3.258 \(\int x^{17} \left (a+b x^3\right )^5 \, dx\)

Optimal. Leaf size=69 \[ \frac{a^5 x^{18}}{18}+\frac{5}{21} a^4 b x^{21}+\frac{5}{12} a^3 b^2 x^{24}+\frac{10}{27} a^2 b^3 x^{27}+\frac{1}{6} a b^4 x^{30}+\frac{b^5 x^{33}}{33} \]

[Out]

(a^5*x^18)/18 + (5*a^4*b*x^21)/21 + (5*a^3*b^2*x^24)/12 + (10*a^2*b^3*x^27)/27 +
(a*b^4*x^30)/6 + (b^5*x^33)/33

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Rubi [A]  time = 0.11091, 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 \[ \frac{a^5 x^{18}}{18}+\frac{5}{21} a^4 b x^{21}+\frac{5}{12} a^3 b^2 x^{24}+\frac{10}{27} a^2 b^3 x^{27}+\frac{1}{6} a b^4 x^{30}+\frac{b^5 x^{33}}{33} \]

Antiderivative was successfully verified.

[In]  Int[x^17*(a + b*x^3)^5,x]

[Out]

(a^5*x^18)/18 + (5*a^4*b*x^21)/21 + (5*a^3*b^2*x^24)/12 + (10*a^2*b^3*x^27)/27 +
(a*b^4*x^30)/6 + (b^5*x^33)/33

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Rubi in Sympy [A]  time = 15.8097, size = 65, normalized size = 0.94 \[ \frac{a^{5} x^{18}}{18} + \frac{5 a^{4} b x^{21}}{21} + \frac{5 a^{3} b^{2} x^{24}}{12} + \frac{10 a^{2} b^{3} x^{27}}{27} + \frac{a b^{4} x^{30}}{6} + \frac{b^{5} x^{33}}{33} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**17*(b*x**3+a)**5,x)

[Out]

a**5*x**18/18 + 5*a**4*b*x**21/21 + 5*a**3*b**2*x**24/12 + 10*a**2*b**3*x**27/27
+ a*b**4*x**30/6 + b**5*x**33/33

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Mathematica [A]  time = 0.00487206, size = 69, normalized size = 1. \[ \frac{a^5 x^{18}}{18}+\frac{5}{21} a^4 b x^{21}+\frac{5}{12} a^3 b^2 x^{24}+\frac{10}{27} a^2 b^3 x^{27}+\frac{1}{6} a b^4 x^{30}+\frac{b^5 x^{33}}{33} \]

Antiderivative was successfully verified.

[In]  Integrate[x^17*(a + b*x^3)^5,x]

[Out]

(a^5*x^18)/18 + (5*a^4*b*x^21)/21 + (5*a^3*b^2*x^24)/12 + (10*a^2*b^3*x^27)/27 +
(a*b^4*x^30)/6 + (b^5*x^33)/33

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Maple [A]  time = 0.002, size = 58, normalized size = 0.8 \[{\frac{{a}^{5}{x}^{18}}{18}}+{\frac{5\,{a}^{4}b{x}^{21}}{21}}+{\frac{5\,{a}^{3}{b}^{2}{x}^{24}}{12}}+{\frac{10\,{a}^{2}{b}^{3}{x}^{27}}{27}}+{\frac{a{b}^{4}{x}^{30}}{6}}+{\frac{{b}^{5}{x}^{33}}{33}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^17*(b*x^3+a)^5,x)

[Out]

1/18*a^5*x^18+5/21*a^4*b*x^21+5/12*a^3*b^2*x^24+10/27*a^2*b^3*x^27+1/6*a*b^4*x^3
0+1/33*b^5*x^33

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Maxima [A]  time = 1.42825, size = 77, normalized size = 1.12 \[ \frac{1}{33} \, b^{5} x^{33} + \frac{1}{6} \, a b^{4} x^{30} + \frac{10}{27} \, a^{2} b^{3} x^{27} + \frac{5}{12} \, a^{3} b^{2} x^{24} + \frac{5}{21} \, a^{4} b x^{21} + \frac{1}{18} \, a^{5} x^{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5*x^17,x, algorithm="maxima")

[Out]

1/33*b^5*x^33 + 1/6*a*b^4*x^30 + 10/27*a^2*b^3*x^27 + 5/12*a^3*b^2*x^24 + 5/21*a
^4*b*x^21 + 1/18*a^5*x^18

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Fricas [A]  time = 0.187771, size = 1, normalized size = 0.01 \[ \frac{1}{33} x^{33} b^{5} + \frac{1}{6} x^{30} b^{4} a + \frac{10}{27} x^{27} b^{3} a^{2} + \frac{5}{12} x^{24} b^{2} a^{3} + \frac{5}{21} x^{21} b a^{4} + \frac{1}{18} x^{18} a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5*x^17,x, algorithm="fricas")

[Out]

1/33*x^33*b^5 + 1/6*x^30*b^4*a + 10/27*x^27*b^3*a^2 + 5/12*x^24*b^2*a^3 + 5/21*x
^21*b*a^4 + 1/18*x^18*a^5

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Sympy [A]  time = 0.131157, size = 65, normalized size = 0.94 \[ \frac{a^{5} x^{18}}{18} + \frac{5 a^{4} b x^{21}}{21} + \frac{5 a^{3} b^{2} x^{24}}{12} + \frac{10 a^{2} b^{3} x^{27}}{27} + \frac{a b^{4} x^{30}}{6} + \frac{b^{5} x^{33}}{33} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**17*(b*x**3+a)**5,x)

[Out]

a**5*x**18/18 + 5*a**4*b*x**21/21 + 5*a**3*b**2*x**24/12 + 10*a**2*b**3*x**27/27
+ a*b**4*x**30/6 + b**5*x**33/33

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GIAC/XCAS [A]  time = 0.229444, size = 77, normalized size = 1.12 \[ \frac{1}{33} \, b^{5} x^{33} + \frac{1}{6} \, a b^{4} x^{30} + \frac{10}{27} \, a^{2} b^{3} x^{27} + \frac{5}{12} \, a^{3} b^{2} x^{24} + \frac{5}{21} \, a^{4} b x^{21} + \frac{1}{18} \, a^{5} x^{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5*x^17,x, algorithm="giac")

[Out]

1/33*b^5*x^33 + 1/6*a*b^4*x^30 + 10/27*a^2*b^3*x^27 + 5/12*a^3*b^2*x^24 + 5/21*a
^4*b*x^21 + 1/18*a^5*x^18