3.4.95 \(\int \frac {1}{\sqrt [3]{a+b x}} \, dx\) [395]

Optimal. Leaf size=16 \[ \frac {3 (a+b x)^{2/3}}{2 b} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {3 (a+b x)^{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 = 16, 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 = 12, normalized size = 0.75 \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 = 13, normalized size = 0.81

method	result	size
gosper	\(\frac {3 \left (b x +a \right )^{\frac {2}{3}}}{2 b}\)	\(13\)
derivativedivides	\(\frac {3 \left (b x +a \right )^{\frac {2}{3}}}{2 b}\)	\(13\)
default	\(\frac {3 \left (b x +a \right )^{\frac {2}{3}}}{2 b}\)	\(13\)
trager	\(\frac {3 \left (b x +a \right )^{\frac {2}{3}}}{2 b}\)	\(13\)
risch	\(\frac {3 \left (b x +a \right )^{\frac {2}{3}}}{2 b}\)	\(13\)

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 = 12, normalized size = 0.75 \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 = 0.30, size = 12, normalized size = 0.75 \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 = 0.75 \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 = 17, normalized size = 1.06 \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 = 12, normalized size = 0.75 \begin {gather*} \frac {3\,{\left (a+b\,x\right )}^{2/3}}{2\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x)^(1/3),x)

[Out]

(3*(a + b*x)^(2/3))/(2*b)

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