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

Optimal. Leaf size=8 \[ 2 \sinh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

2*ArcSinh[Sqrt[x]]

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Rubi [A]  time = 0.0018681, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {54, 215} \[ 2 \sinh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*Sqrt[1 + x]),x]

[Out]

2*ArcSinh[Sqrt[x]]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

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

Mathematica [A]  time = 0.0023085, size = 8, normalized size = 1. \[ 2 \sinh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*Sqrt[1 + x]),x]

[Out]

2*ArcSinh[Sqrt[x]]

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Maple [B]  time = 0.003, size = 28, normalized size = 3.5 \begin{align*}{\sqrt{x \left ( 1+x \right ) }\ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(1/2)/(1+x)^(1/2),x)

[Out]

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

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Maxima [B]  time = 0.980169, size = 36, normalized size = 4.5 \begin{align*} \log \left (\frac{\sqrt{x + 1}}{\sqrt{x}} + 1\right ) - \log \left (\frac{\sqrt{x + 1}}{\sqrt{x}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sqrt(x + 1)/sqrt(x) + 1) - log(sqrt(x + 1)/sqrt(x) - 1)

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Fricas [B]  time = 1.58046, size = 53, normalized size = 6.62 \begin{align*} -\log \left (2 \, \sqrt{x + 1} \sqrt{x} - 2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(2*sqrt(x + 1)*sqrt(x) - 2*x - 1)

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Sympy [A]  time = 0.98345, size = 26, normalized size = 3.25 \begin{align*} \begin{cases} 2 \operatorname{acosh}{\left (\sqrt{x + 1} \right )} & \text{for}\: \left |{x + 1}\right | > 1 \\- 2 i \operatorname{asin}{\left (\sqrt{x + 1} \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*acosh(sqrt(x + 1)), Abs(x + 1) > 1), (-2*I*asin(sqrt(x + 1)), True))

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Giac [B]  time = 1.21662, size = 20, normalized size = 2.5 \begin{align*} -2 \, \log \left ({\left | -\sqrt{x + 1} + \sqrt{x} \right |}\right ) \end{align*}

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(abs(-sqrt(x + 1) + sqrt(x)))