3.367 \(\int \frac{\sqrt{x}}{1+x^3} \, dx\)

Optimal. Leaf size=10 \[ \frac{2}{3} \tan ^{-1}\left (x^{3/2}\right ) \]

[Out]

(2*ArcTan[x^(3/2)])/3

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Rubi [A]  time = 0.0201628, antiderivative size = 10, 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 \[ \frac{2}{3} \tan ^{-1}\left (x^{3/2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(1 + x^3),x]

[Out]

(2*ArcTan[x^(3/2)])/3

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Rubi in Sympy [A]  time = 4.16914, size = 8, normalized size = 0.8 \[ \frac{2 \operatorname{atan}{\left (x^{\frac{3}{2}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)/(x**3+1),x)

[Out]

2*atan(x**(3/2))/3

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Mathematica [A]  time = 0.00759896, size = 10, normalized size = 1. \[ -\frac{2}{3} \tan ^{-1}\left (\frac{1}{x^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]/(1 + x^3),x]

[Out]

(-2*ArcTan[x^(-3/2)])/3

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Maple [A]  time = 0.008, size = 7, normalized size = 0.7 \[{\frac{2}{3}\arctan \left ({x}^{{\frac{3}{2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(x^3+1),x)

[Out]

2/3*arctan(x^(3/2))

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Maxima [A]  time = 1.57841, size = 8, normalized size = 0.8 \[ \frac{2}{3} \, \arctan \left (x^{\frac{3}{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x^3 + 1),x, algorithm="maxima")

[Out]

2/3*arctan(x^(3/2))

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Fricas [A]  time = 0.222299, size = 8, normalized size = 0.8 \[ \frac{2}{3} \, \arctan \left (x^{\frac{3}{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x^3 + 1),x, algorithm="fricas")

[Out]

2/3*arctan(x^(3/2))

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Sympy [A]  time = 5.24713, size = 42, normalized size = 4.2 \[ - \frac{2 \operatorname{atan}{\left (\sqrt{x} \right )}}{3} + \frac{2 \operatorname{atan}{\left (2 \sqrt{x} - \sqrt{3} \right )}}{3} + \frac{2 \operatorname{atan}{\left (2 \sqrt{x} + \sqrt{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(1/2)/(x**3+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.217923, size = 8, normalized size = 0.8 \[ \frac{2}{3} \, \arctan \left (x^{\frac{3}{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x^3 + 1),x, algorithm="giac")

[Out]

2/3*arctan(x^(3/2))