3.2.7 \(\int \sqrt [3]{\frac {b}{x^3}} \, dx\) [107]

Optimal. Leaf size=13 \[ \sqrt [3]{\frac {b}{x^3}} x \log (x) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 13, 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, 29} \begin {gather*} x \sqrt [3]{\frac {b}{x^3}} \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b/x^3)^(1/3)*x*Log[x]

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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \sqrt [3]{\frac {b}{x^3}} x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b/x^3)^(1/3)*x*Log[x]

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Maple [A]
time = 0.04, size = 12, normalized size = 0.92

method	result	size
risch	\(\left (\frac {b}{x^{3}}\right )^{\frac {1}{3}} x \ln \left (x \right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.31, size = 11, normalized size = 0.85 \begin {gather*} x \left (\frac {b}{x^{3}}\right )^{\frac {1}{3}} \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.34, size = 11, normalized size = 0.85 \begin {gather*} x \left (\frac {b}{x^{3}}\right )^{\frac {1}{3}} \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [3]{\frac {b}{x^{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((b/x**3)**(1/3), x)

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Giac [A]
time = 1.99, size = 1, normalized size = 0.08 \begin {gather*} +\infty \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

+Infinity

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.08 \begin {gather*} \int {\left (\frac {b}{x^3}\right )}^{1/3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((b/x^3)^(1/3), x)

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