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

Optimal. Leaf size=14 \[ \frac{3 (b x)^{2/3}}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0013512, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {32} \[ \frac{3 (b x)^{2/3}}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0015152, size = 12, normalized size = 0.86 \[ \frac{3 x}{2 \sqrt [3]{b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 9, normalized size = 0.6 \begin{align*}{\frac{3\,x}{2}{\frac{1}{\sqrt [3]{bx}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.973339, size = 14, normalized size = 1. \begin{align*} \frac{3 \, \left (b x\right )^{\frac{2}{3}}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.74904, size = 26, normalized size = 1.86 \begin{align*} \frac{3 \, \left (b x\right )^{\frac{2}{3}}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.056936, size = 10, normalized size = 0.71 \begin{align*} \frac{3 \left (b x\right )^{\frac{2}{3}}}{2 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1471, size = 14, normalized size = 1. \begin{align*} \frac{3 \, \left (b x\right )^{\frac{2}{3}}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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