3.285 \(\int x^{20} \left (a+b x^3\right )^8 \, dx\)

Optimal. Leaf size=129 \[ \frac{a^6 \left (a+b x^3\right )^9}{27 b^7}-\frac{a^5 \left (a+b x^3\right )^{10}}{5 b^7}+\frac{5 a^4 \left (a+b x^3\right )^{11}}{11 b^7}-\frac{5 a^3 \left (a+b x^3\right )^{12}}{9 b^7}+\frac{5 a^2 \left (a+b x^3\right )^{13}}{13 b^7}+\frac{\left (a+b x^3\right )^{15}}{45 b^7}-\frac{a \left (a+b x^3\right )^{14}}{7 b^7} \]

[Out]

(a^6*(a + b*x^3)^9)/(27*b^7) - (a^5*(a + b*x^3)^10)/(5*b^7) + (5*a^4*(a + b*x^3)
^11)/(11*b^7) - (5*a^3*(a + b*x^3)^12)/(9*b^7) + (5*a^2*(a + b*x^3)^13)/(13*b^7)
- (a*(a + b*x^3)^14)/(7*b^7) + (a + b*x^3)^15/(45*b^7)

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Rubi [A]  time = 0.43871, antiderivative size = 129, 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 \[ \frac{a^6 \left (a+b x^3\right )^9}{27 b^7}-\frac{a^5 \left (a+b x^3\right )^{10}}{5 b^7}+\frac{5 a^4 \left (a+b x^3\right )^{11}}{11 b^7}-\frac{5 a^3 \left (a+b x^3\right )^{12}}{9 b^7}+\frac{5 a^2 \left (a+b x^3\right )^{13}}{13 b^7}+\frac{\left (a+b x^3\right )^{15}}{45 b^7}-\frac{a \left (a+b x^3\right )^{14}}{7 b^7} \]

Antiderivative was successfully verified.

[In]  Int[x^20*(a + b*x^3)^8,x]

[Out]

(a^6*(a + b*x^3)^9)/(27*b^7) - (a^5*(a + b*x^3)^10)/(5*b^7) + (5*a^4*(a + b*x^3)
^11)/(11*b^7) - (5*a^3*(a + b*x^3)^12)/(9*b^7) + (5*a^2*(a + b*x^3)^13)/(13*b^7)
- (a*(a + b*x^3)^14)/(7*b^7) + (a + b*x^3)^15/(45*b^7)

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Rubi in Sympy [A]  time = 25.2957, size = 105, normalized size = 0.81 \[ \frac{a^{8} x^{21}}{21} + \frac{a^{7} b x^{24}}{3} + \frac{28 a^{6} b^{2} x^{27}}{27} + \frac{28 a^{5} b^{3} x^{30}}{15} + \frac{70 a^{4} b^{4} x^{33}}{33} + \frac{14 a^{3} b^{5} x^{36}}{9} + \frac{28 a^{2} b^{6} x^{39}}{39} + \frac{4 a b^{7} x^{42}}{21} + \frac{b^{8} x^{45}}{45} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**20*(b*x**3+a)**8,x)

[Out]

a**8*x**21/21 + a**7*b*x**24/3 + 28*a**6*b**2*x**27/27 + 28*a**5*b**3*x**30/15 +
70*a**4*b**4*x**33/33 + 14*a**3*b**5*x**36/9 + 28*a**2*b**6*x**39/39 + 4*a*b**7
*x**42/21 + b**8*x**45/45

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Mathematica [A]  time = 0.00513093, size = 108, normalized size = 0.84 \[ \frac{a^8 x^{21}}{21}+\frac{1}{3} a^7 b x^{24}+\frac{28}{27} a^6 b^2 x^{27}+\frac{28}{15} a^5 b^3 x^{30}+\frac{70}{33} a^4 b^4 x^{33}+\frac{14}{9} a^3 b^5 x^{36}+\frac{28}{39} a^2 b^6 x^{39}+\frac{4}{21} a b^7 x^{42}+\frac{b^8 x^{45}}{45} \]

Antiderivative was successfully verified.

[In]  Integrate[x^20*(a + b*x^3)^8,x]

[Out]

(a^8*x^21)/21 + (a^7*b*x^24)/3 + (28*a^6*b^2*x^27)/27 + (28*a^5*b^3*x^30)/15 + (
70*a^4*b^4*x^33)/33 + (14*a^3*b^5*x^36)/9 + (28*a^2*b^6*x^39)/39 + (4*a*b^7*x^42
)/21 + (b^8*x^45)/45

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Maple [A]  time = 0.003, size = 91, normalized size = 0.7 \[{\frac{{b}^{8}{x}^{45}}{45}}+{\frac{4\,a{b}^{7}{x}^{42}}{21}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{39}}{39}}+{\frac{14\,{a}^{3}{b}^{5}{x}^{36}}{9}}+{\frac{70\,{a}^{4}{b}^{4}{x}^{33}}{33}}+{\frac{28\,{a}^{5}{b}^{3}{x}^{30}}{15}}+{\frac{28\,{a}^{6}{b}^{2}{x}^{27}}{27}}+{\frac{{a}^{7}b{x}^{24}}{3}}+{\frac{{a}^{8}{x}^{21}}{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^20*(b*x^3+a)^8,x)

