3.9.56 \(\int \frac {1}{\sqrt {1+x}} \, dx\) [856]

Optimal. Leaf size=9 \[ 2 \sqrt {1+x} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \begin {gather*} 2 \sqrt {x+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 + x],x]

[Out]

2*Sqrt[1 + x]

Rule 32

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {1+x}} \, dx &=2 \sqrt {1+x}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 9, normalized size = 1.00 \begin {gather*} 2 \sqrt {1+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 + x],x]

[Out]

2*Sqrt[1 + x]

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Maple [A]
time = 0.54, size = 8, normalized size = 0.89

method	result	size
gosper	\(2 \sqrt {1+x}\)	\(8\)
derivativedivides	\(2 \sqrt {1+x}\)	\(8\)
default	\(2 \sqrt {1+x}\)	\(8\)
trager	\(2 \sqrt {1+x}\)	\(8\)
risch	\(2 \sqrt {1+x}\)	\(8\)
meijerg	\(\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1+x}}{\sqrt {\pi }}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 7, normalized size = 0.78 \begin {gather*} 2 \, \sqrt {x + 1} \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 + 1)

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Fricas [A]
time = 0.32, size = 7, normalized size = 0.78 \begin {gather*} 2 \, \sqrt {x + 1} \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 + 1)

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Sympy [A]
time = 0.01, size = 7, normalized size = 0.78 \begin {gather*} 2 \sqrt {x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x + 1)

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Giac [A]
time = 3.85, size = 7, normalized size = 0.78 \begin {gather*} 2 \, \sqrt {x + 1} \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 + 1)

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Mupad [B]
time = 0.09, size = 7, normalized size = 0.78 \begin {gather*} 2\,\sqrt {x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x + 1)^(1/2),x)

[Out]

2*(x + 1)^(1/2)

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