3.23.51 \(\int \frac {1}{(1+\sqrt {x}) \sqrt {x}} \, dx\) [2251]

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

[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 = {266} \begin {gather*} 2 \log \left (\sqrt {x}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 + Sqrt[x])*Sqrt[x]),x]

[Out]

2*Log[1 + Sqrt[x]]

Rule 266

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 = 0.01, size = 10, normalized size = 1.00 \begin {gather*} 2 \log \left (1+\sqrt {x}\right ) \end {gather*}

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 = 0.18, size = 9, normalized size = 0.90

method	result	size
derivativedivides	\(2 \ln \left (\sqrt {x}+1\right )\)	\(9\)
default	\(2 \ln \left (\sqrt {x}+1\right )\)	\(9\)
meijerg	\(2 \ln \left (\sqrt {x}+1\right )\)	\(9\)
trager	\(\ln \left (x +2 \sqrt {x}+1\right )\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(1/2)/(x^(1/2)+1),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 8, normalized size = 0.80 \begin {gather*} 2 \, \log \left (\sqrt {x} + 1\right ) \end {gather*}

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 = 0.37, size = 8, normalized size = 0.80 \begin {gather*} 2 \, \log \left (\sqrt {x} + 1\right ) \end {gather*}

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 = 0.05, size = 8, normalized size = 0.80 \begin {gather*} 2 \log {\left (\sqrt {x} + 1 \right )} \end {gather*}

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 = 1.04, size = 8, normalized size = 0.80 \begin {gather*} 2 \, \log \left (\sqrt {x} + 1\right ) \end {gather*}

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

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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