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

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Rubi [A]  time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \[ -\frac {2}{n \sqrt {b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.86, size = 20, normalized size = 1.43 \[ -\frac {2 \, \sqrt {b x^{n}}}{b n x^{n}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x^{n}} x}\,{d x} \]

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)

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maple [A]  time = 0.00, size = 13, normalized size = 0.93 \[ -\frac {2}{\sqrt {b \,x^{n}}\, n} \]

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 [A]  time = 1.29, size = 12, normalized size = 0.86 \[ -\frac {2}{\sqrt {b x^{n}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.97, size = 20, normalized size = 1.43 \[ -\frac {2\,\sqrt {b\,x^n}}{b\,n\,x^n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 2.04, size = 24, normalized size = 1.71 \[ \begin {cases} - \frac {2}{\sqrt {b} n \sqrt {x^{n}}} & \text {for}\: n \neq 0 \\\frac {\log {\relax (x )}}{\sqrt {b}} & \text {otherwise} \end {cases} \]

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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