3.6.43 \(\int x^8 (1-x^3)^{6/5} \, dx\) [543]

Optimal. Leaf size=46 \[ -\frac {5}{33} \left (1-x^3\right )^{11/5}+\frac {5}{24} \left (1-x^3\right )^{16/5}-\frac {5}{63} \left (1-x^3\right )^{21/5} \]

[Out]

-5/33*(-x^3+1)^(11/5)+5/24*(-x^3+1)^(16/5)-5/63*(-x^3+1)^(21/5)

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Rubi [A]
time = 0.01, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \begin {gather*} -\frac {5}{63} \left (1-x^3\right )^{21/5}+\frac {5}{24} \left (1-x^3\right )^{16/5}-\frac {5}{33} \left (1-x^3\right )^{11/5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8*(1 - x^3)^(6/5),x]

[Out]

(-5*(1 - x^3)^(11/5))/33 + (5*(1 - x^3)^(16/5))/24 - (5*(1 - x^3)^(21/5))/63

Rule 45

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 272

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^8 \left (1-x^3\right )^{6/5} \, dx &=\frac {1}{3} \text {Subst}\left (\int (1-x)^{6/5} x^2 \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left ((1-x)^{6/5}-2 (1-x)^{11/5}+(1-x)^{16/5}\right ) \, dx,x,x^3\right )\\ &=-\frac {5}{33} \left (1-x^3\right )^{11/5}+\frac {5}{24} \left (1-x^3\right )^{16/5}-\frac {5}{63} \left (1-x^3\right )^{21/5}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 27, normalized size = 0.59 \begin {gather*} -\frac {5 \left (1-x^3\right )^{11/5} \left (25+55 x^3+88 x^6\right )}{5544} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^8*(1 - x^3)^(6/5),x]

[Out]

(-5*(1 - x^3)^(11/5)*(25 + 55*x^3 + 88*x^6))/5544

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 2.
time = 0.15, size = 15, normalized size = 0.33

method	result	size
meijerg	\(\frac {x^{9} \hypergeom \left (\left [-\frac {6}{5}, 3\right ], \left [4\right ], x^{3}\right )}{9}\)	\(15\)
gosper	\(\frac {5 \left (x -1\right ) \left (x^{2}+x +1\right ) \left (88 x^{6}+55 x^{3}+25\right ) \left (-x^{3}+1\right )^{\frac {6}{5}}}{5544}\)	\(33\)
trager	\(\left (-\frac {5}{63} x^{12}+\frac {55}{504} x^{9}-\frac {5}{1848} x^{6}-\frac {25}{5544} x^{3}-\frac {125}{5544}\right ) \left (-x^{3}+1\right )^{\frac {1}{5}}\)	\(33\)
risch	\(\frac {5 \left (88 x^{12}-121 x^{9}+3 x^{6}+5 x^{3}+25\right ) \left (x^{3}-1\right )}{5544 \left (-x^{3}+1\right )^{\frac {4}{5}}}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(-x^3+1)^(6/5),x,method=_RETURNVERBOSE)

[Out]

1/9*x^9*hypergeom([-6/5,3],[4],x^3)

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Maxima [A]
time = 0.29, size = 34, normalized size = 0.74 \begin {gather*} -\frac {5}{63} \, {\left (-x^{3} + 1\right )}^{\frac {21}{5}} + \frac {5}{24} \, {\left (-x^{3} + 1\right )}^{\frac {16}{5}} - \frac {5}{33} \, {\left (-x^{3} + 1\right )}^{\frac {11}{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(-x^3+1)^(6/5),x, algorithm="maxima")

[Out]

-5/63*(-x^3 + 1)^(21/5) + 5/24*(-x^3 + 1)^(16/5) - 5/33*(-x^3 + 1)^(11/5)

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Fricas [A]
time = 0.37, size = 33, normalized size = 0.72 \begin {gather*} -\frac {5}{5544} \, {\left (88 \, x^{12} - 121 \, x^{9} + 3 \, x^{6} + 5 \, x^{3} + 25\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(-x^3+1)^(6/5),x, algorithm="fricas")

[Out]

-5/5544*(88*x^12 - 121*x^9 + 3*x^6 + 5*x^3 + 25)*(-x^3 + 1)^(1/5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (34) = 68\).
time = 0.86, size = 71, normalized size = 1.54 \begin {gather*} - \frac {5 x^{12} \sqrt [5]{1 - x^{3}}}{63} + \frac {55 x^{9} \sqrt [5]{1 - x^{3}}}{504} - \frac {5 x^{6} \sqrt [5]{1 - x^{3}}}{1848} - \frac {25 x^{3} \sqrt [5]{1 - x^{3}}}{5544} - \frac {125 \sqrt [5]{1 - x^{3}}}{5544} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(-x**3+1)**(6/5),x)

[Out]

-5*x**12*(1 - x**3)**(1/5)/63 + 55*x**9*(1 - x**3)**(1/5)/504 - 5*x**6*(1 - x**3)**(1/5)/1848 - 25*x**3*(1 - x
**3)**(1/5)/5544 - 125*(1 - x**3)**(1/5)/5544

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Giac [A]
time = 1.95, size = 55, normalized size = 1.20 \begin {gather*} -\frac {5}{63} \, {\left (x^{3} - 1\right )}^{4} {\left (-x^{3} + 1\right )}^{\frac {1}{5}} - \frac {5}{24} \, {\left (x^{3} - 1\right )}^{3} {\left (-x^{3} + 1\right )}^{\frac {1}{5}} - \frac {5}{33} \, {\left (x^{3} - 1\right )}^{2} {\left (-x^{3} + 1\right )}^{\frac {1}{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(-x^3+1)^(6/5),x, algorithm="giac")

[Out]

-5/63*(x^3 - 1)^4*(-x^3 + 1)^(1/5) - 5/24*(x^3 - 1)^3*(-x^3 + 1)^(1/5) - 5/33*(x^3 - 1)^2*(-x^3 + 1)^(1/5)

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Mupad [B]
time = 1.08, size = 33, normalized size = 0.72 \begin {gather*} -{\left (1-x^3\right )}^{1/5}\,\left (\frac {5\,x^{12}}{63}-\frac {55\,x^9}{504}+\frac {5\,x^6}{1848}+\frac {25\,x^3}{5544}+\frac {125}{5544}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(1 - x^3)^(6/5),x)

[Out]

-(1 - x^3)^(1/5)*((25*x^3)/5544 + (5*x^6)/1848 - (55*x^9)/504 + (5*x^12)/63 + 125/5544)

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