3.152 \(\int \frac{1}{x (b x^n)^{3/2}} \, dx\)

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

[Out]

-2/(3*b*n*x^n*Sqrt[b*x^n])

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Rubi [A]  time = 0.003785, antiderivative size = 24, 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} \[ -\frac{2 x^{-n}}{3 b n \sqrt{b x^n}} \]

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*}

Mathematica [A]  time = 0.0028107, size = 16, normalized size = 0.67 \[ -\frac{2}{3 n \left (b x^n\right )^{3/2}} \]

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 = 0.001, size = 13, normalized size = 0.5 \begin{align*} -{\frac{2}{3\,n} \left ( b{x}^{n} \right ) ^{-{\frac{3}{2}}}} \end{align*}

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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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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.82914, size = 46, normalized size = 1.92 \begin{align*} -\frac{2 \, \sqrt{b x^{n}}}{3 \, b^{2} n x^{2 \, n}} \end{align*}

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 = 3.54714, size = 26, normalized size = 1.08 \begin{align*} \begin{cases} - \frac{2}{3 b^{\frac{3}{2}} n \left (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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2/(3*b**(3/2)*n*(x**n)**(3/2)), Ne(n, 0)), (log(x)/b**(3/2), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b x^{n}\right )^{\frac{3}{2}} x}\,{d x} \end{align*}

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)