\(\int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx\) [70]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx=\frac {\left (-1+x^3\right )^{4/3}}{4 x^4} \]

[Out]

1/4*(x^3-1)^(4/3)/x^4

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx=\frac {\left (x^3-1\right )^{4/3}}{4 x^4} \]

[In]

Int[(-1 + x^3)^(1/3)/x^5,x]

[Out]

(-1 + x^3)^(4/3)/(4*x^4)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (-1+x^3\right )^{4/3}}{4 x^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx=\frac {\left (-1+x^3\right )^{4/3}}{4 x^4} \]

[In]

Integrate[(-1 + x^3)^(1/3)/x^5,x]

[Out]

(-1 + x^3)^(4/3)/(4*x^4)

Maple [A] (verified)

Time = 0.91 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
trager	\(\frac {\left (x^{3}-1\right )^{\frac {4}{3}}}{4 x^{4}}\)	\(13\)
pseudoelliptic	\(\frac {\left (x^{3}-1\right )^{\frac {4}{3}}}{4 x^{4}}\)	\(13\)
gosper	\(\frac {\left (x -1\right ) \left (x^{2}+x +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{4 x^{4}}\)	\(22\)
risch	\(\frac {x^{6}-2 x^{3}+1}{4 \left (x^{3}-1\right )^{\frac {2}{3}} x^{4}}\)	\(23\)
meijerg	\(-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {4}{3}}}{4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x^{4}}\)	\(33\)

[In]

int((x^3-1)^(1/3)/x^5,x,method=_RETURNVERBOSE)

[Out]

1/4*(x^3-1)^(4/3)/x^4

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx=\frac {{\left (x^{3} - 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} \]

[In]

integrate((x^3-1)^(1/3)/x^5,x, algorithm="fricas")

[Out]

1/4*(x^3 - 1)^(4/3)/x^4

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.48 (sec) , antiderivative size = 129, normalized size of antiderivative = 8.06 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx=\begin {cases} \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} - \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases} \]

[In]

integrate((x**3-1)**(1/3)/x**5,x)

[Out]

Piecewise(((-1 + x**(-3))**(1/3)*exp(-2*I*pi/3)*gamma(-4/3)/(3*gamma(-1/3)) - (-1 + x**(-3))**(1/3)*exp(-2*I*p
i/3)*gamma(-4/3)/(3*x**3*gamma(-1/3)), 1/Abs(x**3) > 1), (-(1 - 1/x**3)**(1/3)*gamma(-4/3)/(3*gamma(-1/3)) + (
1 - 1/x**3)**(1/3)*gamma(-4/3)/(3*x**3*gamma(-1/3)), True))

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx=\frac {{\left (x^{3} - 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} \]

[In]

integrate((x^3-1)^(1/3)/x^5,x, algorithm="maxima")

[Out]

1/4*(x^3 - 1)^(4/3)/x^4

Giac [F]

\[ \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{5}} \,d x } \]

[In]

integrate((x^3-1)^(1/3)/x^5,x, algorithm="giac")

[Out]

integrate((x^3 - 1)^(1/3)/x^5, x)

Mupad [B] (verification not implemented)

Time = 5.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx=-\frac {{\left (x^3-1\right )}^{1/3}-x^3\,{\left (x^3-1\right )}^{1/3}}{4\,x^4} \]

[In]

int((x^3 - 1)^(1/3)/x^5,x)

[Out]

-((x^3 - 1)^(1/3) - x^3*(x^3 - 1)^(1/3))/(4*x^4)