3.259 \(\int x^{14} \left (a+b x^3\right )^5 \, dx\)

Optimal. Leaf size=69 \[ \frac{a^5 x^{15}}{15}+\frac{5}{18} a^4 b x^{18}+\frac{10}{21} a^3 b^2 x^{21}+\frac{5}{12} a^2 b^3 x^{24}+\frac{5}{27} a b^4 x^{27}+\frac{b^5 x^{30}}{30} \]

[Out]

(a^5*x^15)/15 + (5*a^4*b*x^18)/18 + (10*a^3*b^2*x^21)/21 + (5*a^2*b^3*x^24)/12 +
(5*a*b^4*x^27)/27 + (b^5*x^30)/30

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Rubi [A]  time = 0.106399, antiderivative size = 69, 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 \[ \frac{a^5 x^{15}}{15}+\frac{5}{18} a^4 b x^{18}+\frac{10}{21} a^3 b^2 x^{21}+\frac{5}{12} a^2 b^3 x^{24}+\frac{5}{27} a b^4 x^{27}+\frac{b^5 x^{30}}{30} \]

Antiderivative was successfully verified.

[In]  Int[x^14*(a + b*x^3)^5,x]

[Out]

(a^5*x^15)/15 + (5*a^4*b*x^18)/18 + (10*a^3*b^2*x^21)/21 + (5*a^2*b^3*x^24)/12 +
(5*a*b^4*x^27)/27 + (b^5*x^30)/30

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Rubi in Sympy [A]  time = 15.5162, size = 66, normalized size = 0.96 \[ \frac{a^{5} x^{15}}{15} + \frac{5 a^{4} b x^{18}}{18} + \frac{10 a^{3} b^{2} x^{21}}{21} + \frac{5 a^{2} b^{3} x^{24}}{12} + \frac{5 a b^{4} x^{27}}{27} + \frac{b^{5} x^{30}}{30} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**14*(b*x**3+a)**5,x)

[Out]

a**5*x**15/15 + 5*a**4*b*x**18/18 + 10*a**3*b**2*x**21/21 + 5*a**2*b**3*x**24/12
+ 5*a*b**4*x**27/27 + b**5*x**30/30

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Mathematica [A]  time = 0.00404075, size = 69, normalized size = 1. \[ \frac{a^5 x^{15}}{15}+\frac{5}{18} a^4 b x^{18}+\frac{10}{21} a^3 b^2 x^{21}+\frac{5}{12} a^2 b^3 x^{24}+\frac{5}{27} a b^4 x^{27}+\frac{b^5 x^{30}}{30} \]

Antiderivative was successfully verified.

[In]  Integrate[x^14*(a + b*x^3)^5,x]

[Out]

(a^5*x^15)/15 + (5*a^4*b*x^18)/18 + (10*a^3*b^2*x^21)/21 + (5*a^2*b^3*x^24)/12 +
(5*a*b^4*x^27)/27 + (b^5*x^30)/30

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Maple [A]  time = 0.003, size = 58, normalized size = 0.8 \[{\frac{{a}^{5}{x}^{15}}{15}}+{\frac{5\,{a}^{4}b{x}^{18}}{18}}+{\frac{10\,{a}^{3}{b}^{2}{x}^{21}}{21}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{24}}{12}}+{\frac{5\,a{b}^{4}{x}^{27}}{27}}+{\frac{{b}^{5}{x}^{30}}{30}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^14*(b*x^3+a)^5,x)

[Out]

1/15*a^5*x^15+5/18*a^4*b*x^18+10/21*a^3*b^2*x^21+5/12*a^2*b^3*x^24+5/27*a*b^4*x^
27+1/30*b^5*x^30

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Maxima [A]  time = 1.42186, size = 77, normalized size = 1.12 \[ \frac{1}{30} \, b^{5} x^{30} + \frac{5}{27} \, a b^{4} x^{27} + \frac{5}{12} \, a^{2} b^{3} x^{24} + \frac{10}{21} \, a^{3} b^{2} x^{21} + \frac{5}{18} \, a^{4} b x^{18} + \frac{1}{15} \, a^{5} x^{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5*x^14,x, algorithm="maxima")

[Out]

1/30*b^5*x^30 + 5/27*a*b^4*x^27 + 5/12*a^2*b^3*x^24 + 10/21*a^3*b^2*x^21 + 5/18*
a^4*b*x^18 + 1/15*a^5*x^15

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Fricas [A]  time = 0.188498, size = 1, normalized size = 0.01 \[ \frac{1}{30} x^{30} b^{5} + \frac{5}{27} x^{27} b^{4} a + \frac{5}{12} x^{24} b^{3} a^{2} + \frac{10}{21} x^{21} b^{2} a^{3} + \frac{5}{18} x^{18} b a^{4} + \frac{1}{15} x^{15} a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5*x^14,x, algorithm="fricas")

[Out]

1/30*x^30*b^5 + 5/27*x^27*b^4*a + 5/12*x^24*b^3*a^2 + 10/21*x^21*b^2*a^3 + 5/18*
x^18*b*a^4 + 1/15*x^15*a^5

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Sympy [A]  time = 0.125735, size = 66, normalized size = 0.96 \[ \frac{a^{5} x^{15}}{15} + \frac{5 a^{4} b x^{18}}{18} + \frac{10 a^{3} b^{2} x^{21}}{21} + \frac{5 a^{2} b^{3} x^{24}}{12} + \frac{5 a b^{4} x^{27}}{27} + \frac{b^{5} x^{30}}{30} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**14*(b*x**3+a)**5,x)

[Out]

a**5*x**15/15 + 5*a**4*b*x**18/18 + 10*a**3*b**2*x**21/21 + 5*a**2*b**3*x**24/12
+ 5*a*b**4*x**27/27 + b**5*x**30/30

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GIAC/XCAS [A]  time = 0.214245, size = 77, normalized size = 1.12 \[ \frac{1}{30} \, b^{5} x^{30} + \frac{5}{27} \, a b^{4} x^{27} + \frac{5}{12} \, a^{2} b^{3} x^{24} + \frac{10}{21} \, a^{3} b^{2} x^{21} + \frac{5}{18} \, a^{4} b x^{18} + \frac{1}{15} \, a^{5} x^{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5*x^14,x, algorithm="giac")

[Out]

1/30*b^5*x^30 + 5/27*a*b^4*x^27 + 5/12*a^2*b^3*x^24 + 10/21*a^3*b^2*x^21 + 5/18*
a^4*b*x^18 + 1/15*a^5*x^15