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 = 0.00848211, 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 \[ x \sqrt{\frac{b}{x^2}} \log (x) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b/x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 1.15439, size = 12, normalized size = 0.92 \[ x \sqrt{\frac{b}{x^{2}}} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b/x**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.00248467, 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 = 0.007, size = 12, normalized size = 0.9 \[ x\ln \left ( x \right ) \sqrt{{\frac{b}{{x}^{2}}}} \]

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 = 1.43562, size = 15, normalized size = 1.15 \[ x \sqrt{\frac{b}{x^{2}}} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b/x^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.22476, size = 15, normalized size = 1.15 \[ x \sqrt{\frac{b}{x^{2}}} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b/x^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. \[ \int \sqrt{\frac{b}{x^{2}}}\, dx \]

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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GIAC/XCAS [A]  time = 0.239839, size = 12, normalized size = 0.92 \[ \sqrt{b}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b/x^2),x, algorithm="giac")

[Out]

sqrt(b)*ln(abs(x))*sign(x)