3.121 \(\int \frac{1}{\sqrt [3]{\frac{b}{x}}} \, dx\)

Optimal. Leaf size=14 \[ \frac{3 x}{4 \sqrt [3]{\frac{b}{x}}} \]

[Out]

(3*x)/(4*(b/x)^(1/3))

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Rubi [A]  time = 0.0015995, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {15, 30} \[ \frac{3 x}{4 \sqrt [3]{\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

Int[(b/x)^(-1/3),x]

[Out]

(3*x)/(4*(b/x)^(1/3))

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}{\sqrt [3]{\frac{b}{x}}} \, dx &=\frac{\int \sqrt [3]{x} \, dx}{\sqrt [3]{\frac{b}{x}} \sqrt [3]{x}}\\ &=\frac{3 x}{4 \sqrt [3]{\frac{b}{x}}}\\ \end{align*}

Mathematica [A]  time = 0.002079, size = 14, normalized size = 1. \[ \frac{3 x}{4 \sqrt [3]{\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b/x)^(-1/3),x]

[Out]

(3*x)/(4*(b/x)^(1/3))

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Maple [A]  time = 0.002, size = 11, normalized size = 0.8 \begin{align*}{\frac{3\,x}{4}{\frac{1}{\sqrt [3]{{\frac{b}{x}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b/x)^(1/3),x)

[Out]

3/4*x/(b/x)^(1/3)

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Maxima [A]  time = 0.997423, size = 14, normalized size = 1. \begin{align*} \frac{3 \, x}{4 \, \left (\frac{b}{x}\right )^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*x/(b/x)^(1/3)

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Fricas [A]  time = 1.66937, size = 31, normalized size = 2.21 \begin{align*} \frac{3 \, x^{2} \left (\frac{b}{x}\right )^{\frac{2}{3}}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*x^2*(b/x)^(2/3)/b

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Sympy [A]  time = 0.371547, size = 15, normalized size = 1.07 \begin{align*} \frac{3 x}{4 \sqrt [3]{b} \sqrt [3]{\frac{1}{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x/(4*b**(1/3)*(1/x)**(1/3))

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Giac [A]  time = 1.16362, size = 14, normalized size = 1. \begin{align*} \frac{3 \, x}{4 \, \left (\frac{b}{x}\right )^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b/x)^(1/3),x, algorithm="giac")

[Out]

3/4*x/(b/x)^(1/3)