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

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

[Out]

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

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Rubi [A]  time = 0.0015289, antiderivative size = 14, normalized size of antiderivative = 1., 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} \[ \frac{2 x}{3 \sqrt{\frac{b}{x}}} \]

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*}

Mathematica [A]  time = 0.0015514, size = 14, normalized size = 1. \[ \frac{2 x}{3 \sqrt{\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 11, normalized size = 0.8 \begin{align*}{\frac{2\,x}{3}{\frac{1}{\sqrt{{\frac{b}{x}}}}}} \end{align*}

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 = 0.953574, size = 14, normalized size = 1. \begin{align*} \frac{2 \, x}{3 \, \sqrt{\frac{b}{x}}} \end{align*}

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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Fricas [A]  time = 1.73286, size = 28, normalized size = 2. \begin{align*} \frac{2 \, x^{2} \sqrt{\frac{b}{x}}}{3 \, b} \end{align*}

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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Sympy [A]  time = 0.519949, size = 15, normalized size = 1.07 \begin{align*} \frac{2 x}{3 \sqrt{b} \sqrt{\frac{1}{x}}} \end{align*}

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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Giac [A]  time = 1.19693, size = 20, normalized size = 1.43 \begin{align*} \frac{2 \, \sqrt{b x} x}{3 \, b \mathrm{sgn}\left (x\right )} \end{align*}

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))