3.286 \(\int x^{17} \left (a+b x^3\right )^8 \, dx\)

Optimal. Leaf size=110 \[ -\frac{a^5 \left (a+b x^3\right )^9}{27 b^6}+\frac{a^4 \left (a+b x^3\right )^{10}}{6 b^6}-\frac{10 a^3 \left (a+b x^3\right )^{11}}{33 b^6}+\frac{5 a^2 \left (a+b x^3\right )^{12}}{18 b^6}+\frac{\left (a+b x^3\right )^{14}}{42 b^6}-\frac{5 a \left (a+b x^3\right )^{13}}{39 b^6} \]

[Out]

-(a^5*(a + b*x^3)^9)/(27*b^6) + (a^4*(a + b*x^3)^10)/(6*b^6) - (10*a^3*(a + b*x^
3)^11)/(33*b^6) + (5*a^2*(a + b*x^3)^12)/(18*b^6) - (5*a*(a + b*x^3)^13)/(39*b^6
) + (a + b*x^3)^14/(42*b^6)

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Rubi [A]  time = 0.346615, antiderivative size = 110, 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 \[ -\frac{a^5 \left (a+b x^3\right )^9}{27 b^6}+\frac{a^4 \left (a+b x^3\right )^{10}}{6 b^6}-\frac{10 a^3 \left (a+b x^3\right )^{11}}{33 b^6}+\frac{5 a^2 \left (a+b x^3\right )^{12}}{18 b^6}+\frac{\left (a+b x^3\right )^{14}}{42 b^6}-\frac{5 a \left (a+b x^3\right )^{13}}{39 b^6} \]

Antiderivative was successfully verified.

[In]  Int[x^17*(a + b*x^3)^8,x]

[Out]

-(a^5*(a + b*x^3)^9)/(27*b^6) + (a^4*(a + b*x^3)^10)/(6*b^6) - (10*a^3*(a + b*x^
3)^11)/(33*b^6) + (5*a^2*(a + b*x^3)^12)/(18*b^6) - (5*a*(a + b*x^3)^13)/(39*b^6
) + (a + b*x^3)^14/(42*b^6)

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Rubi in Sympy [A]  time = 27.7336, size = 100, normalized size = 0.91 \[ - \frac{a^{5} \left (a + b x^{3}\right )^{9}}{27 b^{6}} + \frac{a^{4} \left (a + b x^{3}\right )^{10}}{6 b^{6}} - \frac{10 a^{3} \left (a + b x^{3}\right )^{11}}{33 b^{6}} + \frac{5 a^{2} \left (a + b x^{3}\right )^{12}}{18 b^{6}} - \frac{5 a \left (a + b x^{3}\right )^{13}}{39 b^{6}} + \frac{\left (a + b x^{3}\right )^{14}}{42 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**17*(b*x**3+a)**8,x)

[Out]

-a**5*(a + b*x**3)**9/(27*b**6) + a**4*(a + b*x**3)**10/(6*b**6) - 10*a**3*(a +
b*x**3)**11/(33*b**6) + 5*a**2*(a + b*x**3)**12/(18*b**6) - 5*a*(a + b*x**3)**13
/(39*b**6) + (a + b*x**3)**14/(42*b**6)

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Mathematica [A]  time = 0.00473479, size = 108, normalized size = 0.98 \[ \frac{a^8 x^{18}}{18}+\frac{8}{21} a^7 b x^{21}+\frac{7}{6} a^6 b^2 x^{24}+\frac{56}{27} a^5 b^3 x^{27}+\frac{7}{3} a^4 b^4 x^{30}+\frac{56}{33} a^3 b^5 x^{33}+\frac{7}{9} a^2 b^6 x^{36}+\frac{8}{39} a b^7 x^{39}+\frac{b^8 x^{42}}{42} \]

Antiderivative was successfully verified.

[In]  Integrate[x^17*(a + b*x^3)^8,x]

[Out]

(a^8*x^18)/18 + (8*a^7*b*x^21)/21 + (7*a^6*b^2*x^24)/6 + (56*a^5*b^3*x^27)/27 +
(7*a^4*b^4*x^30)/3 + (56*a^3*b^5*x^33)/33 + (7*a^2*b^6*x^36)/9 + (8*a*b^7*x^39)/
39 + (b^8*x^42)/42

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Maple [A]  time = 0.003, size = 91, normalized size = 0.8 \[{\frac{{b}^{8}{x}^{42}}{42}}+{\frac{8\,a{b}^{7}{x}^{39}}{39}}+{\frac{7\,{a}^{2}{b}^{6}{x}^{36}}{9}}+{\frac{56\,{a}^{3}{b}^{5}{x}^{33}}{33}}+{\frac{7\,{a}^{4}{b}^{4}{x}^{30}}{3}}+{\frac{56\,{a}^{5}{b}^{3}{x}^{27}}{27}}+{\frac{7\,{a}^{6}{b}^{2}{x}^{24}}{6}}+{\frac{8\,{a}^{7}b{x}^{21}}{21}}+{\frac{{a}^{8}{x}^{18}}{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^17*(b*x^3+a)^8,x)

