3.2.46 \(\int \frac {1}{x \sqrt {b x^n}} \, dx\) [146]

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} \begin {gather*} -\frac {2}{n \sqrt {b x^n}} \end {gather*}

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 \begin {gather*} -\frac {2}{n \sqrt {b x^n}} \end {gather*}

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 = 0.01, size = 13, normalized size = 0.93

method	result	size
gosper	\(-\frac {2}{n \sqrt {b \,x^{n}}}\)	\(13\)
derivativedivides	\(-\frac {2}{n \sqrt {b \,x^{n}}}\)	\(13\)
default	\(-\frac {2}{n \sqrt {b \,x^{n}}}\)	\(13\)
risch	\(-\frac {2}{n \sqrt {b \,x^{n}}}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.31, size = 12, normalized size = 0.86 \begin {gather*} -\frac {2}{\sqrt {b x^{n}} n} \end {gather*}

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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Fricas [A]
time = 0.36, size = 20, normalized size = 1.43 \begin {gather*} -\frac {2 \, \sqrt {b x^{n}}}{b n x^{n}} \end {gather*}

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 = 0.52, size = 20, normalized size = 1.43 \begin {gather*} \begin {cases} - \frac {2}{n \sqrt {b x^{n}}} & \text {for}\: n \neq 0 \\\frac {\log {\left (x \right )}}{\sqrt {b}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x**n)**(1/2),x)

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Mupad [B]
time = 0.97, size = 20, normalized size = 1.43 \begin {gather*} -\frac {2\,\sqrt {b\,x^n}}{b\,n\,x^n} \end {gather*}

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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