3.2.5 \(\int \sqrt [3]{\frac {b}{x}} \, dx\) [105]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {3}{2} x \sqrt [3]{\frac {b}{x}} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {3}{2} \sqrt [3]{\frac {b}{x}} x \end {gather*}

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 = 0.03, size = 11, normalized size = 0.79

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 10, normalized size = 0.71 \begin {gather*} \frac {3}{2} \, x \left (\frac {b}{x}\right )^{\frac {1}{3}} \end {gather*}

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 = 0.35, size = 10, normalized size = 0.71 \begin {gather*} \frac {3}{2} \, x \left (\frac {b}{x}\right )^{\frac {1}{3}} \end {gather*}

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 = 0.06, size = 10, normalized size = 0.71 \begin {gather*} \frac {3 x \sqrt [3]{\frac {b}{x}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.51, size = 10, normalized size = 0.71 \begin {gather*} \frac {3 \, b}{2 \, \left (\frac {b}{x}\right )^{\frac {2}{3}}} \end {gather*}

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)

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Mupad [B]
time = 0.92, size = 10, normalized size = 0.71 \begin {gather*} \frac {3\,x\,{\left (\frac {b}{x}\right )}^{1/3}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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