\(\int \sqrt [3]{\frac {b}{x^3}} \, dx\) [107]

Optimal result

Integrand size = 9, antiderivative size = 13 \[ \int \sqrt [3]{\frac {b}{x^3}} \, dx=\sqrt [3]{\frac {b}{x^3}} x \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 29} \[ \int \sqrt [3]{\frac {b}{x^3}} \, dx=x \sqrt [3]{\frac {b}{x^3}} \log (x) \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \sqrt [3]{\frac {b}{x^3}} \, dx=\sqrt [3]{\frac {b}{x^3}} x \log (x) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92

method	result	size
risch	\(\left (\frac {b}{x^{3}}\right )^{\frac {1}{3}} x \ln \left (x \right )\)	\(12\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \sqrt [3]{\frac {b}{x^3}} \, dx=x \left (\frac {b}{x^{3}}\right )^{\frac {1}{3}} \log \left (x\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \sqrt [3]{\frac {b}{x^3}} \, dx=x \sqrt [3]{\frac {b}{x^{3}}} \log {\left (x \right )} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \sqrt [3]{\frac {b}{x^3}} \, dx=x \left (\frac {b}{x^{3}}\right )^{\frac {1}{3}} \log \left (x\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.08 \[ \int \sqrt [3]{\frac {b}{x^3}} \, dx=+\infty \]

[In]

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

[Out]

+Infinity

Mupad [F(-1)]

Timed out. \[ \int \sqrt [3]{\frac {b}{x^3}} \, dx=\int {\left (\frac {b}{x^3}\right )}^{1/3} \,d x \]

[In]

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

[Out]

int((b/x^3)^(1/3), x)