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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0014171, 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} \[ 3 x \left (\frac{b}{x}\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0011418, size = 12, normalized size = 1. \[ 3 x \left (\frac{b}{x}\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0., size = 11, normalized size = 0.9 \begin{align*} 3\, \left ({\frac{b}{x}} \right ) ^{2/3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.979768, size = 14, normalized size = 1.17 \begin{align*} 3 \, x \left (\frac{b}{x}\right )^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.72077, size = 23, normalized size = 1.92 \begin{align*} 3 \, x \left (\frac{b}{x}\right )^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.286561, size = 14, normalized size = 1.17 \begin{align*} 3 b^{\frac{2}{3}} x \left (\frac{1}{x}\right )^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.17897, size = 14, normalized size = 1.17 \begin{align*} \frac{3 \, b}{\left (\frac{b}{x}\right )^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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