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 = 0.0029323, 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, Rules used = {15, 30} \[ -\frac{2}{n \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[b*x^n]),x]

[Out]

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

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}{x \sqrt{b x^n}} \, dx &=\frac{x^{n/2} \int x^{-1-\frac{n}{2}} \, dx}{\sqrt{b x^n}}\\ &=-\frac{2}{n \sqrt{b x^n}}\\ \end{align*}

Mathematica [A]  time = 0.0023034, 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., size = 13, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{n\sqrt{b{x}^{n}}}} \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.69845, size = 35, normalized size = 2.5 \begin{align*} -\frac{2 \, \sqrt{b x^{n}}}{b n x^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.65637, size = 24, normalized size = 1.71 \begin{align*} \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} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{n}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(b*x^n)*x), x)