3.526 \(\int x^{11} \left (a+b x^3\right )^{2/3} \, dx\)

Optimal. Leaf size=80 \[ -\frac{a^3 \left (a+b x^3\right )^{5/3}}{5 b^4}+\frac{3 a^2 \left (a+b x^3\right )^{8/3}}{8 b^4}+\frac{\left (a+b x^3\right )^{14/3}}{14 b^4}-\frac{3 a \left (a+b x^3\right )^{11/3}}{11 b^4} \]

[Out]

-(a^3*(a + b*x^3)^(5/3))/(5*b^4) + (3*a^2*(a + b*x^3)^(8/3))/(8*b^4) - (3*a*(a +
 b*x^3)^(11/3))/(11*b^4) + (a + b*x^3)^(14/3)/(14*b^4)

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Rubi [A]  time = 0.105074, antiderivative size = 80, 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 \[ -\frac{a^3 \left (a+b x^3\right )^{5/3}}{5 b^4}+\frac{3 a^2 \left (a+b x^3\right )^{8/3}}{8 b^4}+\frac{\left (a+b x^3\right )^{14/3}}{14 b^4}-\frac{3 a \left (a+b x^3\right )^{11/3}}{11 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^11*(a + b*x^3)^(2/3),x]

[Out]

-(a^3*(a + b*x^3)^(5/3))/(5*b^4) + (3*a^2*(a + b*x^3)^(8/3))/(8*b^4) - (3*a*(a +
 b*x^3)^(11/3))/(11*b^4) + (a + b*x^3)^(14/3)/(14*b^4)

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Rubi in Sympy [A]  time = 14.5799, size = 71, normalized size = 0.89 \[ - \frac{a^{3} \left (a + b x^{3}\right )^{\frac{5}{3}}}{5 b^{4}} + \frac{3 a^{2} \left (a + b x^{3}\right )^{\frac{8}{3}}}{8 b^{4}} - \frac{3 a \left (a + b x^{3}\right )^{\frac{11}{3}}}{11 b^{4}} + \frac{\left (a + b x^{3}\right )^{\frac{14}{3}}}{14 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11*(b*x**3+a)**(2/3),x)

[Out]

-a**3*(a + b*x**3)**(5/3)/(5*b**4) + 3*a**2*(a + b*x**3)**(8/3)/(8*b**4) - 3*a*(
a + b*x**3)**(11/3)/(11*b**4) + (a + b*x**3)**(14/3)/(14*b**4)

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Mathematica [A]  time = 0.0304697, size = 61, normalized size = 0.76 \[ \frac{\left (a+b x^3\right )^{2/3} \left (-81 a^4+54 a^3 b x^3-45 a^2 b^2 x^6+40 a b^3 x^9+220 b^4 x^{12}\right )}{3080 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11*(a + b*x^3)^(2/3),x]

[Out]

((a + b*x^3)^(2/3)*(-81*a^4 + 54*a^3*b*x^3 - 45*a^2*b^2*x^6 + 40*a*b^3*x^9 + 220
*b^4*x^12))/(3080*b^4)

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Maple [A]  time = 0.008, size = 47, normalized size = 0.6 \[ -{\frac{-220\,{b}^{3}{x}^{9}+180\,a{b}^{2}{x}^{6}-135\,{a}^{2}b{x}^{3}+81\,{a}^{3}}{3080\,{b}^{4}} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11*(b*x^3+a)^(2/3),x)

[Out]

-1/3080*(b*x^3+a)^(5/3)*(-220*b^3*x^9+180*a*b^2*x^6-135*a^2*b*x^3+81*a^3)/b^4

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Maxima [A]  time = 1.44635, size = 86, normalized size = 1.08 \[ \frac{{\left (b x^{3} + a\right )}^{\frac{14}{3}}}{14 \, b^{4}} - \frac{3 \,{\left (b x^{3} + a\right )}^{\frac{11}{3}} a}{11 \, b^{4}} + \frac{3 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} a^{2}}{8 \, b^{4}} - \frac{{\left (b x^{3} + a\right )}^{\frac{5}{3}} a^{3}}{5 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(2/3)*x^11,x, algorithm="maxima")

[Out]

1/14*(b*x^3 + a)^(14/3)/b^4 - 3/11*(b*x^3 + a)^(11/3)*a/b^4 + 3/8*(b*x^3 + a)^(8
/3)*a^2/b^4 - 1/5*(b*x^3 + a)^(5/3)*a^3/b^4

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Fricas [A]  time = 0.263038, size = 77, normalized size = 0.96 \[ \frac{{\left (220 \, b^{4} x^{12} + 40 \, a b^{3} x^{9} - 45 \, a^{2} b^{2} x^{6} + 54 \, a^{3} b x^{3} - 81 \, a^{4}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{3080 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(2/3)*x^11,x, algorithm="fricas")

[Out]

1/3080*(220*b^4*x^12 + 40*a*b^3*x^9 - 45*a^2*b^2*x^6 + 54*a^3*b*x^3 - 81*a^4)*(b
*x^3 + a)^(2/3)/b^4

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Sympy [A]  time = 20.9894, size = 110, normalized size = 1.38 \[ \begin{cases} - \frac{81 a^{4} \left (a + b x^{3}\right )^{\frac{2}{3}}}{3080 b^{4}} + \frac{27 a^{3} x^{3} \left (a + b x^{3}\right )^{\frac{2}{3}}}{1540 b^{3}} - \frac{9 a^{2} x^{6} \left (a + b x^{3}\right )^{\frac{2}{3}}}{616 b^{2}} + \frac{a x^{9} \left (a + b x^{3}\right )^{\frac{2}{3}}}{77 b} + \frac{x^{12} \left (a + b x^{3}\right )^{\frac{2}{3}}}{14} & \text{for}\: b \neq 0 \\\frac{a^{\frac{2}{3}} x^{12}}{12} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11*(b*x**3+a)**(2/3),x)

[Out]

Piecewise((-81*a**4*(a + b*x**3)**(2/3)/(3080*b**4) + 27*a**3*x**3*(a + b*x**3)*
*(2/3)/(1540*b**3) - 9*a**2*x**6*(a + b*x**3)**(2/3)/(616*b**2) + a*x**9*(a + b*
x**3)**(2/3)/(77*b) + x**12*(a + b*x**3)**(2/3)/14, Ne(b, 0)), (a**(2/3)*x**12/1
2, True))

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GIAC/XCAS [A]  time = 0.229222, size = 77, normalized size = 0.96 \[ \frac{220 \,{\left (b x^{3} + a\right )}^{\frac{14}{3}} - 840 \,{\left (b x^{3} + a\right )}^{\frac{11}{3}} a + 1155 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} a^{2} - 616 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} a^{3}}{3080 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(2/3)*x^11,x, algorithm="giac")

[Out]

1/3080*(220*(b*x^3 + a)^(14/3) - 840*(b*x^3 + a)^(11/3)*a + 1155*(b*x^3 + a)^(8/
3)*a^2 - 616*(b*x^3 + a)^(5/3)*a^3)/b^4