[Out]

1/45*b^8*x^45+4/21*a*b^7*x^42+28/39*a^2*b^6*x^39+14/9*a^3*b^5*x^36+70/33*a^4*b^4
*x^33+28/15*a^5*b^3*x^30+28/27*a^6*b^2*x^27+1/3*a^7*b*x^24+1/21*a^8*x^21

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Maxima [A]  time = 1.429, size = 122, normalized size = 0.95 \[ \frac{1}{45} \, b^{8} x^{45} + \frac{4}{21} \, a b^{7} x^{42} + \frac{28}{39} \, a^{2} b^{6} x^{39} + \frac{14}{9} \, a^{3} b^{5} x^{36} + \frac{70}{33} \, a^{4} b^{4} x^{33} + \frac{28}{15} \, a^{5} b^{3} x^{30} + \frac{28}{27} \, a^{6} b^{2} x^{27} + \frac{1}{3} \, a^{7} b x^{24} + \frac{1}{21} \, a^{8} x^{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8*x^20,x, algorithm="maxima")

[Out]

1/45*b^8*x^45 + 4/21*a*b^7*x^42 + 28/39*a^2*b^6*x^39 + 14/9*a^3*b^5*x^36 + 70/33
*a^4*b^4*x^33 + 28/15*a^5*b^3*x^30 + 28/27*a^6*b^2*x^27 + 1/3*a^7*b*x^24 + 1/21*
a^8*x^21

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Fricas [A]  time = 0.190702, size = 1, normalized size = 0.01 \[ \frac{1}{45} x^{45} b^{8} + \frac{4}{21} x^{42} b^{7} a + \frac{28}{39} x^{39} b^{6} a^{2} + \frac{14}{9} x^{36} b^{5} a^{3} + \frac{70}{33} x^{33} b^{4} a^{4} + \frac{28}{15} x^{30} b^{3} a^{5} + \frac{28}{27} x^{27} b^{2} a^{6} + \frac{1}{3} x^{24} b a^{7} + \frac{1}{21} x^{21} a^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8*x^20,x, algorithm="fricas")

[Out]

1/45*x^45*b^8 + 4/21*x^42*b^7*a + 28/39*x^39*b^6*a^2 + 14/9*x^36*b^5*a^3 + 70/33
*x^33*b^4*a^4 + 28/15*x^30*b^3*a^5 + 28/27*x^27*b^2*a^6 + 1/3*x^24*b*a^7 + 1/21*
x^21*a^8

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Sympy [A]  time = 0.179298, size = 105, normalized size = 0.81 \[ \frac{a^{8} x^{21}}{21} + \frac{a^{7} b x^{24}}{3} + \frac{28 a^{6} b^{2} x^{27}}{27} + \frac{28 a^{5} b^{3} x^{30}}{15} + \frac{70 a^{4} b^{4} x^{33}}{33} + \frac{14 a^{3} b^{5} x^{36}}{9} + \frac{28 a^{2} b^{6} x^{39}}{39} + \frac{4 a b^{7} x^{42}}{21} + \frac{b^{8} x^{45}}{45} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**20*(b*x**3+a)**8,x)

[Out]

a**8*x**21/21 + a**7*b*x**24/3 + 28*a**6*b**2*x**27/27 + 28*a**5*b**3*x**30/15 +
70*a**4*b**4*x**33/33 + 14*a**3*b**5*x**36/9 + 28*a**2*b**6*x**39/39 + 4*a*b**7
*x**42/21 + b**8*x**45/45

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GIAC/XCAS [A]  time = 0.217363, size = 122, normalized size = 0.95 \[ \frac{1}{45} \, b^{8} x^{45} + \frac{4}{21} \, a b^{7} x^{42} + \frac{28}{39} \, a^{2} b^{6} x^{39} + \frac{14}{9} \, a^{3} b^{5} x^{36} + \frac{70}{33} \, a^{4} b^{4} x^{33} + \frac{28}{15} \, a^{5} b^{3} x^{30} + \frac{28}{27} \, a^{6} b^{2} x^{27} + \frac{1}{3} \, a^{7} b x^{24} + \frac{1}{21} \, a^{8} x^{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8*x^20,x, algorithm="giac")

[Out]

1/45*b^8*x^45 + 4/21*a*b^7*x^42 + 28/39*a^2*b^6*x^39 + 14/9*a^3*b^5*x^36 + 70/33
*a^4*b^4*x^33 + 28/15*a^5*b^3*x^30 + 28/27*a^6*b^2*x^27 + 1/3*a^7*b*x^24 + 1/21*
a^8*x^21