3.2277 \(\int \frac{\sqrt{x}}{1+x^{3/2}} \, dx\)

Optimal. Leaf size=12 \[ \frac{2}{3} \log \left (x^{3/2}+1\right ) \]

[Out]

(2*Log[1 + x^(3/2)])/3

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Rubi [A]  time = 0.0121616, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2}{3} \log \left (x^{3/2}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(1 + x^(3/2)),x]

[Out]

(2*Log[1 + x^(3/2)])/3

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Rubi in Sympy [A]  time = 1.68948, size = 10, normalized size = 0.83 \[ \frac{2 \log{\left (x^{\frac{3}{2}} + 1 \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(x**(3/2) + 1)/3

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Mathematica [A]  time = 0.00328783, size = 12, normalized size = 1. \[ \frac{2}{3} \log \left (x^{3/2}+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]/(1 + x^(3/2)),x]

[Out]

(2*Log[1 + x^(3/2)])/3

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Maple [A]  time = 0.003, size = 9, normalized size = 0.8 \[{\frac{2}{3}\ln \left ( 1+{x}^{{\frac{3}{2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(1+x^(3/2)),x)

[Out]

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

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Maxima [A]  time = 1.42787, size = 11, normalized size = 0.92 \[ \frac{2}{3} \, \log \left (x^{\frac{3}{2}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x^(3/2) + 1),x, algorithm="maxima")

[Out]

2/3*log(x^(3/2) + 1)

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Fricas [A]  time = 0.226772, size = 11, normalized size = 0.92 \[ \frac{2}{3} \, \log \left (x^{\frac{3}{2}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x^(3/2) + 1),x, algorithm="fricas")

[Out]

2/3*log(x^(3/2) + 1)

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Sympy [A]  time = 0.520411, size = 24, normalized size = 2. \[ \frac{2 \log{\left (\sqrt{x} + 1 \right )}}{3} + \frac{2 \log{\left (- \sqrt{x} + x + 1 \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.234105, size = 27, normalized size = 2.25 \[ \frac{2}{3} \,{\rm ln}\left (x - \sqrt{x} + 1\right ) + \frac{2}{3} \,{\rm ln}\left (\sqrt{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x^(3/2) + 1),x, algorithm="giac")

[Out]

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