3.686 \(\int x^5 \sqrt{1-x^3} (1+x^9)^2 \, dx\)

Optimal. Leaf size=121 \[ \frac{2}{51} \left (1-x^3\right )^{17/2}-\frac{14}{45} \left (1-x^3\right )^{15/2}+\frac{14}{13} \left (1-x^3\right )^{13/2}-\frac{74}{33} \left (1-x^3\right )^{11/2}+\frac{86}{27} \left (1-x^3\right )^{9/2}-\frac{22}{7} \left (1-x^3\right )^{7/2}+\frac{32}{15} \left (1-x^3\right )^{5/2}-\frac{8}{9} \left (1-x^3\right )^{3/2} \]

[Out]

(-8*(1 - x^3)^(3/2))/9 + (32*(1 - x^3)^(5/2))/15 - (22*(1 - x^3)^(7/2))/7 + (86*(1 - x^3)^(9/2))/27 - (74*(1 -
x^3)^(11/2))/33 + (14*(1 - x^3)^(13/2))/13 - (14*(1 - x^3)^(15/2))/45 + (2*(1 - x^3)^(17/2))/51

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Rubi [A]  time = 0.111314, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1821, 1620} \[ \frac{2}{51} \left (1-x^3\right )^{17/2}-\frac{14}{45} \left (1-x^3\right )^{15/2}+\frac{14}{13} \left (1-x^3\right )^{13/2}-\frac{74}{33} \left (1-x^3\right )^{11/2}+\frac{86}{27} \left (1-x^3\right )^{9/2}-\frac{22}{7} \left (1-x^3\right )^{7/2}+\frac{32}{15} \left (1-x^3\right )^{5/2}-\frac{8}{9} \left (1-x^3\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x^5*Sqrt[1 - x^3]*(1 + x^9)^2,x]

[Out]

(-8*(1 - x^3)^(3/2))/9 + (32*(1 - x^3)^(5/2))/15 - (22*(1 - x^3)^(7/2))/7 + (86*(1 - x^3)^(9/2))/27 - (74*(1 -
x^3)^(11/2))/33 + (14*(1 - x^3)^(13/2))/13 - (14*(1 - x^3)^(15/2))/45 + (2*(1 - x^3)^(17/2))/51

Rule 1821

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

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int x^5 \sqrt{1-x^3} \left (1+x^9\right )^2 \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \sqrt{1-x} x \left (1+x^3\right )^2 \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (4 \sqrt{1-x}-16 (1-x)^{3/2}+33 (1-x)^{5/2}-43 (1-x)^{7/2}+37 (1-x)^{9/2}-21 (1-x)^{11/2}+7 (1-x)^{13/2}-(1-x)^{15/2}\right ) \, dx,x,x^3\right )\\ &=-\frac{8}{9} \left (1-x^3\right )^{3/2}+\frac{32}{15} \left (1-x^3\right )^{5/2}-\frac{22}{7} \left (1-x^3\right )^{7/2}+\frac{86}{27} \left (1-x^3\right )^{9/2}-\frac{74}{33} \left (1-x^3\right )^{11/2}+\frac{14}{13} \left (1-x^3\right )^{13/2}-\frac{14}{45} \left (1-x^3\right )^{15/2}+\frac{2}{51} \left (1-x^3\right )^{17/2}\\ \end{align*}

Mathematica [A]  time = 0.0427483, size = 57, normalized size = 0.47 \[ \frac{2 \sqrt{1-x^3} \left (45045 x^{24}-3003 x^{21}-3234 x^{18}+135702 x^{15}-19390 x^{12}-22160 x^9+126561 x^6-86507 x^3-173014\right )}{2297295} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*Sqrt[1 - x^3]*(1 + x^9)^2,x]

[Out]

(2*Sqrt[1 - x^3]*(-173014 - 86507*x^3 + 126561*x^6 - 22160*x^9 - 19390*x^12 + 135702*x^15 - 3234*x^18 - 3003*x
^21 + 45045*x^24))/2297295

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Maple [A]  time = 0.012, size = 58, normalized size = 0.5 \begin{align*}{\frac{ \left ( 90090\,{x}^{21}+84084\,{x}^{18}+77616\,{x}^{15}+349020\,{x}^{12}+310240\,{x}^{9}+265920\,{x}^{6}+519042\,{x}^{3}+346028 \right ) \left ( x-1 \right ) \left ({x}^{2}+x+1 \right ) }{2297295}\sqrt{-{x}^{3}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(x^9+1)^2*(-x^3+1)^(1/2),x)

[Out]

2/2297295*(-x^3+1)^(1/2)*(45045*x^21+42042*x^18+38808*x^15+174510*x^12+155120*x^9+132960*x^6+259521*x^3+173014
)*(x-1)*(x^2+x+1)

