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

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

[Out]

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

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Rubi [A]  time = 0.002619, antiderivative size = 13, normalized size of antiderivative = 1., 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} \[ x \sqrt [3]{\frac{b}{x^3}} \log (x) \]

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*}

Mathematica [A]  time = 0.0010816, size = 13, normalized size = 1. \[ x \sqrt [3]{\frac{b}{x^3}} \log (x) \]

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 = 0.008, size = 12, normalized size = 0.9 \begin{align*} \sqrt [3]{{\frac{b}{{x}^{3}}}}x\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.968207, size = 15, normalized size = 1.15 \begin{align*} x \left (\frac{b}{x^{3}}\right )^{\frac{1}{3}} \log \left (x\right ) \end{align*}

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 = 1.68131, size = 32, normalized size = 2.46 \begin{align*} x \left (\frac{b}{x^{3}}\right )^{\frac{1}{3}} \log \left (x\right ) \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{\frac{b}{x^{3}}}\, dx \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\frac{b}{x^{3}}\right )^{\frac{1}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b/x^3)^(1/3), x)