3.146 \(\int \frac{1}{x \sqrt{b x^n}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.00903984, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{2}{n \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[b*x^n]),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.79048, size = 12, normalized size = 0.86 \[ - \frac{2}{n \sqrt{b x^{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00456392, size = 14, normalized size = 1. \[ -\frac{2}{n \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*Sqrt[b*x^n]),x]

[Out]

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

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Maple [A]  time = 0.002, size = 13, normalized size = 0.9 \[ -2\,{\frac{1}{n\sqrt{b{x}^{n}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^n)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^n)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22817, size = 16, normalized size = 1.14 \[ -\frac{2}{\sqrt{b x^{n}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^n)*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.72571, size = 24, normalized size = 1.71 \[ \begin{cases} - \frac{2}{\sqrt{b} n \sqrt{x^{n}}} & \text{for}\: n \neq 0 \\\frac{\log{\left (x \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2/(sqrt(b)*n*sqrt(x**n)), Ne(n, 0)), (log(x)/sqrt(b), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{n}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^n)*x),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x^n)*x), x)