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

Optimal. Leaf size=14 \[ \frac{3 x}{5 \left (\frac{b}{x}\right )^{2/3}} \]

[Out]

(3*x)/(5*(b/x)^(2/3))

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Rubi [A]  time = 0.001393, 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} \[ \frac{3 x}{5 \left (\frac{b}{x}\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x)/(5*(b/x)^(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 (\frac{b}{x}\right )^{2/3}} \, dx &=\frac{\int x^{2/3} \, dx}{\left (\frac{b}{x}\right )^{2/3} x^{2/3}}\\ &=\frac{3 x}{5 \left (\frac{b}{x}\right )^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.0017159, size = 14, normalized size = 1. \[ \frac{3 x}{5 \left (\frac{b}{x}\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x)/(5*(b/x)^(2/3))

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Maple [A]  time = 0.002, size = 11, normalized size = 0.8 \begin{align*}{\frac{3\,x}{5} \left ({\frac{b}{x}} \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*x/(b/x)^(2/3)

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Maxima [A]  time = 0.973182, size = 14, normalized size = 1. \begin{align*} \frac{3 \, x}{5 \, \left (\frac{b}{x}\right )^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*x/(b/x)^(2/3)

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Fricas [A]  time = 1.63053, size = 31, normalized size = 2.21 \begin{align*} \frac{3 \, x^{2} \left (\frac{b}{x}\right )^{\frac{1}{3}}}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*x^2*(b/x)^(1/3)/b

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Sympy [A]  time = 0.572401, size = 15, normalized size = 1.07 \begin{align*} \frac{3 x}{5 b^{\frac{2}{3}} \left (\frac{1}{x}\right )^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x/(5*b**(2/3)*(1/x)**(2/3))

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Giac [A]  time = 1.14345, size = 14, normalized size = 1. \begin{align*} \frac{3 \, x}{5 \, \left (\frac{b}{x}\right )^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*x/(b/x)^(2/3)