3.231 \(\int \frac{\sqrt{x}}{x+x^2} \, dx\)

Optimal. Leaf size=8 \[ 2 \tan ^{-1}\left (\sqrt{x}\right ) \]

[Out]

2*ArcTan[Sqrt[x]]

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Rubi [A]  time = 0.0100113, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ 2 \tan ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(x + x^2),x]

[Out]

2*ArcTan[Sqrt[x]]

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Rubi in Sympy [A]  time = 1.23645, size = 7, normalized size = 0.88 \[ 2 \operatorname{atan}{\left (\sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*atan(sqrt(x))

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Mathematica [A]  time = 0.00403947, size = 8, normalized size = 1. \[ 2 \tan ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]/(x + x^2),x]

[Out]

2*ArcTan[Sqrt[x]]

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Maple [A]  time = 0.006, size = 7, normalized size = 0.9 \[ 2\,\arctan \left ( \sqrt{x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(x^2+x),x)

[Out]

2*arctan(x^(1/2))

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Maxima [A]  time = 1.52463, size = 8, normalized size = 1. \[ 2 \, \arctan \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x^2 + x),x, algorithm="maxima")

[Out]

2*arctan(sqrt(x))

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Fricas [A]  time = 0.205759, size = 8, normalized size = 1. \[ 2 \, \arctan \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x^2 + x),x, algorithm="fricas")

[Out]

2*arctan(sqrt(x))

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Sympy [A]  time = 0.971066, size = 7, normalized size = 0.88 \[ 2 \operatorname{atan}{\left (\sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(1/2)/(x**2+x),x)

[Out]

2*atan(sqrt(x))

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GIAC/XCAS [A]  time = 0.208505, size = 8, normalized size = 1. \[ 2 \, \arctan \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x^2 + x),x, algorithm="giac")

[Out]

2*arctan(sqrt(x))