3.2.27 \(\int \frac {1}{(b x)^{2/3}} \, dx\) [127]

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 = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {3 \sqrt [3]{b x}}{b} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 10, normalized size = 0.83 \begin {gather*} \frac {3 x}{(b x)^{2/3}} \end {gather*}

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 = 0.02, size = 11, normalized size = 0.92

method	result	size
gosper	\(\frac {3 x}{\left (b x \right )^{\frac {2}{3}}}\)	\(9\)
risch	\(\frac {3 x}{\left (b x \right )^{\frac {2}{3}}}\)	\(9\)
derivativedivides	\(\frac {3 \left (b x \right )^{\frac {1}{3}}}{b}\)	\(11\)
default	\(\frac {3 \left (b x \right )^{\frac {1}{3}}}{b}\)	\(11\)
trager	\(\frac {3 \left (b x \right )^{\frac {1}{3}}}{b}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 10, normalized size = 0.83 \begin {gather*} \frac {3 \, \left (b x\right )^{\frac {1}{3}}}{b} \end {gather*}

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 = 0.35, size = 10, normalized size = 0.83 \begin {gather*} \frac {3 \, \left (b x\right )^{\frac {1}{3}}}{b} \end {gather*}

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 = 0.01, size = 8, normalized size = 0.67 \begin {gather*} \frac {3 \sqrt [3]{b x}}{b} \end {gather*}

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 = 1.64, size = 10, normalized size = 0.83 \begin {gather*} \frac {3 \, \left (b x\right )^{\frac {1}{3}}}{b} \end {gather*}

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

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Mupad [B]
time = 0.01, size = 10, normalized size = 0.83 \begin {gather*} \frac {3\,{\left (b\,x\right )}^{1/3}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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