3.2270 \(\int \frac{\sqrt{x}}{1+x^{3/2}} \, dx\)

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 = 0.0025368, 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, Rules used = {260} \[ \frac{2}{3} \log \left (x^{3/2}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(1 + x^(3/2)),x]

[Out]

(2*Log[1 + x^(3/2)])/3

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{\sqrt{x}}{1+x^{3/2}} \, dx &=\frac{2}{3} \log \left (1+x^{3/2}\right )\\ \end{align*}

Mathematica [A]  time = 0.001608, 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 = 0.001, size = 9, normalized size = 0.8 \begin{align*}{\frac{2}{3}\ln \left ( 1+{x}^{{\frac{3}{2}}} \right ) } \end{align*}

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 = 0.973684, size = 11, normalized size = 0.92 \begin{align*} \frac{2}{3} \, \log \left (x^{\frac{3}{2}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(1+x^(3/2)),x, algorithm="maxima")

[Out]

2/3*log(x^(3/2) + 1)

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Fricas [A]  time = 1.17086, size = 30, normalized size = 2.5 \begin{align*} \frac{2}{3} \, \log \left (x^{\frac{3}{2}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(1+x^(3/2)),x, algorithm="fricas")

[Out]

2/3*log(x^(3/2) + 1)

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Sympy [B]  time = 0.185694, size = 24, normalized size = 2. \begin{align*} \frac{2 \log{\left (\sqrt{x} + 1 \right )}}{3} + \frac{2 \log{\left (- \sqrt{x} + x + 1 \right )}}{3} \end{align*}

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 [B]  time = 1.11511, size = 27, normalized size = 2.25 \begin{align*} \frac{2}{3} \, \log \left (x - \sqrt{x} + 1\right ) + \frac{2}{3} \, \log \left (\sqrt{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(1+x^(3/2)),x, algorithm="giac")

[Out]

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