3.3.22 \(\int \frac {1}{1+\sqrt {x}} \, dx\) [222]

Optimal. Leaf size=18 \[ 2 \sqrt {x}-2 \log \left (1+\sqrt {x}\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {196, 45} \begin {gather*} 2 \sqrt {x}-2 \log \left (\sqrt {x}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + Sqrt[x])^(-1),x]

[Out]

2*Sqrt[x] - 2*Log[1 + Sqrt[x]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 196

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rubi steps

\begin {align*} \int \frac {1}{1+\sqrt {x}} \, dx &=2 \text {Subst}\left (\int \frac {x}{1+x} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (1+\frac {1}{-1-x}\right ) \, dx,x,\sqrt {x}\right )\\ &=2 \sqrt {x}-2 \log \left (1+\sqrt {x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} 2 \sqrt {x}-2 \log \left (1+\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sqrt[x])^(-1),x]

[Out]

2*Sqrt[x] - 2*Log[1 + Sqrt[x]]

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Maple [A]
time = 0.04, size = 27, normalized size = 1.50

method	result	size
derivativedivides	\(-2 \ln \left (1+\sqrt {x}\right )+2 \sqrt {x}\)	\(15\)
meijerg	\(-2 \ln \left (1+\sqrt {x}\right )+2 \sqrt {x}\)	\(15\)
trager	\(2 \sqrt {x}-\ln \left (2 \sqrt {x}+1+x \right )\)	\(18\)
default	\(2 \sqrt {x}+\ln \left (\sqrt {x}-1\right )-\ln \left (1+\sqrt {x}\right )-\ln \left (-1+x \right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 1.94, size = 15, normalized size = 0.83 \begin {gather*} 2 \, \sqrt {x} - 2 \, \log \left (\sqrt {x} + 1\right ) + 2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x) - 2*log(sqrt(x) + 1) + 2

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Fricas [A]
time = 0.48, size = 14, normalized size = 0.78 \begin {gather*} 2 \, \sqrt {x} - 2 \, \log \left (\sqrt {x} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x) - 2*log(sqrt(x) + 1)

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Sympy [A]
time = 0.04, size = 15, normalized size = 0.83 \begin {gather*} 2 \sqrt {x} - 2 \log {\left (\sqrt {x} + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x) - 2*log(sqrt(x) + 1)

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Giac [A]
time = 0.44, size = 14, normalized size = 0.78 \begin {gather*} 2 \, \sqrt {x} - 2 \, \log \left (\sqrt {x} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x) - 2*log(sqrt(x) + 1)

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Mupad [B]
time = 0.06, size = 14, normalized size = 0.78 \begin {gather*} 2\,\sqrt {x}-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) + 1),x)

[Out]

2*x^(1/2) - 2*log(x^(1/2) + 1)

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