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

Optimal. Leaf size=13 \[ \frac {1}{8} \left (x^6-1\right )^{4/3} \]

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {261} \begin {gather*} \frac {1}{8} \left (x^6-1\right )^{4/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(-1 + x^6)^(1/3),x]

[Out]

(-1 + x^6)^(4/3)/8

Rule 261

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*} \int x^5 \sqrt [3]{-1+x^6} \, dx &=\frac {1}{8} \left (-1+x^6\right )^{4/3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {1}{8} \left (x^6-1\right )^{4/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(-1 + x^6)^(1/3),x]

[Out]

(-1 + x^6)^(4/3)/8

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IntegrateAlgebraic [A]  time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {1}{8} \left (-1+x^6\right )^{4/3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^5*(-1 + x^6)^(1/3),x]

[Out]

(-1 + x^6)^(4/3)/8

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fricas [A]  time = 0.46, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{8} \, {\left (x^{6} - 1\right )}^{\frac {4}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(x^6 - 1)^(4/3)

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giac [A]  time = 0.32, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{8} \, {\left (x^{6} - 1\right )}^{\frac {4}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(x^6 - 1)^(4/3)

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maple [A]  time = 0.07, size = 10, normalized size = 0.77

method	result	size
derivativedivides	\(\frac {\left (x^{6}-1\right )^{\frac {4}{3}}}{8}\)	\(10\)
default	\(\frac {\left (x^{6}-1\right )^{\frac {4}{3}}}{8}\)	\(10\)
risch	\(\frac {\left (x^{6}-1\right )^{\frac {4}{3}}}{8}\)	\(10\)
trager	\(\left (\frac {x^{6}}{8}-\frac {1}{8}\right ) \left (x^{6}-1\right )^{\frac {1}{3}}\)	\(16\)
gosper	\(\frac {\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}}{8}\)	\(30\)
meijerg	\(\frac {\mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [-\frac {1}{3}, 1\right ], \relax [2], x^{6}\right ) x^{6}}{6 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(x^6-1)^(4/3)

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maxima [A]  time = 0.34, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{8} \, {\left (x^{6} - 1\right )}^{\frac {4}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(x^6 - 1)^(4/3)

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mupad [B]  time = 0.14, size = 9, normalized size = 0.69 \begin {gather*} \frac {{\left (x^6-1\right )}^{4/3}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(x^6 - 1)^(1/3),x)

[Out]

(x^6 - 1)^(4/3)/8

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sympy [B]  time = 0.35, size = 22, normalized size = 1.69 \begin {gather*} \frac {x^{6} \sqrt [3]{x^{6} - 1}}{8} - \frac {\sqrt [3]{x^{6} - 1}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**6*(x**6 - 1)**(1/3)/8 - (x**6 - 1)**(1/3)/8

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