3.1.92 \(\int \frac {1}{\sqrt {\frac {b}{x}}} \, dx\)

Optimal. Leaf size=14 \[ \frac {2 x}{3 \sqrt {\frac {b}{x}}} \]

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Rubi [A]  time = 0.00, antiderivative size = 14, 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*} \frac {2 x}{3 \sqrt {\frac {b}{x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[b/x],x]

[Out]

(2*x)/(3*Sqrt[b/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 \frac {1}{\sqrt {\frac {b}{x}}} \, dx &=\frac {\int \sqrt {x} \, dx}{\sqrt {\frac {b}{x}} \sqrt {x}}\\ &=\frac {2 x}{3 \sqrt {\frac {b}{x}}}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {2 x}{3 \sqrt {\frac {b}{x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[b/x],x]

[Out]

(2*x)/(3*Sqrt[b/x])

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IntegrateAlgebraic [A]  time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} \frac {2 b}{3 \left (\frac {b}{x}\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/Sqrt[b/x],x]

[Out]

(2*b)/(3*(b/x)^(3/2))

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fricas [A]  time = 1.11, size = 15, normalized size = 1.07 \begin {gather*} \frac {2 \, x^{2} \sqrt {\frac {b}{x}}}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^2*sqrt(b/x)/b

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giac [A]  time = 0.15, size = 15, normalized size = 1.07 \begin {gather*} \frac {2 \, \sqrt {b x} x}{3 \, b \mathrm {sgn}\relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(b*x)*x/(b*sgn(x))

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maple [A]  time = 0.00, size = 11, normalized size = 0.79 \begin {gather*} \frac {2 x}{3 \sqrt {\frac {b}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x/(b/x)^(1/2)

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maxima [A]  time = 1.33, size = 10, normalized size = 0.71 \begin {gather*} \frac {2 \, x}{3 \, \sqrt {\frac {b}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x/sqrt(b/x)

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mupad [B]  time = 0.95, size = 15, normalized size = 1.07 \begin {gather*} \frac {2\,x^2\,\sqrt {\frac {b}{x}}}{3\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x^2*(b/x)^(1/2))/(3*b)

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sympy [A]  time = 0.39, size = 15, normalized size = 1.07 \begin {gather*} \frac {2 x}{3 \sqrt {b} \sqrt {\frac {1}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x/(3*sqrt(b)*sqrt(1/x))

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