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

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 1.00 \[ 2 \log \left (\sqrt {x}+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*Log[1 + Sqrt[x]]

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fricas [A]  time = 0.78, size = 8, normalized size = 0.80 \[ 2 \, \log \left (\sqrt {x} + 1\right ) \]

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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giac [A]  time = 0.17, size = 8, normalized size = 0.80 \[ 2 \, \log \left (\sqrt {x} + 1\right ) \]

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)

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maple [A]  time = 0.00, size = 9, normalized size = 0.90 \[ 2 \ln \left (\sqrt {x}+1\right ) \]

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 = 0.86, size = 8, normalized size = 0.80 \[ 2 \, \log \left (\sqrt {x} + 1\right ) \]

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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mupad [B]  time = 0.05, size = 8, normalized size = 0.80 \[ 2\,\ln \left (\sqrt {x}+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(x^(1/2) + 1)),x)

[Out]

2*log(x^(1/2) + 1)

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sympy [A]  time = 0.15, size = 8, normalized size = 0.80 \[ 2 \log {\left (\sqrt {x} + 1 \right )} \]

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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