3.3.57 \(\int x^5 (1+x^3)^{2/3} \, dx\)

Optimal. Leaf size=25 \[ \frac {1}{40} \left (x^3+1\right )^{2/3} \left (5 x^6+2 x^3-3\right ) \]

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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}{8} \left (x^3+1\right )^{8/3}-\frac {1}{5} \left (x^3+1\right )^{5/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(1 + x^3)^(2/3),x]

[Out]

-1/5*(1 + x^3)^(5/3) + (1 + x^3)^(8/3)/8

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 \left (1+x^3\right )^{2/3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int x (1+x)^{2/3} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-(1+x)^{2/3}+(1+x)^{5/3}\right ) \, dx,x,x^3\right )\\ &=-\frac {1}{5} \left (1+x^3\right )^{5/3}+\frac {1}{8} \left (1+x^3\right )^{8/3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 0.80 \begin {gather*} \frac {1}{40} \left (x^3+1\right )^{5/3} \left (5 x^3-3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(1 + x^3)^(2/3),x]

[Out]

((1 + x^3)^(5/3)*(-3 + 5*x^3))/40

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IntegrateAlgebraic [A]  time = 0.02, size = 20, normalized size = 0.80 \begin {gather*} \frac {1}{40} \left (1+x^3\right )^{5/3} \left (-3+5 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^5*(1 + x^3)^(2/3),x]

[Out]

((1 + x^3)^(5/3)*(-3 + 5*x^3))/40

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fricas [A]  time = 0.45, size = 21, normalized size = 0.84 \begin {gather*} \frac {1}{40} \, {\left (5 \, x^{6} + 2 \, x^{3} - 3\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(x^3+1)^(2/3),x, algorithm="fricas")

[Out]

1/40*(5*x^6 + 2*x^3 - 3)*(x^3 + 1)^(2/3)

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giac [A]  time = 0.41, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{8} \, {\left (x^{3} + 1\right )}^{\frac {8}{3}} - \frac {1}{5} \, {\left (x^{3} + 1\right )}^{\frac {5}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(x^3+1)^(2/3),x, algorithm="giac")

[Out]

1/8*(x^3 + 1)^(8/3) - 1/5*(x^3 + 1)^(5/3)

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maple [C]  time = 0.06, size = 17, normalized size = 0.68

method	result	size
meijerg	\(\frac {\hypergeom \left (\left [-\frac {2}{3}, 2\right ], \relax [3], -x^{3}\right ) x^{6}}{6}\)	\(17\)
trager	\(\left (\frac {1}{8} x^{6}+\frac {1}{20} x^{3}-\frac {3}{40}\right ) \left (x^{3}+1\right )^{\frac {2}{3}}\)	\(21\)
risch	\(\frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (5 x^{6}+2 x^{3}-3\right )}{40}\)	\(22\)
gosper	\(\frac {\left (1+x \right ) \left (x^{2}-x +1\right ) \left (5 x^{3}-3\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{40}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(x^3+1)^(2/3),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.42, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{8} \, {\left (x^{3} + 1\right )}^{\frac {8}{3}} - \frac {1}{5} \, {\left (x^{3} + 1\right )}^{\frac {5}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(x^3+1)^(2/3),x, algorithm="maxima")

[Out]

1/8*(x^3 + 1)^(8/3) - 1/5*(x^3 + 1)^(5/3)

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mupad [B]  time = 0.19, size = 20, normalized size = 0.80 \begin {gather*} {\left (x^3+1\right )}^{2/3}\,\left (\frac {x^6}{8}+\frac {x^3}{20}-\frac {3}{40}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(x^3 + 1)^(2/3),x)

[Out]

(x^3 + 1)^(2/3)*(x^3/20 + x^6/8 - 3/40)

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sympy [A]  time = 0.63, size = 37, normalized size = 1.48 \begin {gather*} \frac {x^{6} \left (x^{3} + 1\right )^{\frac {2}{3}}}{8} + \frac {x^{3} \left (x^{3} + 1\right )^{\frac {2}{3}}}{20} - \frac {3 \left (x^{3} + 1\right )^{\frac {2}{3}}}{40} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(x**3+1)**(2/3),x)

[Out]

x**6*(x**3 + 1)**(2/3)/8 + x**3*(x**3 + 1)**(2/3)/20 - 3*(x**3 + 1)**(2/3)/40

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