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

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 13, 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, 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*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ x \sqrt {\frac {b}{x^2}} \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.84, size = 11, normalized size = 0.85 \[ x \sqrt {\frac {b}{x^{2}}} \log \relax (x) \]

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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giac [A]  time = 0.17, size = 9, normalized size = 0.69 \[ \sqrt {b} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\relax (x) \]

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)

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maple [A]  time = 0.00, size = 12, normalized size = 0.92 \[ \sqrt {\frac {b}{x^{2}}}\, x \ln \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.33, size = 11, normalized size = 0.85 \[ x \sqrt {\frac {b}{x^{2}}} \log \relax (x) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.08 \[ \int \sqrt {\frac {b}{x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((b/x^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\frac {b}{x^{2}}}\, dx \]

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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