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

Optimal. Leaf size=25 \[ \frac {1}{28} \sqrt [3]{x^3-1} \left (4 x^6-x^3-3\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \begin {gather*} \frac {1}{7} \left (x^3-1\right )^{7/3}+\frac {1}{4} \left (x^3-1\right )^{4/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(-1 + x^3)^(1/3),x]

[Out]

(-1 + x^3)^(4/3)/4 + (-1 + x^3)^(7/3)/7

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

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

Rubi steps

\begin {align*} \int x^5 \sqrt [3]{-1+x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \sqrt [3]{-1+x} x \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\sqrt [3]{-1+x}+(-1+x)^{4/3}\right ) \, dx,x,x^3\right )\\ &=\frac {1}{4} \left (-1+x^3\right )^{4/3}+\frac {1}{7} \left (-1+x^3\right )^{7/3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 0.80 \begin {gather*} \frac {1}{28} \left (x^3-1\right )^{4/3} \left (4 x^3+3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(-1 + x^3)^(1/3),x]

[Out]

((-1 + x^3)^(4/3)*(3 + 4*x^3))/28

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IntegrateAlgebraic [A]  time = 0.01, size = 20, normalized size = 0.80 \begin {gather*} \frac {1}{28} \left (-1+x^3\right )^{4/3} \left (3+4 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + x^3)^(4/3)*(3 + 4*x^3))/28

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fricas [A]  time = 0.44, size = 21, normalized size = 0.84 \begin {gather*} \frac {1}{28} \, {\left (4 \, x^{6} - x^{3} - 3\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/28*(4*x^6 - x^3 - 3)*(x^3 - 1)^(1/3)

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giac [A]  time = 0.28, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{7} \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + \frac {1}{4} \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*(x^3 - 1)^(7/3) + 1/4*(x^3 - 1)^(4/3)

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maple [A]  time = 0.08, size = 21, normalized size = 0.84

method	result	size
trager	\(\left (\frac {1}{7} x^{6}-\frac {1}{28} x^{3}-\frac {3}{28}\right ) \left (x^{3}-1\right )^{\frac {1}{3}}\)	\(21\)
risch	\(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (4 x^{6}-x^{3}-3\right )}{28}\)	\(22\)
gosper	\(\frac {\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (4 x^{3}+3\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{28}\)	\(26\)
meijerg	\(\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [-\frac {1}{3}, 2\right ], \relax [3], x^{3}\right ) x^{6}}{6 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/7*x^6-1/28*x^3-3/28)*(x^3-1)^(1/3)

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maxima [A]  time = 0.57, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{7} \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + \frac {1}{4} \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*(x^3 - 1)^(7/3) + 1/4*(x^3 - 1)^(4/3)

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mupad [B]  time = 0.20, size = 21, normalized size = 0.84 \begin {gather*} -{\left (x^3-1\right )}^{1/3}\,\left (-\frac {x^6}{7}+\frac {x^3}{28}+\frac {3}{28}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(x^3 - 1)^(1/3),x)

[Out]

-(x^3 - 1)^(1/3)*(x^3/28 - x^6/7 + 3/28)

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sympy [A]  time = 0.41, size = 37, normalized size = 1.48 \begin {gather*} \frac {x^{6} \sqrt [3]{x^{3} - 1}}{7} - \frac {x^{3} \sqrt [3]{x^{3} - 1}}{28} - \frac {3 \sqrt [3]{x^{3} - 1}}{28} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**6*(x**3 - 1)**(1/3)/7 - x**3*(x**3 - 1)**(1/3)/28 - 3*(x**3 - 1)**(1/3)/28

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