3.2.25 \(\int \frac {1}{(b x^3)^{2/3}} \, dx\) [125]

Optimal. Leaf size=12 \[ -\frac {x}{\left (b x^3\right )^{2/3}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 30} \begin {gather*} -\frac {x}{\left (b x^3\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^3)^(-2/3),x]

[Out]

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{\left (b x^3\right )^{2/3}} \, dx &=\frac {x^2 \int \frac {1}{x^2} \, dx}{\left (b x^3\right )^{2/3}}\\ &=-\frac {x}{\left (b x^3\right )^{2/3}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} -\frac {x}{\left (b x^3\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^3)^(-2/3),x]

[Out]

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

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Maple [A]
time = 0.02, size = 11, normalized size = 0.92

method	result	size
gosper	\(-\frac {x}{\left (b \,x^{3}\right )^{\frac {2}{3}}}\)	\(11\)
risch	\(-\frac {x}{\left (b \,x^{3}\right )^{\frac {2}{3}}}\)	\(11\)
trager	\(\frac {\left (x -1\right ) \left (b \,x^{3}\right )^{\frac {1}{3}}}{b \,x^{2}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 10, normalized size = 0.83 \begin {gather*} -\frac {x}{\left (b x^{3}\right )^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3)^(2/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.34, size = 15, normalized size = 1.25 \begin {gather*} -\frac {\left (b x^{3}\right )^{\frac {1}{3}}}{b x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3)^(2/3),x, algorithm="fricas")

[Out]

-(b*x^3)^(1/3)/(b*x^2)

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Sympy [A]
time = 0.17, size = 10, normalized size = 0.83 \begin {gather*} - \frac {x}{\left (b x^{3}\right )^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**3)**(2/3),x)

[Out]

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

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Giac [A]
time = 1.05, size = 10, normalized size = 0.83 \begin {gather*} -\frac {x}{\left (b x^{3}\right )^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3)^(2/3),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.93, size = 15, normalized size = 1.25 \begin {gather*} -\frac {{\left (b\,x^3\right )}^{1/3}}{b\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^3)^(2/3),x)

[Out]

-(b*x^3)^(1/3)/(b*x^2)

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