3.93 \(\int \frac{1}{\sqrt{\frac{b}{x^2}}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0014358, 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} \[ \frac{x}{2 \sqrt{\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[b/x^2],x]

[Out]

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

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^2}}} \, dx &=\frac{\int x \, dx}{\sqrt{\frac{b}{x^2}} x}\\ &=\frac{x}{2 \sqrt{\frac{b}{x^2}}}\\ \end{align*}

Mathematica [A]  time = 0.0016363, size = 14, normalized size = 1. \[ \frac{x}{2 \sqrt{\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[b/x^2],x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00709, size = 14, normalized size = 1. \begin{align*} \frac{x}{2 \, \sqrt{\frac{b}{x^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.72275, size = 31, normalized size = 2.21 \begin{align*} \frac{x^{3} \sqrt{\frac{b}{x^{2}}}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.463038, size = 15, normalized size = 1.07 \begin{align*} \frac{x}{2 \sqrt{b} \sqrt{\frac{1}{x^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18242, size = 16, normalized size = 1.14 \begin{align*} \frac{x^{2}}{2 \, \sqrt{b} \mathrm{sgn}\left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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