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

Optimal. Leaf size=13 \[ x \sqrt{\frac{b}{x^2}} \log (x) \]

[Out]

Sqrt[b/x^2]*x*Log[x]

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Rubi [A]  time = 0.0025132, antiderivative size = 13, 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, 29} \[ x \sqrt{\frac{b}{x^2}} \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[b/x^2]*x*Log[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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.0012014, size = 13, normalized size = 1. \[ x \sqrt{\frac{b}{x^2}} \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[b/x^2]*x*Log[x]

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Maple [A]  time = 0.003, size = 12, normalized size = 0.9 \begin{align*} x\ln \left ( x \right ) \sqrt{{\frac{b}{{x}^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.39201, size = 15, normalized size = 1.15 \begin{align*} x \sqrt{\frac{b}{x^{2}}} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.68759, size = 30, normalized size = 2.31 \begin{align*} x \sqrt{\frac{b}{x^{2}}} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{b}{x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(b/x**2), x)

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Giac [A]  time = 1.1727, size = 12, normalized size = 0.92 \begin{align*} \sqrt{b} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b)*log(abs(x))*sgn(x)