3.402 \(\int \frac{1}{\sqrt [3]{-a+b x}} \, dx\)

Optimal. Leaf size=18 \[ \frac{3 (b x-a)^{2/3}}{2 b} \]

[Out]

(3*(-a + b*x)^(2/3))/(2*b)

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Rubi [A]  time = 0.0014808, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {32} \[ \frac{3 (b x-a)^{2/3}}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[(-a + b*x)^(-1/3),x]

[Out]

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

Mathematica [A]  time = 0.0078637, size = 18, normalized size = 1. \[ \frac{3 (b x-a)^{2/3}}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(-a + b*x)^(-1/3),x]

[Out]

(3*(-a + b*x)^(2/3))/(2*b)

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Maple [A]  time = 0.002, size = 15, normalized size = 0.8 \begin{align*}{\frac{3}{2\,b} \left ( bx-a \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.986556, size = 19, normalized size = 1.06 \begin{align*} \frac{3 \,{\left (b x - a\right )}^{\frac{2}{3}}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(b*x - a)^(2/3)/b

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Fricas [A]  time = 1.43817, size = 31, normalized size = 1.72 \begin{align*} \frac{3 \,{\left (b x - a\right )}^{\frac{2}{3}}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(b*x - a)^(2/3)/b

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Sympy [A]  time = 0.068408, size = 12, normalized size = 0.67 \begin{align*} \frac{3 \left (- a + b x\right )^{\frac{2}{3}}}{2 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x-a)**(1/3),x)

[Out]

3*(-a + b*x)**(2/3)/(2*b)

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Giac [A]  time = 1.19638, size = 19, normalized size = 1.06 \begin{align*} \frac{3 \,{\left (b x - a\right )}^{\frac{2}{3}}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(b*x - a)^(2/3)/b