\(\int (\frac {b}{x^3})^{2/3} \, dx\) [114]

Optimal result

Integrand size = 9, antiderivative size = 12 \[ \int \left (\frac {b}{x^3}\right )^{2/3} \, dx=-\left (\frac {b}{x^3}\right )^{2/3} x \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, 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, 30} \[ \int \left (\frac {b}{x^3}\right )^{2/3} \, dx=x \left (-\left (\frac {b}{x^3}\right )^{2/3}\right ) \]

[In]

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

[Out]

-((b/x^3)^(2/3)*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \left (\left (\frac {b}{x^3}\right )^{2/3} x^2\right ) \int \frac {1}{x^2} \, dx \\ & = -\left (\frac {b}{x^3}\right )^{2/3} x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \left (\frac {b}{x^3}\right )^{2/3} \, dx=-\left (\frac {b}{x^3}\right )^{2/3} x \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92

method	result	size
gosper	\(-\left (\frac {b}{x^{3}}\right )^{\frac {2}{3}} x\)	\(11\)
risch	\(-\left (\frac {b}{x^{3}}\right )^{\frac {2}{3}} x\)	\(11\)
trager	\(\left (-1+x \right ) x \left (\frac {b}{x^{3}}\right )^{\frac {2}{3}}\)	\(13\)

[In]

int((b/x^3)^(2/3),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \left (\frac {b}{x^3}\right )^{2/3} \, dx=-x \left (\frac {b}{x^{3}}\right )^{\frac {2}{3}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \left (\frac {b}{x^3}\right )^{2/3} \, dx=- x \left (\frac {b}{x^{3}}\right )^{\frac {2}{3}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \left (\frac {b}{x^3}\right )^{2/3} \, dx=-x \left (\frac {b}{x^{3}}\right )^{\frac {2}{3}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \left (\frac {b}{x^3}\right )^{2/3} \, dx=-x \left (\frac {b}{x^{3}}\right )^{\frac {2}{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.34 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \left (\frac {b}{x^3}\right )^{2/3} \, dx=-x\,{\left (\frac {b}{x^3}\right )}^{2/3} \]

[In]

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

[Out]

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