3.287 \(\int x^{14} \left (a+b x^3\right )^8 \, dx\)

Optimal. Leaf size=91 \[ \frac{a^4 \left (a+b x^3\right )^9}{27 b^5}-\frac{2 a^3 \left (a+b x^3\right )^{10}}{15 b^5}+\frac{2 a^2 \left (a+b x^3\right )^{11}}{11 b^5}+\frac{\left (a+b x^3\right )^{13}}{39 b^5}-\frac{a \left (a+b x^3\right )^{12}}{9 b^5} \]

[Out]

(a^4*(a + b*x^3)^9)/(27*b^5) - (2*a^3*(a + b*x^3)^10)/(15*b^5) + (2*a^2*(a + b*x
^3)^11)/(11*b^5) - (a*(a + b*x^3)^12)/(9*b^5) + (a + b*x^3)^13/(39*b^5)

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Rubi [A]  time = 0.289569, antiderivative size = 91, 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 \[ \frac{a^4 \left (a+b x^3\right )^9}{27 b^5}-\frac{2 a^3 \left (a+b x^3\right )^{10}}{15 b^5}+\frac{2 a^2 \left (a+b x^3\right )^{11}}{11 b^5}+\frac{\left (a+b x^3\right )^{13}}{39 b^5}-\frac{a \left (a+b x^3\right )^{12}}{9 b^5} \]

Antiderivative was successfully verified.

[In]  Int[x^14*(a + b*x^3)^8,x]

[Out]

(a^4*(a + b*x^3)^9)/(27*b^5) - (2*a^3*(a + b*x^3)^10)/(15*b^5) + (2*a^2*(a + b*x
^3)^11)/(11*b^5) - (a*(a + b*x^3)^12)/(9*b^5) + (a + b*x^3)^13/(39*b^5)

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Rubi in Sympy [A]  time = 23.79, size = 82, normalized size = 0.9 \[ \frac{a^{4} \left (a + b x^{3}\right )^{9}}{27 b^{5}} - \frac{2 a^{3} \left (a + b x^{3}\right )^{10}}{15 b^{5}} + \frac{2 a^{2} \left (a + b x^{3}\right )^{11}}{11 b^{5}} - \frac{a \left (a + b x^{3}\right )^{12}}{9 b^{5}} + \frac{\left (a + b x^{3}\right )^{13}}{39 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**14*(b*x**3+a)**8,x)

[Out]

a**4*(a + b*x**3)**9/(27*b**5) - 2*a**3*(a + b*x**3)**10/(15*b**5) + 2*a**2*(a +
b*x**3)**11/(11*b**5) - a*(a + b*x**3)**12/(9*b**5) + (a + b*x**3)**13/(39*b**5
)

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Mathematica [A]  time = 0.00473191, size = 108, normalized size = 1.19 \[ \frac{a^8 x^{15}}{15}+\frac{4}{9} a^7 b x^{18}+\frac{4}{3} a^6 b^2 x^{21}+\frac{7}{3} a^5 b^3 x^{24}+\frac{70}{27} a^4 b^4 x^{27}+\frac{28}{15} a^3 b^5 x^{30}+\frac{28}{33} a^2 b^6 x^{33}+\frac{2}{9} a b^7 x^{36}+\frac{b^8 x^{39}}{39} \]

Antiderivative was successfully verified.

[In]  Integrate[x^14*(a + b*x^3)^8,x]

[Out]

(a^8*x^15)/15 + (4*a^7*b*x^18)/9 + (4*a^6*b^2*x^21)/3 + (7*a^5*b^3*x^24)/3 + (70
*a^4*b^4*x^27)/27 + (28*a^3*b^5*x^30)/15 + (28*a^2*b^6*x^33)/33 + (2*a*b^7*x^36)
/9 + (b^8*x^39)/39

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Maple [A]  time = 0.001, size = 91, normalized size = 1. \[{\frac{{b}^{8}{x}^{39}}{39}}+{\frac{2\,a{b}^{7}{x}^{36}}{9}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{33}}{33}}+{\frac{28\,{a}^{3}{b}^{5}{x}^{30}}{15}}+{\frac{70\,{a}^{4}{b}^{4}{x}^{27}}{27}}+{\frac{7\,{a}^{5}{b}^{3}{x}^{24}}{3}}+{\frac{4\,{a}^{6}{b}^{2}{x}^{21}}{3}}+{\frac{4\,{a}^{7}b{x}^{18}}{9}}+{\frac{{a}^{8}{x}^{15}}{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^14*(b*x^3+a)^8,x)

[Out]

