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

Optimal. Leaf size=12 \[ \frac{3 \sqrt [3]{b x}}{b} \]

[Out]

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

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Rubi [A]  time = 0.0013403, antiderivative size = 12, 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} \[ \frac{3 \sqrt [3]{b x}}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0012122, size = 10, normalized size = 0.83 \[ \frac{3 x}{(b x)^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 9, normalized size = 0.8 \begin{align*} 3\,{\frac{x}{ \left ( bx \right ) ^{2/3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.988783, size = 14, normalized size = 1.17 \begin{align*} \frac{3 \, \left (b x\right )^{\frac{1}{3}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.68667, size = 23, normalized size = 1.92 \begin{align*} \frac{3 \, \left (b x\right )^{\frac{1}{3}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.061202, size = 8, normalized size = 0.67 \begin{align*} \frac{3 \sqrt [3]{b x}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.57076, size = 14, normalized size = 1.17 \begin{align*} \frac{3 \, \left (b x\right )^{\frac{1}{3}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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