3.2.23 \(\int \frac {1}{\sqrt [3]{\frac {b}{x^3}}} \, dx\) [123]

Optimal. Leaf size=14 \[ \frac {x}{2 \sqrt [3]{\frac {b}{x^3}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 14, 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} \begin {gather*} \frac {x}{2 \sqrt [3]{\frac {b}{x^3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/(2*(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 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]{\frac {b}{x^3}}} \, dx &=\frac {\int x \, dx}{\sqrt [3]{\frac {b}{x^3}} x}\\ &=\frac {x}{2 \sqrt [3]{\frac {b}{x^3}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {x}{2 \sqrt [3]{\frac {b}{x^3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 11, normalized size = 0.79

method	result	size
gosper	\(\frac {x}{2 \left (\frac {b}{x^{3}}\right )^{\frac {1}{3}}}\)	\(11\)
risch	\(\frac {x}{2 \left (\frac {b}{x^{3}}\right )^{\frac {1}{3}}}\)	\(11\)
trager	\(\frac {x^{2} \left (x +1\right ) \left (x -1\right ) \left (\frac {b}{x^{3}}\right )^{\frac {2}{3}}}{2 b}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b/x^3)^(1/3),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 10, normalized size = 0.71 \begin {gather*} \frac {x}{2 \, \left (\frac {b}{x^{3}}\right )^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.34, size = 15, normalized size = 1.07 \begin {gather*} \frac {x^{4} \left (\frac {b}{x^{3}}\right )^{\frac {2}{3}}}{2 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^4*(b/x^3)^(2/3)/b

________________________________________________________________________________________

Sympy [A]
time = 0.16, size = 10, normalized size = 0.71 \begin {gather*} \frac {x}{2 \sqrt [3]{\frac {b}{x^{3}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 1.52, size = 10, normalized size = 0.71 \begin {gather*} \frac {x}{2 \, \left (\frac {b}{x^{3}}\right )^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.94, size = 15, normalized size = 1.07 \begin {gather*} \frac {x^4\,{\left (\frac {b}{x^3}\right )}^{2/3}}{2\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________