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

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 = 0.17454, 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 \[ \frac{2}{51} \left (1-x^3\right )^{17/2}-\frac{14}{45} \left (1-x^3\right )^{15/2}+\frac{14}{13} \left (1-x^3\right )^{13/2}-\frac{74}{33} \left (1-x^3\right )^{11/2}+\frac{86}{27} \left (1-x^3\right )^{9/2}-\frac{22}{7} \left (1-x^3\right )^{7/2}+\frac{32}{15} \left (1-x^3\right )^{5/2}-\frac{8}{9} \left (1-x^3\right )^{3/2} \]

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

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Rubi in Sympy [A]  time = 12.639, size = 94, normalized size = 0.78 \[ \frac{2 \left (- x^{3} + 1\right )^{\frac{17}{2}}}{51} - \frac{14 \left (- x^{3} + 1\right )^{\frac{15}{2}}}{45} + \frac{14 \left (- x^{3} + 1\right )^{\frac{13}{2}}}{13} - \frac{74 \left (- x^{3} + 1\right )^{\frac{11}{2}}}{33} + \frac{86 \left (- x^{3} + 1\right )^{\frac{9}{2}}}{27} - \frac{22 \left (- x^{3} + 1\right )^{\frac{7}{2}}}{7} + \frac{32 \left (- x^{3} + 1\right )^{\frac{5}{2}}}{15} - \frac{8 \left (- x^{3} + 1\right )^{\frac{3}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(x**9+1)**2*(-x**3+1)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0361146, size = 52, normalized size = 0.43 \[ -\frac{2 \left (1-x^3\right )^{3/2} \left (45045 x^{21}+42042 x^{18}+38808 x^{15}+174510 x^{12}+155120 x^9+132960 x^6+259521 x^3+173014\right )}{2297295} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.019, size = 58, normalized size = 0.5 \[{\frac{ \left ( 90090\,{x}^{21}+84084\,{x}^{18}+77616\,{x}^{15}+349020\,{x}^{12}+310240\,{x}^{9}+265920\,{x}^{6}+519042\,{x}^{3}+346028 \right ) \left ( -1+x \right ) \left ({x}^{2}+x+1 \right ) }{2297295}\sqrt{-{x}^{3}+1}} \]

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)*(-1+x)*(x^2+x+1)

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Maxima [A]  time = 0.808153, size = 120, normalized size = 0.99 \[ \frac{2}{51} \,{\left (-x^{3} + 1\right )}^{\frac{17}{2}} - \frac{14}{45} \,{\left (-x^{3} + 1\right )}^{\frac{15}{2}} + \frac{14}{13} \,{\left (-x^{3} + 1\right )}^{\frac{13}{2}} - \frac{74}{33} \,{\left (-x^{3} + 1\right )}^{\frac{11}{2}} + \frac{86}{27} \,{\left (-x^{3} + 1\right )}^{\frac{9}{2}} - \frac{22}{7} \,{\left (-x^{3} + 1\right )}^{\frac{7}{2}} + \frac{32}{15} \,{\left (-x^{3} + 1\right )}^{\frac{5}{2}} - \frac{8}{9} \,{\left (-x^{3} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^9 + 1)^2*sqrt(-x^3 + 1)*x^5,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 = 0.294157, size = 72, normalized size = 0.6 \[ \frac{2}{2297295} \,{\left (45045 \, x^{24} - 3003 \, x^{21} - 3234 \, x^{18} + 135702 \, x^{15} - 19390 \, x^{12} - 22160 \, x^{9} + 126561 \, x^{6} - 86507 \, x^{3} - 173014\right )} \sqrt{-x^{3} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^9 + 1)^2*sqrt(-x^3 + 1)*x^5,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 = 54.8838, size = 133, normalized size = 1.1 \[ \frac{2 x^{24} \sqrt{- x^{3} + 1}}{51} - \frac{2 x^{21} \sqrt{- x^{3} + 1}}{765} - \frac{28 x^{18} \sqrt{- x^{3} + 1}}{9945} + \frac{1436 x^{15} \sqrt{- x^{3} + 1}}{12155} - \frac{1108 x^{12} \sqrt{- x^{3} + 1}}{65637} - \frac{8864 x^{9} \sqrt{- x^{3} + 1}}{459459} + \frac{84374 x^{6} \sqrt{- x^{3} + 1}}{765765} - \frac{173014 x^{3} \sqrt{- x^{3} + 1}}{2297295} - \frac{346028 \sqrt{- x^{3} + 1}}{2297295} \]

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(-x**3 + 1)/51 - 2*x**21*sqrt(-x**3 + 1)/765 - 28*x**18*sqrt(-x**3 +
1)/9945 + 1436*x**15*sqrt(-x**3 + 1)/12155 - 1108*x**12*sqrt(-x**3 + 1)/65637 -
8864*x**9*sqrt(-x**3 + 1)/459459 + 84374*x**6*sqrt(-x**3 + 1)/765765 - 173014*x
**3*sqrt(-x**3 + 1)/2297295 - 346028*sqrt(-x**3 + 1)/2297295

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GIAC/XCAS [A]  time = 0.267418, size = 186, normalized size = 1.54 \[ \frac{2}{51} \,{\left (x^{3} - 1\right )}^{8} \sqrt{-x^{3} + 1} + \frac{14}{45} \,{\left (x^{3} - 1\right )}^{7} \sqrt{-x^{3} + 1} + \frac{14}{13} \,{\left (x^{3} - 1\right )}^{6} \sqrt{-x^{3} + 1} + \frac{74}{33} \,{\left (x^{3} - 1\right )}^{5} \sqrt{-x^{3} + 1} + \frac{86}{27} \,{\left (x^{3} - 1\right )}^{4} \sqrt{-x^{3} + 1} + \frac{22}{7} \,{\left (x^{3} - 1\right )}^{3} \sqrt{-x^{3} + 1} + \frac{32}{15} \,{\left (x^{3} - 1\right )}^{2} \sqrt{-x^{3} + 1} - \frac{8}{9} \,{\left (-x^{3} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^9 + 1)^2*sqrt(-x^3 + 1)*x^5,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*sqr
t(-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)