3.125 \(\int \frac{1}{(b x^3)^{2/3}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0013575, 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} \[ -\frac{x}{\left (b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0015347, size = 12, normalized size = 1. \[ -\frac{x}{\left (b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03529, size = 14, normalized size = 1.17 \begin{align*} -\frac{x}{\left (b x^{3}\right )^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.65802, size = 32, normalized size = 2.67 \begin{align*} -\frac{\left (b x^{3}\right )^{\frac{1}{3}}}{b x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.598726, size = 14, normalized size = 1.17 \begin{align*} - \frac{x}{b^{\frac{2}{3}} \left (x^{3}\right )^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^3)^(-2/3), x)