\(\int \frac {x^5}{\sqrt [3]{-1+x^6}} \, dx\) [38]

Optimal result

Integrand size = 13, antiderivative size = 13 \[ \int \frac {x^5}{\sqrt [3]{-1+x^6}} \, dx=\frac {1}{4} \left (-1+x^6\right )^{2/3} \]

[Out]

1/4*(x^6-1)^(2/3)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, 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 = {267} \[ \int \frac {x^5}{\sqrt [3]{-1+x^6}} \, dx=\frac {1}{4} \left (x^6-1\right )^{2/3} \]

[In]

Int[x^5/(-1 + x^6)^(1/3),x]

[Out]

(-1 + x^6)^(2/3)/4

Rule 267

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \left (-1+x^6\right )^{2/3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\sqrt [3]{-1+x^6}} \, dx=\frac {1}{4} \left (-1+x^6\right )^{2/3} \]

[In]

Integrate[x^5/(-1 + x^6)^(1/3),x]

[Out]

(-1 + x^6)^(2/3)/4

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(\frac {\left (x^{6}-1\right )^{\frac {2}{3}}}{4}\)	\(10\)
default	\(\frac {\left (x^{6}-1\right )^{\frac {2}{3}}}{4}\)	\(10\)
trager	\(\frac {\left (x^{6}-1\right )^{\frac {2}{3}}}{4}\)	\(10\)
risch	\(\frac {\left (x^{6}-1\right )^{\frac {2}{3}}}{4}\)	\(10\)
pseudoelliptic	\(\frac {\left (x^{6}-1\right )^{\frac {2}{3}}}{4}\)	\(10\)
gosper	\(\frac {\left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{4 \left (x^{6}-1\right )^{\frac {1}{3}}}\)	\(30\)
meijerg	\(\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{6} \operatorname {hypergeom}\left (\left [\frac {1}{3}, 1\right ], \left [2\right ], x^{6}\right )}{6 \operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\)	\(33\)

[In]

int(x^5/(x^6-1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/4*(x^6-1)^(2/3)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x^5}{\sqrt [3]{-1+x^6}} \, dx=\frac {1}{4} \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} \]

[In]

integrate(x^5/(x^6-1)^(1/3),x, algorithm="fricas")

[Out]

1/4*(x^6 - 1)^(2/3)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int \frac {x^5}{\sqrt [3]{-1+x^6}} \, dx=\frac {\left (x^{6} - 1\right )^{\frac {2}{3}}}{4} \]

[In]

integrate(x**5/(x**6-1)**(1/3),x)

[Out]

(x**6 - 1)**(2/3)/4

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x^5}{\sqrt [3]{-1+x^6}} \, dx=\frac {1}{4} \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} \]

[In]

integrate(x^5/(x^6-1)^(1/3),x, algorithm="maxima")

[Out]

1/4*(x^6 - 1)^(2/3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x^5}{\sqrt [3]{-1+x^6}} \, dx=\frac {1}{4} \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} \]

[In]

integrate(x^5/(x^6-1)^(1/3),x, algorithm="giac")

[Out]

1/4*(x^6 - 1)^(2/3)

Mupad [B] (verification not implemented)

Time = 5.23 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {x^5}{\sqrt [3]{-1+x^6}} \, dx=\frac {{\left (x^6-1\right )}^{2/3}}{4} \]

[In]

int(x^5/(x^6 - 1)^(1/3),x)

[Out]

(x^6 - 1)^(2/3)/4