3.444 \(\int x^8 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx\)

Optimal. Leaf size=82 \[ \frac{a^6 x^9}{9}+\frac{6}{11} a^5 b x^{11}+\frac{15}{13} a^4 b^2 x^{13}+\frac{4}{3} a^3 b^3 x^{15}+\frac{15}{17} a^2 b^4 x^{17}+\frac{6}{19} a b^5 x^{19}+\frac{b^6 x^{21}}{21} \]

[Out]

(a^6*x^9)/9 + (6*a^5*b*x^11)/11 + (15*a^4*b^2*x^13)/13 + (4*a^3*b^3*x^15)/3 + (1
5*a^2*b^4*x^17)/17 + (6*a*b^5*x^19)/19 + (b^6*x^21)/21

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Rubi [A]  time = 0.116048, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{a^6 x^9}{9}+\frac{6}{11} a^5 b x^{11}+\frac{15}{13} a^4 b^2 x^{13}+\frac{4}{3} a^3 b^3 x^{15}+\frac{15}{17} a^2 b^4 x^{17}+\frac{6}{19} a b^5 x^{19}+\frac{b^6 x^{21}}{21} \]

Antiderivative was successfully verified.

[In]  Int[x^8*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(a^6*x^9)/9 + (6*a^5*b*x^11)/11 + (15*a^4*b^2*x^13)/13 + (4*a^3*b^3*x^15)/3 + (1
5*a^2*b^4*x^17)/17 + (6*a*b^5*x^19)/19 + (b^6*x^21)/21

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Rubi in Sympy [A]  time = 23.2947, size = 80, normalized size = 0.98 \[ \frac{a^{6} x^{9}}{9} + \frac{6 a^{5} b x^{11}}{11} + \frac{15 a^{4} b^{2} x^{13}}{13} + \frac{4 a^{3} b^{3} x^{15}}{3} + \frac{15 a^{2} b^{4} x^{17}}{17} + \frac{6 a b^{5} x^{19}}{19} + \frac{b^{6} x^{21}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

a**6*x**9/9 + 6*a**5*b*x**11/11 + 15*a**4*b**2*x**13/13 + 4*a**3*b**3*x**15/3 +
15*a**2*b**4*x**17/17 + 6*a*b**5*x**19/19 + b**6*x**21/21

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Mathematica [A]  time = 0.00525348, size = 82, normalized size = 1. \[ \frac{a^6 x^9}{9}+\frac{6}{11} a^5 b x^{11}+\frac{15}{13} a^4 b^2 x^{13}+\frac{4}{3} a^3 b^3 x^{15}+\frac{15}{17} a^2 b^4 x^{17}+\frac{6}{19} a b^5 x^{19}+\frac{b^6 x^{21}}{21} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(a^6*x^9)/9 + (6*a^5*b*x^11)/11 + (15*a^4*b^2*x^13)/13 + (4*a^3*b^3*x^15)/3 + (1
5*a^2*b^4*x^17)/17 + (6*a*b^5*x^19)/19 + (b^6*x^21)/21

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Maple [A]  time = 0.001, size = 69, normalized size = 0.8 \[{\frac{{a}^{6}{x}^{9}}{9}}+{\frac{6\,{a}^{5}b{x}^{11}}{11}}+{\frac{15\,{a}^{4}{b}^{2}{x}^{13}}{13}}+{\frac{4\,{a}^{3}{b}^{3}{x}^{15}}{3}}+{\frac{15\,{a}^{2}{b}^{4}{x}^{17}}{17}}+{\frac{6\,a{b}^{5}{x}^{19}}{19}}+{\frac{{b}^{6}{x}^{21}}{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

1/9*a^6*x^9+6/11*a^5*b*x^11+15/13*a^4*b^2*x^13+4/3*a^3*b^3*x^15+15/17*a^2*b^4*x^
17+6/19*a*b^5*x^19+1/21*b^6*x^21

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Maxima [A]  time = 0.703172, size = 92, normalized size = 1.12 \[ \frac{1}{21} \, b^{6} x^{21} + \frac{6}{19} \, a b^{5} x^{19} + \frac{15}{17} \, a^{2} b^{4} x^{17} + \frac{4}{3} \, a^{3} b^{3} x^{15} + \frac{15}{13} \, a^{4} b^{2} x^{13} + \frac{6}{11} \, a^{5} b x^{11} + \frac{1}{9} \, a^{6} x^{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^3*x^8,x, algorithm="maxima")

[Out]

1/21*b^6*x^21 + 6/19*a*b^5*x^19 + 15/17*a^2*b^4*x^17 + 4/3*a^3*b^3*x^15 + 15/13*
a^4*b^2*x^13 + 6/11*a^5*b*x^11 + 1/9*a^6*x^9

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Fricas [A]  time = 0.235543, size = 1, normalized size = 0.01 \[ \frac{1}{21} x^{21} b^{6} + \frac{6}{19} x^{19} b^{5} a + \frac{15}{17} x^{17} b^{4} a^{2} + \frac{4}{3} x^{15} b^{3} a^{3} + \frac{15}{13} x^{13} b^{2} a^{4} + \frac{6}{11} x^{11} b a^{5} + \frac{1}{9} x^{9} a^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^3*x^8,x, algorithm="fricas")

[Out]

1/21*x^21*b^6 + 6/19*x^19*b^5*a + 15/17*x^17*b^4*a^2 + 4/3*x^15*b^3*a^3 + 15/13*
x^13*b^2*a^4 + 6/11*x^11*b*a^5 + 1/9*x^9*a^6

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Sympy [A]  time = 0.131709, size = 80, normalized size = 0.98 \[ \frac{a^{6} x^{9}}{9} + \frac{6 a^{5} b x^{11}}{11} + \frac{15 a^{4} b^{2} x^{13}}{13} + \frac{4 a^{3} b^{3} x^{15}}{3} + \frac{15 a^{2} b^{4} x^{17}}{17} + \frac{6 a b^{5} x^{19}}{19} + \frac{b^{6} x^{21}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

a**6*x**9/9 + 6*a**5*b*x**11/11 + 15*a**4*b**2*x**13/13 + 4*a**3*b**3*x**15/3 +
15*a**2*b**4*x**17/17 + 6*a*b**5*x**19/19 + b**6*x**21/21

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GIAC/XCAS [A]  time = 0.267199, size = 92, normalized size = 1.12 \[ \frac{1}{21} \, b^{6} x^{21} + \frac{6}{19} \, a b^{5} x^{19} + \frac{15}{17} \, a^{2} b^{4} x^{17} + \frac{4}{3} \, a^{3} b^{3} x^{15} + \frac{15}{13} \, a^{4} b^{2} x^{13} + \frac{6}{11} \, a^{5} b x^{11} + \frac{1}{9} \, a^{6} x^{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^3*x^8,x, algorithm="giac")

[Out]

1/21*b^6*x^21 + 6/19*a*b^5*x^19 + 15/17*a^2*b^4*x^17 + 4/3*a^3*b^3*x^15 + 15/13*
a^4*b^2*x^13 + 6/11*a^5*b*x^11 + 1/9*a^6*x^9