3.117 \(\int \frac{1}{\sqrt [3]{b x^4}} \, dx\)

Optimal. Leaf size=12 \[ -\frac{3 x}{\sqrt [3]{b x^4}} \]

[Out]

(-3*x)/(b*x^4)^(1/3)

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Rubi [A]  time = 0.0014454, 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} \[ -\frac{3 x}{\sqrt [3]{b x^4}} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^4)^(-1/3),x]

[Out]

(-3*x)/(b*x^4)^(1/3)

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}{\sqrt [3]{b x^4}} \, dx &=\frac{x^{4/3} \int \frac{1}{x^{4/3}} \, dx}{\sqrt [3]{b x^4}}\\ &=-\frac{3 x}{\sqrt [3]{b x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0018404, size = 12, normalized size = 1. \[ -\frac{3 x}{\sqrt [3]{b x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^4)^(-1/3),x]

[Out]

(-3*x)/(b*x^4)^(1/3)

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Maple [A]  time = 0.002, size = 11, normalized size = 0.9 \begin{align*} -3\,{\frac{x}{\sqrt [3]{b{x}^{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^4)^(1/3),x)

[Out]

-3*x/(b*x^4)^(1/3)

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Maxima [A]  time = 0.978753, size = 14, normalized size = 1.17 \begin{align*} -\frac{3 \, x}{\left (b x^{4}\right )^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4)^(1/3),x, algorithm="maxima")

[Out]

-3*x/(b*x^4)^(1/3)

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Fricas [A]  time = 1.6545, size = 35, normalized size = 2.92 \begin{align*} -\frac{3 \, \left (b x^{4}\right )^{\frac{2}{3}}}{b x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4)^(1/3),x, algorithm="fricas")

[Out]

-3*(b*x^4)^(2/3)/(b*x^3)

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Sympy [A]  time = 0.364231, size = 15, normalized size = 1.25 \begin{align*} - \frac{3 x}{\sqrt [3]{b} \sqrt [3]{x^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**4)**(1/3),x)

[Out]

-3*x/(b**(1/3)*(x**4)**(1/3))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b x^{4}\right )^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4)^(1/3),x, algorithm="giac")

[Out]

integrate((b*x^4)^(-1/3), x)