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Maxima [A]  time = 1.85608, size = 120, normalized size = 0.99 \begin{align*} \frac{2}{51} \,{\left (-x^{3} + 1\right )}^{\frac{17}{2}} - \frac{14}{45} \,{\left (-x^{3} + 1\right )}^{\frac{15}{2}} + \frac{14}{13} \,{\left (-x^{3} + 1\right )}^{\frac{13}{2}} - \frac{74}{33} \,{\left (-x^{3} + 1\right )}^{\frac{11}{2}} + \frac{86}{27} \,{\left (-x^{3} + 1\right )}^{\frac{9}{2}} - \frac{22}{7} \,{\left (-x^{3} + 1\right )}^{\frac{7}{2}} + \frac{32}{15} \,{\left (-x^{3} + 1\right )}^{\frac{5}{2}} - \frac{8}{9} \,{\left (-x^{3} + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(x^9+1)^2*(-x^3+1)^(1/2),x, algorithm="maxima")

[Out]

2/51*(-x^3 + 1)^(17/2) - 14/45*(-x^3 + 1)^(15/2) + 14/13*(-x^3 + 1)^(13/2) - 74/33*(-x^3 + 1)^(11/2) + 86/27*(
-x^3 + 1)^(9/2) - 22/7*(-x^3 + 1)^(7/2) + 32/15*(-x^3 + 1)^(5/2) - 8/9*(-x^3 + 1)^(3/2)

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Fricas [A]  time = 1.8591, size = 184, normalized size = 1.52 \begin{align*} \frac{2}{2297295} \,{\left (45045 \, x^{24} - 3003 \, x^{21} - 3234 \, x^{18} + 135702 \, x^{15} - 19390 \, x^{12} - 22160 \, x^{9} + 126561 \, x^{6} - 86507 \, x^{3} - 173014\right )} \sqrt{-x^{3} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(x^9+1)^2*(-x^3+1)^(1/2),x, algorithm="fricas")

[Out]

2/2297295*(45045*x^24 - 3003*x^21 - 3234*x^18 + 135702*x^15 - 19390*x^12 - 22160*x^9 + 126561*x^6 - 86507*x^3
- 173014)*sqrt(-x^3 + 1)

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Sympy [A]  time = 14.7357, size = 133, normalized size = 1.1 \begin{align*} \frac{2 x^{24} \sqrt{1 - x^{3}}}{51} - \frac{2 x^{21} \sqrt{1 - x^{3}}}{765} - \frac{28 x^{18} \sqrt{1 - x^{3}}}{9945} + \frac{1436 x^{15} \sqrt{1 - x^{3}}}{12155} - \frac{1108 x^{12} \sqrt{1 - x^{3}}}{65637} - \frac{8864 x^{9} \sqrt{1 - x^{3}}}{459459} + \frac{84374 x^{6} \sqrt{1 - x^{3}}}{765765} - \frac{173014 x^{3} \sqrt{1 - x^{3}}}{2297295} - \frac{346028 \sqrt{1 - x^{3}}}{2297295} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(x**9+1)**2*(-x**3+1)**(1/2),x)

[Out]

2*x**24*sqrt(1 - x**3)/51 - 2*x**21*sqrt(1 - x**3)/765 - 28*x**18*sqrt(1 - x**3)/9945 + 1436*x**15*sqrt(1 - x*
*3)/12155 - 1108*x**12*sqrt(1 - x**3)/65637 - 8864*x**9*sqrt(1 - x**3)/459459 + 84374*x**6*sqrt(1 - x**3)/7657
65 - 173014*x**3*sqrt(1 - x**3)/2297295 - 346028*sqrt(1 - x**3)/2297295

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Giac [A]  time = 1.14174, size = 186, normalized size = 1.54 \begin{align*} \frac{2}{51} \,{\left (x^{3} - 1\right )}^{8} \sqrt{-x^{3} + 1} + \frac{14}{45} \,{\left (x^{3} - 1\right )}^{7} \sqrt{-x^{3} + 1} + \frac{14}{13} \,{\left (x^{3} - 1\right )}^{6} \sqrt{-x^{3} + 1} + \frac{74}{33} \,{\left (x^{3} - 1\right )}^{5} \sqrt{-x^{3} + 1} + \frac{86}{27} \,{\left (x^{3} - 1\right )}^{4} \sqrt{-x^{3} + 1} + \frac{22}{7} \,{\left (x^{3} - 1\right )}^{3} \sqrt{-x^{3} + 1} + \frac{32}{15} \,{\left (x^{3} - 1\right )}^{2} \sqrt{-x^{3} + 1} - \frac{8}{9} \,{\left (-x^{3} + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(x^9+1)^2*(-x^3+1)^(1/2),x, algorithm="giac")

[Out]

2/51*(x^3 - 1)^8*sqrt(-x^3 + 1) + 14/45*(x^3 - 1)^7*sqrt(-x^3 + 1) + 14/13*(x^3 - 1)^6*sqrt(-x^3 + 1) + 74/33*
(x^3 - 1)^5*sqrt(-x^3 + 1) + 86/27*(x^3 - 1)^4*sqrt(-x^3 + 1) + 22/7*(x^3 - 1)^3*sqrt(-x^3 + 1) + 32/15*(x^3 -
1)^2*sqrt(-x^3 + 1) - 8/9*(-x^3 + 1)^(3/2)