3.2.52 \(\int \frac {1}{x (b x^n)^{3/2}} \, dx\) [152]

Optimal. Leaf size=24 \[ -\frac {2 x^{-n}}{3 b n \sqrt {b x^n}} \]

[Out]

-2/3/b/n/(x^n)/(b*x^n)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 24, 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 x^{-n}}{3 b n \sqrt {b x^n}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(b*x^n)^(3/2)),x]

[Out]

-2/(3*b*n*x^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 \left (b x^n\right )^{3/2}} \, dx &=\frac {x^{n/2} \int x^{-1-\frac {3 n}{2}} \, dx}{b \sqrt {b x^n}}\\ &=-\frac {2 x^{-n}}{3 b n \sqrt {b x^n}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 16, normalized size = 0.67 \begin {gather*} -\frac {2}{3 n \left (b x^n\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(b*x^n)^(3/2)),x]

[Out]

-2/(3*n*(b*x^n)^(3/2))

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Maple [A]
time = 0.06, size = 13, normalized size = 0.54

method	result	size
gosper	\(-\frac {2}{3 n \left (b \,x^{n}\right )^{\frac {3}{2}}}\)	\(13\)
derivativedivides	\(-\frac {2}{3 n \left (b \,x^{n}\right )^{\frac {3}{2}}}\)	\(13\)
default	\(-\frac {2}{3 n \left (b \,x^{n}\right )^{\frac {3}{2}}}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(b*x^n)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/3/n/(b*x^n)^(3/2)

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Maxima [A]
time = 0.30, size = 12, normalized size = 0.50 \begin {gather*} -\frac {2}{3 \, \left (b x^{n}\right )^{\frac {3}{2}} n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^n)^(3/2),x, algorithm="maxima")

[Out]

-2/3/((b*x^n)^(3/2)*n)

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Fricas [A]
time = 0.36, size = 22, normalized size = 0.92 \begin {gather*} -\frac {2 \, \sqrt {b x^{n}}}{3 \, b^{2} n x^{2 \, n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^n)^(3/2),x, algorithm="fricas")

[Out]

-2/3*sqrt(b*x^n)/(b^2*n*x^(2*n))

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Sympy [A]
time = 0.77, size = 22, normalized size = 0.92 \begin {gather*} \begin {cases} - \frac {2}{3 n \left (b x^{n}\right )^{\frac {3}{2}}} & \text {for}\: n \neq 0 \\\frac {\log {\left (x \right )}}{b^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x**n)**(3/2),x)

[Out]

Piecewise((-2/(3*n*(b*x**n)**(3/2)), Ne(n, 0)), (log(x)/b**(3/2), 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)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((b*x^n)^(3/2)*x), x)

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Mupad [B]
time = 0.93, size = 12, normalized size = 0.50 \begin {gather*} -\frac {2}{3\,n\,{\left (b\,x^n\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(b*x^n)^(3/2)),x)

[Out]

-2/(3*n*(b*x^n)^(3/2))

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