3.2251 \(\int \frac{1}{(1+\sqrt{x}) \sqrt{x}} \, dx\)

Optimal. Leaf size=10 \[ 2 \log \left (\sqrt{x}+1\right ) \]

[Out]

2*Log[1 + Sqrt[x]]

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Rubi [A]  time = 0.0021274, antiderivative size = 10, 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} \[ 2 \log \left (\sqrt{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 + Sqrt[x])*Sqrt[x]),x]

[Out]

2*Log[1 + Sqrt[x]]

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{1}{\left (1+\sqrt{x}\right ) \sqrt{x}} \, dx &=2 \log \left (1+\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0012391, size = 10, normalized size = 1. \[ 2 \log \left (\sqrt{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*Log[1 + Sqrt[x]]

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Maple [A]  time = 0.001, size = 9, normalized size = 0.9 \begin{align*} 2\,\ln \left ( \sqrt{x}+1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(1/2)/(x^(1/2)+1),x)

[Out]

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

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Maxima [A]  time = 0.972923, size = 11, normalized size = 1.1 \begin{align*} 2 \, \log \left (\sqrt{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(sqrt(x) + 1)

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Fricas [A]  time = 1.20862, size = 27, normalized size = 2.7 \begin{align*} 2 \, \log \left (\sqrt{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(sqrt(x) + 1)

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Sympy [A]  time = 0.132629, size = 8, normalized size = 0.8 \begin{align*} 2 \log{\left (\sqrt{x} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(1/2)/(1+x**(1/2)),x)

[Out]

2*log(sqrt(x) + 1)

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Giac [A]  time = 1.10207, size = 11, normalized size = 1.1 \begin{align*} 2 \, \log \left (\sqrt{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(sqrt(x) + 1)