3.5.2 \(\int \frac {1}{\sqrt [3]{-a+b x}} \, dx\) [402]

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

[Out]

3/2*(b*x-a)^(2/3)/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 = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {3 (-a+b x)^{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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Mathics [A]
time = 1.57, size = 14, normalized size = 0.78 \begin {gather*} \frac {3 \left (-a+b x\right )^{\frac {2}{3}}}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(-a + b*x)^(-1/3),x]')

[Out]

3 (-a + b x) ^ (2 / 3) / (2 b)

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Maple [A]
time = 0.11, size = 15, normalized size = 0.83

method	result	size
gosper	\(\frac {3 \left (b x -a \right )^{\frac {2}{3}}}{2 b}\)	\(15\)
derivativedivides	\(\frac {3 \left (b x -a \right )^{\frac {2}{3}}}{2 b}\)	\(15\)
default	\(\frac {3 \left (b x -a \right )^{\frac {2}{3}}}{2 b}\)	\(15\)
trager	\(\frac {3 \left (b x -a \right )^{\frac {2}{3}}}{2 b}\)	\(15\)
risch	\(-\frac {3 \left (-b x +a \right )}{2 b \left (b x -a \right )^{\frac {1}{3}}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, 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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Fricas [A]
time = 0.31, 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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Sympy [A]
time = 0.03, 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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Giac [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \frac {3 \left (\left (-a+b x\right )^{\frac {1}{3}}\right )^{2}}{b\cdot 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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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