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

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

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {3 (b x-a)^{2/3}}{2 b} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \frac {3 (b x-a)^{2/3}}{2 b} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {3 (b x-a)^{2/3}}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-a + b*x)^(-1/3),x]

[Out]

(3*(-a + b*x)^(2/3))/(2*b)

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fricas [A]  time = 0.71, size = 14, normalized size = 0.78 \begin {gather*} \frac {3 \, {\left (b x - a\right )}^{\frac {2}{3}}}{2 \, b} \end {gather*}

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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giac [A]  time = 1.02, size = 14, normalized size = 0.78 \begin {gather*} \frac {3 \, {\left (b x - a\right )}^{\frac {2}{3}}}{2 \, b} \end {gather*}

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

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maple [A]  time = 0.00, size = 15, normalized size = 0.83 \begin {gather*} \frac {3 \left (b x -a \right )^{\frac {2}{3}}}{2 b} \end {gather*}

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 = 1.32, size = 14, normalized size = 0.78 \begin {gather*} \frac {3 \, {\left (b x - a\right )}^{\frac {2}{3}}}{2 \, b} \end {gather*}

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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mupad [B]  time = 0.02, size = 14, normalized size = 0.78 \begin {gather*} \frac {3\,{\left (b\,x-a\right )}^{2/3}}{2\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x - a)^(1/3),x)

[Out]

(3*(b*x - a)^(2/3))/(2*b)

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sympy [A]  time = 0.07, size = 12, normalized size = 0.67 \begin {gather*} \frac {3 \left (- a + b x\right )^{\frac {2}{3}}}{2 b} \end {gather*}

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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