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

Optimal. Leaf size=23 \[ \frac {1}{28} \sqrt [3]{x^3+1} \left (4 x^6+x^3-3\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.17, 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]

-1/4*(1 + x^3)^(4/3) + (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 x \sqrt [3]{1+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 = 0.87 \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 = 0.87 \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 = 19, normalized size = 0.83 \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 = 0.62, size = 19, normalized size = 0.83 \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 [C]  time = 0.06, size = 17, normalized size = 0.74

method	result	size
meijerg	\(\frac {\hypergeom \left (\left [-\frac {1}{3}, 2\right ], \relax [3], -x^{3}\right ) x^{6}}{6}\)	\(17\)
risch	\(\frac {\left (x^{3}+1\right )^{\frac {1}{3}} \left (4 x^{6}+x^{3}-3\right )}{28}\)	\(20\)
trager	\(\left (\frac {1}{7} x^{6}+\frac {1}{28} x^{3}-\frac {3}{28}\right ) \left (x^{3}+1\right )^{\frac {1}{3}}\)	\(21\)
gosper	\(\frac {\left (1+x \right ) \left (x^{2}-x +1\right ) \left (4 x^{3}-3\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{28}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*hypergeom([-1/3,2],[3],-x^3)*x^6

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maxima [A]  time = 0.37, size = 19, normalized size = 0.83 \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 = 0.19, size = 20, normalized size = 0.87 \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 = 0.36, size = 37, normalized size = 1.61 \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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