3.445 \(\int x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx\)

Optimal. Leaf size=72 \[ -\frac{a^3 \left (a+b x^2\right )^7}{14 b^4}+\frac{3 a^2 \left (a+b x^2\right )^8}{16 b^4}+\frac{\left (a+b x^2\right )^{10}}{20 b^4}-\frac{a \left (a+b x^2\right )^9}{6 b^4} \]

[Out]

-(a^3*(a + b*x^2)^7)/(14*b^4) + (3*a^2*(a + b*x^2)^8)/(16*b^4) - (a*(a + b*x^2)^
9)/(6*b^4) + (a + b*x^2)^10/(20*b^4)

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Rubi [A]  time = 0.255441, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{a^3 \left (a+b x^2\right )^7}{14 b^4}+\frac{3 a^2 \left (a+b x^2\right )^8}{16 b^4}+\frac{\left (a+b x^2\right )^{10}}{20 b^4}-\frac{a \left (a+b x^2\right )^9}{6 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-(a^3*(a + b*x^2)^7)/(14*b^4) + (3*a^2*(a + b*x^2)^8)/(16*b^4) - (a*(a + b*x^2)^
9)/(6*b^4) + (a + b*x^2)^10/(20*b^4)

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Rubi in Sympy [A]  time = 27.5963, size = 63, normalized size = 0.88 \[ - \frac{a^{3} \left (a + b x^{2}\right )^{7}}{14 b^{4}} + \frac{3 a^{2} \left (a + b x^{2}\right )^{8}}{16 b^{4}} - \frac{a \left (a + b x^{2}\right )^{9}}{6 b^{4}} + \frac{\left (a + b x^{2}\right )^{10}}{20 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

-a**3*(a + b*x**2)**7/(14*b**4) + 3*a**2*(a + b*x**2)**8/(16*b**4) - a*(a + b*x*
*2)**9/(6*b**4) + (a + b*x**2)**10/(20*b**4)

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Mathematica [A]  time = 0.00438665, size = 82, normalized size = 1.14 \[ \frac{a^6 x^8}{8}+\frac{3}{5} a^5 b x^{10}+\frac{5}{4} a^4 b^2 x^{12}+\frac{10}{7} a^3 b^3 x^{14}+\frac{15}{16} a^2 b^4 x^{16}+\frac{1}{3} a b^5 x^{18}+\frac{b^6 x^{20}}{20} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(a^6*x^8)/8 + (3*a^5*b*x^10)/5 + (5*a^4*b^2*x^12)/4 + (10*a^3*b^3*x^14)/7 + (15*
a^2*b^4*x^16)/16 + (a*b^5*x^18)/3 + (b^6*x^20)/20

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Maple [A]  time = 0.002, size = 69, normalized size = 1. \[{\frac{{b}^{6}{x}^{20}}{20}}+{\frac{a{b}^{5}{x}^{18}}{3}}+{\frac{15\,{a}^{2}{b}^{4}{x}^{16}}{16}}+{\frac{10\,{a}^{3}{b}^{3}{x}^{14}}{7}}+{\frac{5\,{a}^{4}{b}^{2}{x}^{12}}{4}}+{\frac{3\,{a}^{5}b{x}^{10}}{5}}+{\frac{{a}^{6}{x}^{8}}{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

1/20*b^6*x^20+1/3*a*b^5*x^18+15/16*a^2*b^4*x^16+10/7*a^3*b^3*x^14+5/4*a^4*b^2*x^
12+3/5*a^5*b*x^10+1/8*a^6*x^8

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Maxima [A]  time = 0.704549, size = 92, normalized size = 1.28 \[ \frac{1}{20} \, b^{6} x^{20} + \frac{1}{3} \, a b^{5} x^{18} + \frac{15}{16} \, a^{2} b^{4} x^{16} + \frac{10}{7} \, a^{3} b^{3} x^{14} + \frac{5}{4} \, a^{4} b^{2} x^{12} + \frac{3}{5} \, a^{5} b x^{10} + \frac{1}{8} \, a^{6} x^{8} \]

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^7,x, algorithm="maxima")

[Out]

1/20*b^6*x^20 + 1/3*a*b^5*x^18 + 15/16*a^2*b^4*x^16 + 10/7*a^3*b^3*x^14 + 5/4*a^
4*b^2*x^12 + 3/5*a^5*b*x^10 + 1/8*a^6*x^8

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Fricas [A]  time = 0.236289, size = 1, normalized size = 0.01 \[ \frac{1}{20} x^{20} b^{6} + \frac{1}{3} x^{18} b^{5} a + \frac{15}{16} x^{16} b^{4} a^{2} + \frac{10}{7} x^{14} b^{3} a^{3} + \frac{5}{4} x^{12} b^{2} a^{4} + \frac{3}{5} x^{10} b a^{5} + \frac{1}{8} x^{8} a^{6} \]

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^7,x, algorithm="fricas")

[Out]

1/20*x^20*b^6 + 1/3*x^18*b^5*a + 15/16*x^16*b^4*a^2 + 10/7*x^14*b^3*a^3 + 5/4*x^
12*b^2*a^4 + 3/5*x^10*b*a^5 + 1/8*x^8*a^6

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Sympy [A]  time = 0.13885, size = 78, normalized size = 1.08 \[ \frac{a^{6} x^{8}}{8} + \frac{3 a^{5} b x^{10}}{5} + \frac{5 a^{4} b^{2} x^{12}}{4} + \frac{10 a^{3} b^{3} x^{14}}{7} + \frac{15 a^{2} b^{4} x^{16}}{16} + \frac{a b^{5} x^{18}}{3} + \frac{b^{6} x^{20}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

a**6*x**8/8 + 3*a**5*b*x**10/5 + 5*a**4*b**2*x**12/4 + 10*a**3*b**3*x**14/7 + 15
*a**2*b**4*x**16/16 + a*b**5*x**18/3 + b**6*x**20/20

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GIAC/XCAS [A]  time = 0.268042, size = 92, normalized size = 1.28 \[ \frac{1}{20} \, b^{6} x^{20} + \frac{1}{3} \, a b^{5} x^{18} + \frac{15}{16} \, a^{2} b^{4} x^{16} + \frac{10}{7} \, a^{3} b^{3} x^{14} + \frac{5}{4} \, a^{4} b^{2} x^{12} + \frac{3}{5} \, a^{5} b x^{10} + \frac{1}{8} \, a^{6} x^{8} \]

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^7,x, algorithm="giac")

[Out]

1/20*b^6*x^20 + 1/3*a*b^5*x^18 + 15/16*a^2*b^4*x^16 + 10/7*a^3*b^3*x^14 + 5/4*a^
4*b^2*x^12 + 3/5*a^5*b*x^10 + 1/8*a^6*x^8