3.543 \(\int x^8 (1-x^3)^{6/5} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.019694, antiderivative size = 46, normalized size of antiderivative = 1., 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 = {266, 43} \[ -\frac{5}{63} \left (1-x^3\right )^{21/5}+\frac{5}{24} \left (1-x^3\right )^{16/5}-\frac{5}{33} \left (1-x^3\right )^{11/5} \]

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 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]]

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])

Rubi steps

\begin{align*} \int x^8 \left (1-x^3\right )^{6/5} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int (1-x)^{6/5} x^2 \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{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*}

Mathematica [A]  time = 0.0134046, size = 27, normalized size = 0.59 \[ -\frac{5 \left (1-x^3\right )^{11/5} \left (88 x^6+55 x^3+25\right )}{5544} \]

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 [A]  time = 0.004, size = 33, normalized size = 0.7 \begin{align*}{\frac{ \left ( -5+5\,x \right ) \left ({x}^{2}+x+1 \right ) \left ( 88\,{x}^{6}+55\,{x}^{3}+25 \right ) }{5544} \left ( -{x}^{3}+1 \right ) ^{{\frac{6}{5}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/5544*(-1+x)*(x^2+x+1)*(88*x^6+55*x^3+25)*(-x^3+1)^(6/5)

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Maxima [A]  time = 1.02281, size = 46, normalized size = 1. \begin{align*} -\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{align*}

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 = 1.73452, size = 90, normalized size = 1.96 \begin{align*} -\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{align*}

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]  time = 12.2684, size = 71, normalized size = 1.54 \begin{align*} - \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{align*}

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 = 1.14739, size = 74, normalized size = 1.61 \begin{align*} -\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{align*}

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)