3.123 \(\int \frac{1}{\sqrt [3]{\frac{b}{x^3}}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0013602, antiderivative size = 14, 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, 30} \[ \frac{x}{2 \sqrt [3]{\frac{b}{x^3}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0017628, size = 14, normalized size = 1. \[ \frac{x}{2 \sqrt [3]{\frac{b}{x^3}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 11, normalized size = 0.8 \begin{align*}{\frac{x}{2}{\frac{1}{\sqrt [3]{{\frac{b}{{x}^{3}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.949881, size = 14, normalized size = 1. \begin{align*} \frac{x}{2 \, \left (\frac{b}{x^{3}}\right )^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.73494, size = 34, normalized size = 2.43 \begin{align*} \frac{x^{4} \left (\frac{b}{x^{3}}\right )^{\frac{2}{3}}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^4*(b/x^3)^(2/3)/b

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Sympy [A]  time = 0.392303, size = 15, normalized size = 1.07 \begin{align*} \frac{x}{2 \sqrt [3]{b} \sqrt [3]{\frac{1}{x^{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(b/x^3)^(1/3),x, algorithm="giac")

[Out]

integrate((b/x^3)^(-1/3), x)