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

Optimal. Leaf size=43 \[ \frac{a^3 x^{12}}{12}+\frac{1}{5} a^2 b x^{15}+\frac{1}{6} a b^2 x^{18}+\frac{b^3 x^{21}}{21} \]

[Out]

(a^3*x^12)/12 + (a^2*b*x^15)/5 + (a*b^2*x^18)/6 + (b^3*x^21)/21

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Rubi [A]  time = 0.072255, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{a^3 x^{12}}{12}+\frac{1}{5} a^2 b x^{15}+\frac{1}{6} a b^2 x^{18}+\frac{b^3 x^{21}}{21} \]

Antiderivative was successfully verified.

[In]  Int[x^11*(a + b*x^3)^3,x]

[Out]

(a^3*x^12)/12 + (a^2*b*x^15)/5 + (a*b^2*x^18)/6 + (b^3*x^21)/21

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Rubi in Sympy [A]  time = 9.84814, size = 36, normalized size = 0.84 \[ \frac{a^{3} x^{12}}{12} + \frac{a^{2} b x^{15}}{5} + \frac{a b^{2} x^{18}}{6} + \frac{b^{3} x^{21}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x**12/12 + a**2*b*x**15/5 + a*b**2*x**18/6 + b**3*x**21/21

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Mathematica [A]  time = 0.00312111, size = 43, normalized size = 1. \[ \frac{a^3 x^{12}}{12}+\frac{1}{5} a^2 b x^{15}+\frac{1}{6} a b^2 x^{18}+\frac{b^3 x^{21}}{21} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11*(a + b*x^3)^3,x]

[Out]

(a^3*x^12)/12 + (a^2*b*x^15)/5 + (a*b^2*x^18)/6 + (b^3*x^21)/21

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Maple [A]  time = 0., size = 36, normalized size = 0.8 \[{\frac{{a}^{3}{x}^{12}}{12}}+{\frac{{a}^{2}b{x}^{15}}{5}}+{\frac{a{b}^{2}{x}^{18}}{6}}+{\frac{{b}^{3}{x}^{21}}{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11*(b*x^3+a)^3,x)

[Out]

1/12*a^3*x^12+1/5*a^2*b*x^15+1/6*a*b^2*x^18+1/21*b^3*x^21

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Maxima [A]  time = 1.4392, size = 47, normalized size = 1.09 \[ \frac{1}{21} \, b^{3} x^{21} + \frac{1}{6} \, a b^{2} x^{18} + \frac{1}{5} \, a^{2} b x^{15} + \frac{1}{12} \, a^{3} x^{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21*b^3*x^21 + 1/6*a*b^2*x^18 + 1/5*a^2*b*x^15 + 1/12*a^3*x^12

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Fricas [A]  time = 0.19088, size = 1, normalized size = 0.02 \[ \frac{1}{21} x^{21} b^{3} + \frac{1}{6} x^{18} b^{2} a + \frac{1}{5} x^{15} b a^{2} + \frac{1}{12} x^{12} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21*x^21*b^3 + 1/6*x^18*b^2*a + 1/5*x^15*b*a^2 + 1/12*x^12*a^3

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Sympy [A]  time = 0.098185, size = 36, normalized size = 0.84 \[ \frac{a^{3} x^{12}}{12} + \frac{a^{2} b x^{15}}{5} + \frac{a b^{2} x^{18}}{6} + \frac{b^{3} x^{21}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x**12/12 + a**2*b*x**15/5 + a*b**2*x**18/6 + b**3*x**21/21

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GIAC/XCAS [A]  time = 0.220302, size = 47, normalized size = 1.09 \[ \frac{1}{21} \, b^{3} x^{21} + \frac{1}{6} \, a b^{2} x^{18} + \frac{1}{5} \, a^{2} b x^{15} + \frac{1}{12} \, a^{3} x^{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21*b^3*x^21 + 1/6*a*b^2*x^18 + 1/5*a^2*b*x^15 + 1/12*a^3*x^12