3.5.9 \(\int \frac {1}{(a+b x)^{2/3}} \, dx\) [409]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 14, 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 \sqrt [3]{a+b x}}{b} \end {gather*}

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*}

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Mathematica [A]
time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{a+b x}}{b} \end {gather*}

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 = 0.12, size = 13, normalized size = 0.93

method	result	size
gosper	\(\frac {3 \left (b x +a \right )^{\frac {1}{3}}}{b}\)	\(13\)
derivativedivides	\(\frac {3 \left (b x +a \right )^{\frac {1}{3}}}{b}\)	\(13\)
default	\(\frac {3 \left (b x +a \right )^{\frac {1}{3}}}{b}\)	\(13\)
trager	\(\frac {3 \left (b x +a \right )^{\frac {1}{3}}}{b}\)	\(13\)
risch	\(\frac {3 \left (b x +a \right )^{\frac {1}{3}}}{b}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 12, normalized size = 0.86 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {1}{3}}}{b} \end {gather*}

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 = 0.76, size = 12, normalized size = 0.86 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {1}{3}}}{b} \end {gather*}

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 = 0.01, size = 10, normalized size = 0.71 \begin {gather*} \frac {3 \sqrt [3]{a + b x}}{b} \end {gather*}

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 = 0.78, size = 12, normalized size = 0.86 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {1}{3}}}{b} \end {gather*}

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

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Mupad [B]
time = 0.02, size = 12, normalized size = 0.86 \begin {gather*} \frac {3\,{\left (a+b\,x\right )}^{1/3}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x)^(2/3),x)

[Out]

(3*(a + b*x)^(1/3))/b

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