[Out]

1/42*b^8*x^42+8/39*a*b^7*x^39+7/9*a^2*b^6*x^36+56/33*a^3*b^5*x^33+7/3*a^4*b^4*x^
30+56/27*a^5*b^3*x^27+7/6*a^6*b^2*x^24+8/21*a^7*b*x^21+1/18*a^8*x^18

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Maxima [A]  time = 1.43434, size = 122, normalized size = 1.11 \[ \frac{1}{42} \, b^{8} x^{42} + \frac{8}{39} \, a b^{7} x^{39} + \frac{7}{9} \, a^{2} b^{6} x^{36} + \frac{56}{33} \, a^{3} b^{5} x^{33} + \frac{7}{3} \, a^{4} b^{4} x^{30} + \frac{56}{27} \, a^{5} b^{3} x^{27} + \frac{7}{6} \, a^{6} b^{2} x^{24} + \frac{8}{21} \, a^{7} b x^{21} + \frac{1}{18} \, a^{8} x^{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8*x^17,x, algorithm="maxima")

[Out]

1/42*b^8*x^42 + 8/39*a*b^7*x^39 + 7/9*a^2*b^6*x^36 + 56/33*a^3*b^5*x^33 + 7/3*a^
4*b^4*x^30 + 56/27*a^5*b^3*x^27 + 7/6*a^6*b^2*x^24 + 8/21*a^7*b*x^21 + 1/18*a^8*
x^18

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Fricas [A]  time = 0.192949, size = 1, normalized size = 0.01 \[ \frac{1}{42} x^{42} b^{8} + \frac{8}{39} x^{39} b^{7} a + \frac{7}{9} x^{36} b^{6} a^{2} + \frac{56}{33} x^{33} b^{5} a^{3} + \frac{7}{3} x^{30} b^{4} a^{4} + \frac{56}{27} x^{27} b^{3} a^{5} + \frac{7}{6} x^{24} b^{2} a^{6} + \frac{8}{21} x^{21} b a^{7} + \frac{1}{18} x^{18} a^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8*x^17,x, algorithm="fricas")

[Out]

1/42*x^42*b^8 + 8/39*x^39*b^7*a + 7/9*x^36*b^6*a^2 + 56/33*x^33*b^5*a^3 + 7/3*x^
30*b^4*a^4 + 56/27*x^27*b^3*a^5 + 7/6*x^24*b^2*a^6 + 8/21*x^21*b*a^7 + 1/18*x^18
*a^8

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Sympy [A]  time = 0.162676, size = 107, normalized size = 0.97 \[ \frac{a^{8} x^{18}}{18} + \frac{8 a^{7} b x^{21}}{21} + \frac{7 a^{6} b^{2} x^{24}}{6} + \frac{56 a^{5} b^{3} x^{27}}{27} + \frac{7 a^{4} b^{4} x^{30}}{3} + \frac{56 a^{3} b^{5} x^{33}}{33} + \frac{7 a^{2} b^{6} x^{36}}{9} + \frac{8 a b^{7} x^{39}}{39} + \frac{b^{8} x^{42}}{42} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**17*(b*x**3+a)**8,x)

[Out]

a**8*x**18/18 + 8*a**7*b*x**21/21 + 7*a**6*b**2*x**24/6 + 56*a**5*b**3*x**27/27
+ 7*a**4*b**4*x**30/3 + 56*a**3*b**5*x**33/33 + 7*a**2*b**6*x**36/9 + 8*a*b**7*x
**39/39 + b**8*x**42/42

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GIAC/XCAS [A]  time = 0.21549, size = 122, normalized size = 1.11 \[ \frac{1}{42} \, b^{8} x^{42} + \frac{8}{39} \, a b^{7} x^{39} + \frac{7}{9} \, a^{2} b^{6} x^{36} + \frac{56}{33} \, a^{3} b^{5} x^{33} + \frac{7}{3} \, a^{4} b^{4} x^{30} + \frac{56}{27} \, a^{5} b^{3} x^{27} + \frac{7}{6} \, a^{6} b^{2} x^{24} + \frac{8}{21} \, a^{7} b x^{21} + \frac{1}{18} \, a^{8} x^{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8*x^17,x, algorithm="giac")

[Out]

1/42*b^8*x^42 + 8/39*a*b^7*x^39 + 7/9*a^2*b^6*x^36 + 56/33*a^3*b^5*x^33 + 7/3*a^
4*b^4*x^30 + 56/27*a^5*b^3*x^27 + 7/6*a^6*b^2*x^24 + 8/21*a^7*b*x^21 + 1/18*a^8*
x^18