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

Optimal. Leaf size=14 \[ \frac{3 \sqrt [3]{a+b x}}{b} \]

[Out]

(3*(a + b*x)^(1/3))/b

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Rubi [A]  time = 0.001359, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {32} \[ \frac{3 \sqrt [3]{a+b x}}{b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(-2/3),x]

[Out]

(3*(a + b*x)^(1/3))/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 \frac{1}{(a+b x)^{2/3}} \, dx &=\frac{3 \sqrt [3]{a+b x}}{b}\\ \end{align*}

Mathematica [A]  time = 0.0052719, size = 14, normalized size = 1. \[ \frac{3 \sqrt [3]{a+b x}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(-2/3),x]

[Out]

(3*(a + b*x)^(1/3))/b

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Maple [A]  time = 0.002, size = 13, normalized size = 0.9 \begin{align*} 3\,{\frac{\sqrt [3]{bx+a}}{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07606, size = 16, normalized size = 1.14 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{1}{3}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(b*x + a)^(1/3)/b

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Fricas [A]  time = 1.51201, size = 28, normalized size = 2. \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{1}{3}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(b*x + a)^(1/3)/b

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Sympy [A]  time = 0.102487, size = 10, normalized size = 0.71 \begin{align*} \frac{3 \sqrt [3]{a + b x}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(a + b*x)**(1/3)/b

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Giac [A]  time = 1.24887, size = 16, normalized size = 1.14 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{1}{3}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(b*x + a)^(1/3)/b