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

Optimal. Leaf size=13 \[ \frac{x \log (x)}{\sqrt [3]{b x^3}} \]

[Out]

(x*Log[x])/(b*x^3)^(1/3)

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Rubi [A]  time = 0.0013001, antiderivative size = 13, 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, 29} \[ \frac{x \log (x)}{\sqrt [3]{b x^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Log[x])/(b*x^3)^(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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [3]{b x^3}} \, dx &=\frac{x \int \frac{1}{x} \, dx}{\sqrt [3]{b x^3}}\\ &=\frac{x \log (x)}{\sqrt [3]{b x^3}}\\ \end{align*}

Mathematica [A]  time = 0.0013723, size = 13, normalized size = 1. \[ \frac{x \log (x)}{\sqrt [3]{b x^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Log[x])/(b*x^3)^(1/3)

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Maple [A]  time = 0.007, size = 12, normalized size = 0.9 \begin{align*}{x\ln \left ( x \right ){\frac{1}{\sqrt [3]{b{x}^{3}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*ln(x)/(b*x^3)^(1/3)

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Maxima [A]  time = 0.95983, size = 15, normalized size = 1.15 \begin{align*} \frac{x \log \left (x\right )}{\left (b x^{3}\right )^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x)/(b*x^3)^(1/3)

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Fricas [A]  time = 1.74394, size = 41, normalized size = 3.15 \begin{align*} \frac{\left (b x^{3}\right )^{\frac{2}{3}} \log \left (x\right )}{b x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{b x^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((b*x**3)**(-1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^3)^(-1/3), x)