3.115 \(\int (\frac{b}{x^4})^{2/3} \, dx\)

Optimal. Leaf size=14 \[ -\frac{3}{5} x \left (\frac{b}{x^4}\right )^{2/3} \]

[Out]

(-3*(b/x^4)^(2/3)*x)/5

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Rubi [A]  time = 0.0015081, antiderivative size = 14, 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} \[ -\frac{3}{5} x \left (\frac{b}{x^4}\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

Int[(b/x^4)^(2/3),x]

[Out]

(-3*(b/x^4)^(2/3)*x)/5

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 \left (\frac{b}{x^4}\right )^{2/3} \, dx &=\left (\left (\frac{b}{x^4}\right )^{2/3} x^{8/3}\right ) \int \frac{1}{x^{8/3}} \, dx\\ &=-\frac{3}{5} \left (\frac{b}{x^4}\right )^{2/3} x\\ \end{align*}

Mathematica [A]  time = 0.0013227, size = 14, normalized size = 1. \[ -\frac{3}{5} x \left (\frac{b}{x^4}\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(b/x^4)^(2/3),x]

[Out]

(-3*(b/x^4)^(2/3)*x)/5

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Maple [A]  time = 0.001, size = 11, normalized size = 0.8 \begin{align*} -{\frac{3\,x}{5} \left ({\frac{b}{{x}^{4}}} \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b/x^4)^(2/3),x)

[Out]

-3/5*(b/x^4)^(2/3)*x

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Maxima [A]  time = 0.973675, size = 14, normalized size = 1. \begin{align*} -\frac{3}{5} \, x \left (\frac{b}{x^{4}}\right )^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b/x^4)^(2/3),x, algorithm="maxima")

[Out]

-3/5*x*(b/x^4)^(2/3)

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Fricas [A]  time = 1.71728, size = 30, normalized size = 2.14 \begin{align*} -\frac{3}{5} \, x \left (\frac{b}{x^{4}}\right )^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b/x^4)^(2/3),x, algorithm="fricas")

[Out]

-3/5*x*(b/x^4)^(2/3)

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Sympy [A]  time = 0.443952, size = 19, normalized size = 1.36 \begin{align*} - \frac{3 b^{\frac{2}{3}} x \left (\frac{1}{x^{4}}\right )^{\frac{2}{3}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b/x**4)**(2/3),x)

[Out]

-3*b**(2/3)*x*(x**(-4))**(2/3)/5

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\frac{b}{x^{4}}\right )^{\frac{2}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b/x^4)^(2/3),x, algorithm="giac")

[Out]

integrate((b/x^4)^(2/3), x)