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

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

[Out]

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

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Rubi [A]  time = 0.0011246, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {32} \[ \frac{3 (b x)^{5/3}}{5 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0010097, size = 12, normalized size = 0.86 \[ \frac{3}{5} x (b x)^{2/3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 9, normalized size = 0.6 \begin{align*}{\frac{3\,x}{5} \left ( bx \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.992125, size = 14, normalized size = 1. \begin{align*} \frac{3 \, \left (b x\right )^{\frac{5}{3}}}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.66858, size = 26, normalized size = 1.86 \begin{align*} \frac{3}{5} \, \left (b x\right )^{\frac{2}{3}} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.075967, size = 10, normalized size = 0.71 \begin{align*} \frac{3 \left (b x\right )^{\frac{5}{3}}}{5 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18019, size = 11, normalized size = 0.79 \begin{align*} \frac{3}{5} \, \left (b x\right )^{\frac{2}{3}} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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