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

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

[Out]

2*Sqrt[b/x]*x

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Rubi [A]  time = 0.001497, antiderivative size = 12, 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} \[ 2 x \sqrt{\frac{b}{x}} \]

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*}

Mathematica [A]  time = 0.0011144, size = 12, normalized size = 1. \[ 2 x \sqrt{\frac{b}{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b/x],x]

[Out]

2*Sqrt[b/x]*x

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Maple [A]  time = 0.001, size = 11, normalized size = 0.9 \begin{align*} 2\,x\sqrt{{\frac{b}{x}}} \end{align*}

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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Maxima [A]  time = 0.983577, size = 14, normalized size = 1.17 \begin{align*} 2 \, x \sqrt{\frac{b}{x}} \end{align*}

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 = 1.69507, size = 20, normalized size = 1.67 \begin{align*} 2 \, x \sqrt{\frac{b}{x}} \end{align*}

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 = 0.18345, size = 14, normalized size = 1.17 \begin{align*} 2 \sqrt{b} x \sqrt{\frac{1}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.20002, size = 12, normalized size = 1. \begin{align*} 2 \, \sqrt{b x} \mathrm{sgn}\left (x\right ) \end{align*}

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)