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

Optimal. Leaf size=14 \[ \frac{3}{2} x \sqrt [3]{\frac{b}{x}} \]

[Out]

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

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Rubi [A]  time = 0.0016445, 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} \[ \frac{3}{2} x \sqrt [3]{\frac{b}{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 \sqrt [3]{\frac{b}{x}} \, dx &=\left (\sqrt [3]{\frac{b}{x}} \sqrt [3]{x}\right ) \int \frac{1}{\sqrt [3]{x}} \, dx\\ &=\frac{3}{2} \sqrt [3]{\frac{b}{x}} x\\ \end{align*}

Mathematica [A]  time = 0.0009351, size = 14, normalized size = 1. \[ \frac{3}{2} x \sqrt [3]{\frac{b}{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 11, normalized size = 0.8 \begin{align*}{\frac{3\,x}{2}\sqrt [3]{{\frac{b}{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.995227, size = 14, normalized size = 1. \begin{align*} \frac{3}{2} \, x \left (\frac{b}{x}\right )^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.6553, size = 26, normalized size = 1.86 \begin{align*} \frac{3}{2} \, x \left (\frac{b}{x}\right )^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.222572, size = 15, normalized size = 1.07 \begin{align*} \frac{3 \sqrt [3]{b} x \sqrt [3]{\frac{1}{x}}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*b**(1/3)*x*(1/x)**(1/3)/2

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Giac [A]  time = 1.1913, size = 14, normalized size = 1. \begin{align*} \frac{3 \, b}{2 \, \left (\frac{b}{x}\right )^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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