1/39*b^8*x^39+2/9*a*b^7*x^36+28/33*a^2*b^6*x^33+28/15*a^3*b^5*x^30+70/27*a^4*b^4
*x^27+7/3*a^5*b^3*x^24+4/3*a^6*b^2*x^21+4/9*a^7*b*x^18+1/15*a^8*x^15

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Maxima [A]  time = 1.44251, size = 122, normalized size = 1.34 \[ \frac{1}{39} \, b^{8} x^{39} + \frac{2}{9} \, a b^{7} x^{36} + \frac{28}{33} \, a^{2} b^{6} x^{33} + \frac{28}{15} \, a^{3} b^{5} x^{30} + \frac{70}{27} \, a^{4} b^{4} x^{27} + \frac{7}{3} \, a^{5} b^{3} x^{24} + \frac{4}{3} \, a^{6} b^{2} x^{21} + \frac{4}{9} \, a^{7} b x^{18} + \frac{1}{15} \, a^{8} x^{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8*x^14,x, algorithm="maxima")

[Out]

1/39*b^8*x^39 + 2/9*a*b^7*x^36 + 28/33*a^2*b^6*x^33 + 28/15*a^3*b^5*x^30 + 70/27
*a^4*b^4*x^27 + 7/3*a^5*b^3*x^24 + 4/3*a^6*b^2*x^21 + 4/9*a^7*b*x^18 + 1/15*a^8*
x^15

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Fricas [A]  time = 0.190995, size = 1, normalized size = 0.01 \[ \frac{1}{39} x^{39} b^{8} + \frac{2}{9} x^{36} b^{7} a + \frac{28}{33} x^{33} b^{6} a^{2} + \frac{28}{15} x^{30} b^{5} a^{3} + \frac{70}{27} x^{27} b^{4} a^{4} + \frac{7}{3} x^{24} b^{3} a^{5} + \frac{4}{3} x^{21} b^{2} a^{6} + \frac{4}{9} x^{18} b a^{7} + \frac{1}{15} x^{15} a^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8*x^14,x, algorithm="fricas")

[Out]

1/39*x^39*b^8 + 2/9*x^36*b^7*a + 28/33*x^33*b^6*a^2 + 28/15*x^30*b^5*a^3 + 70/27
*x^27*b^4*a^4 + 7/3*x^24*b^3*a^5 + 4/3*x^21*b^2*a^6 + 4/9*x^18*b*a^7 + 1/15*x^15
*a^8

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Sympy [A]  time = 0.158714, size = 107, normalized size = 1.18 \[ \frac{a^{8} x^{15}}{15} + \frac{4 a^{7} b x^{18}}{9} + \frac{4 a^{6} b^{2} x^{21}}{3} + \frac{7 a^{5} b^{3} x^{24}}{3} + \frac{70 a^{4} b^{4} x^{27}}{27} + \frac{28 a^{3} b^{5} x^{30}}{15} + \frac{28 a^{2} b^{6} x^{33}}{33} + \frac{2 a b^{7} x^{36}}{9} + \frac{b^{8} x^{39}}{39} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**14*(b*x**3+a)**8,x)

[Out]

a**8*x**15/15 + 4*a**7*b*x**18/9 + 4*a**6*b**2*x**21/3 + 7*a**5*b**3*x**24/3 + 7
0*a**4*b**4*x**27/27 + 28*a**3*b**5*x**30/15 + 28*a**2*b**6*x**33/33 + 2*a*b**7*
x**36/9 + b**8*x**39/39

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GIAC/XCAS [A]  time = 0.213722, size = 122, normalized size = 1.34 \[ \frac{1}{39} \, b^{8} x^{39} + \frac{2}{9} \, a b^{7} x^{36} + \frac{28}{33} \, a^{2} b^{6} x^{33} + \frac{28}{15} \, a^{3} b^{5} x^{30} + \frac{70}{27} \, a^{4} b^{4} x^{27} + \frac{7}{3} \, a^{5} b^{3} x^{24} + \frac{4}{3} \, a^{6} b^{2} x^{21} + \frac{4}{9} \, a^{7} b x^{18} + \frac{1}{15} \, a^{8} x^{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8*x^14,x, algorithm="giac")

[Out]

1/39*b^8*x^39 + 2/9*a*b^7*x^36 + 28/33*a^2*b^6*x^33 + 28/15*a^3*b^5*x^30 + 70/27
*a^4*b^4*x^27 + 7/3*a^5*b^3*x^24 + 4/3*a^6*b^2*x^21 + 4/9*a^7*b*x^18 + 1/15*a^8*
x^15