3.106 \(\int \sqrt [3]{\frac{b}{x^2}} \, dx\)

Optimal. Leaf size=12 \[ 3 x \sqrt [3]{\frac{b}{x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0015893, antiderivative size = 12, 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} \[ 3 x \sqrt [3]{\frac{b}{x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(b/x^2)^(1/3),x]

[Out]

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

Mathematica [A]  time = 0.0009864, size = 12, normalized size = 1. \[ 3 x \sqrt [3]{\frac{b}{x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b/x^2)^(1/3),x]

[Out]

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

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Maple [A]  time = 0.001, size = 11, normalized size = 0.9 \begin{align*} 3\,\sqrt [3]{{\frac{b}{{x}^{2}}}}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.973353, size = 14, normalized size = 1.17 \begin{align*} 3 \, x \left (\frac{b}{x^{2}}\right )^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b/x^2)^(1/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.63219, size = 26, normalized size = 2.17 \begin{align*} 3 \, x \left (\frac{b}{x^{2}}\right )^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b/x^2)^(1/3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.45948, size = 15, normalized size = 1.25 \begin{align*} 3 \sqrt [3]{b} x \sqrt [3]{\frac{1}{x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\frac{b}{x^{2}}\right )^{\frac{1}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b/x^2)^(1/3),x, algorithm="giac")

[Out]

integrate((b/x^2)^(1/3), x)