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.272774, size = 49, normalized size = 1.53 \[ \frac{1}{3} \, \sqrt{3}{\left (\sqrt{3} x - 2 \, \sqrt{3} \sqrt{x} + 4 \, \arctan \left (\frac{2}{3} \, \sqrt{3} \sqrt{x} + \frac{1}{3} \, \sqrt{3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{x} + x + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(1+x+x**(1/2)),x)

[Out]

Integral(x/(sqrt(x) + x + 1), x)

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GIAC/XCAS [A]  time = 0.261984, size = 34, normalized size = 1.06 \[ \frac{4}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \sqrt{x} + 1\right )}\right ) + x - 2 \, \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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