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

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

[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} \[ 2 \sqrt {x+1} \]

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 \[ 2 \sqrt {x+1} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 + x],x]

[Out]

2*Sqrt[1 + x]

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fricas [A]  time = 0.40, size = 7, normalized size = 0.78 \[ 2 \, \sqrt {x + 1} \]

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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giac [A]  time = 0.34, size = 7, normalized size = 0.78 \[ 2 \, \sqrt {x + 1} \]

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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maple [A]  time = 0.00, size = 8, normalized size = 0.89 \[ 2 \sqrt {x +1} \]

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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maxima [A]  time = 0.88, size = 7, normalized size = 0.78 \[ 2 \, \sqrt {x + 1} \]

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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mupad [B]  time = 0.09, size = 7, normalized size = 0.78 \[ 2\,\sqrt {x+1} \]

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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sympy [A]  time = 0.06, size = 7, normalized size = 0.78 \[ 2 \sqrt {x + 1} \]

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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