3.1.80 \(\int \sqrt {\frac {b}{x}} \, dx\) [80]

Optimal. Leaf size=12 \[ 2 \sqrt {\frac {b}{x}} x \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 30} \begin {gather*} 2 x \sqrt {\frac {b}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[b/x],x]

[Out]

2*Sqrt[b/x]*x

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \sqrt {\frac {b}{x}} \, dx &=\left (\sqrt {\frac {b}{x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x}} \, dx\\ &=2 \sqrt {\frac {b}{x}} x\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} 2 \sqrt {\frac {b}{x}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b/x],x]

[Out]

2*Sqrt[b/x]*x

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Maple [A]
time = 0.02, size = 11, normalized size = 0.92

method	result	size
gosper	\(2 x \sqrt {\frac {b}{x}}\)	\(11\)
default	\(2 x \sqrt {\frac {b}{x}}\)	\(11\)
trager	\(2 x \sqrt {\frac {b}{x}}\)	\(11\)
risch	\(2 x \sqrt {\frac {b}{x}}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 10, normalized size = 0.83 \begin {gather*} 2 \, x \sqrt {\frac {b}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x*sqrt(b/x)

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Fricas [A]
time = 0.34, size = 10, normalized size = 0.83 \begin {gather*} 2 \, x \sqrt {\frac {b}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x*sqrt(b/x)

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Sympy [A]
time = 0.05, size = 8, normalized size = 0.67 \begin {gather*} 2 x \sqrt {\frac {b}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x*sqrt(b/x)

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Giac [A]
time = 1.57, size = 9, normalized size = 0.75 \begin {gather*} 2 \, \sqrt {b x} \mathrm {sgn}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*x)*sgn(x)

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Mupad [B]
time = 0.97, size = 10, normalized size = 0.83 \begin {gather*} 2\,x\,\sqrt {\frac {b}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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