3.125 \(\int \frac {1}{(b x^3)^{2/3}} \, dx\)

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} \[ -\frac {x}{\left (b x^3\right )^{2/3}} \]

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 \[ -\frac {x}{\left (b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.74, size = 15, normalized size = 1.25 \[ -\frac {\left (b x^{3}\right )^{\frac {1}{3}}}{b x^{2}} \]

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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giac [A]  time = 0.15, size = 8, normalized size = 0.67 \[ -\frac {1}{b^{\frac {2}{3}} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 11, normalized size = 0.92 \[ -\frac {x}{\left (b \,x^{3}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.29, size = 10, normalized size = 0.83 \[ -\frac {x}{\left (b x^{3}\right )^{\frac {2}{3}}} \]

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

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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sympy [A]  time = 0.54, size = 14, normalized size = 1.17 \[ - \frac {x}{b^{\frac {2}{3}} \left (x^{3}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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