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

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

[Out]

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

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Rubi [A]  time = 0.0013542, antiderivative size = 12, normalized size of antiderivative = 1., 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} \[ -\frac{3 x}{\left (b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x)/(b*x^2)^(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^2\right )^{2/3}} \, dx &=\frac{x^{4/3} \int \frac{1}{x^{4/3}} \, dx}{\left (b x^2\right )^{2/3}}\\ &=-\frac{3 x}{\left (b x^2\right )^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.0012241, size = 12, normalized size = 1. \[ -\frac{3 x}{\left (b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 11, normalized size = 0.9 \begin{align*} -3\,{\frac{x}{ \left ( b{x}^{2} \right ) ^{2/3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.960016, size = 14, normalized size = 1.17 \begin{align*} -\frac{3 \, x}{\left (b x^{2}\right )^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.74738, size = 32, normalized size = 2.67 \begin{align*} -\frac{3 \, \left (b x^{2}\right )^{\frac{1}{3}}}{b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.571284, size = 15, normalized size = 1.25 \begin{align*} - \frac{3 x}{b^{\frac{2}{3}} \left (x^{2}\right )^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b x^{2}\right )^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^2)^(-2/3), x)