3.114 \(\int (\frac{b}{x^3})^{2/3} \, dx\)

Optimal. Leaf size=12 \[ x \left (-\left (\frac{b}{x^3}\right )^{2/3}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0014428, 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} \[ x \left (-\left (\frac{b}{x^3}\right )^{2/3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0011566, size = 12, normalized size = 1. \[ x \left (-\left (\frac{b}{x^3}\right )^{2/3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.992274, size = 14, normalized size = 1.17 \begin{align*} -x \left (\frac{b}{x^{3}}\right )^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.67286, size = 24, normalized size = 2. \begin{align*} -x \left (\frac{b}{x^{3}}\right )^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.424078, size = 15, normalized size = 1.25 \begin{align*} - b^{\frac{2}{3}} x \left (\frac{1}{x^{3}}\right )^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b/x^3)^(2/3), x)