3.620 \(\int x^4 \left (a+b x^4\right )^2 \, dx\)

Optimal. Leaf size=30 \[ \frac{a^2 x^5}{5}+\frac{2}{9} a b x^9+\frac{b^2 x^{13}}{13} \]

[Out]

(a^2*x^5)/5 + (2*a*b*x^9)/9 + (b^2*x^13)/13

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Rubi [A]  time = 0.0322847, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{a^2 x^5}{5}+\frac{2}{9} a b x^9+\frac{b^2 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a + b*x^4)^2,x]

[Out]

(a^2*x^5)/5 + (2*a*b*x^9)/9 + (b^2*x^13)/13

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Rubi in Sympy [A]  time = 5.32679, size = 26, normalized size = 0.87 \[ \frac{a^{2} x^{5}}{5} + \frac{2 a b x^{9}}{9} + \frac{b^{2} x^{13}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**5/5 + 2*a*b*x**9/9 + b**2*x**13/13

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Mathematica [A]  time = 0.00139993, size = 30, normalized size = 1. \[ \frac{a^2 x^5}{5}+\frac{2}{9} a b x^9+\frac{b^2 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(a + b*x^4)^2,x]

[Out]

(a^2*x^5)/5 + (2*a*b*x^9)/9 + (b^2*x^13)/13

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Maple [A]  time = 0.002, size = 25, normalized size = 0.8 \[{\frac{{x}^{5}{a}^{2}}{5}}+{\frac{2\,ab{x}^{9}}{9}}+{\frac{{b}^{2}{x}^{13}}{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x^4+a)^2,x)

[Out]

1/5*x^5*a^2+2/9*a*b*x^9+1/13*b^2*x^13

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Maxima [A]  time = 1.43832, size = 32, normalized size = 1.07 \[ \frac{1}{13} \, b^{2} x^{13} + \frac{2}{9} \, a b x^{9} + \frac{1}{5} \, a^{2} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/13*b^2*x^13 + 2/9*a*b*x^9 + 1/5*a^2*x^5

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Fricas [A]  time = 0.198883, size = 1, normalized size = 0.03 \[ \frac{1}{13} x^{13} b^{2} + \frac{2}{9} x^{9} b a + \frac{1}{5} x^{5} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/13*x^13*b^2 + 2/9*x^9*b*a + 1/5*x^5*a^2

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Sympy [A]  time = 0.081982, size = 26, normalized size = 0.87 \[ \frac{a^{2} x^{5}}{5} + \frac{2 a b x^{9}}{9} + \frac{b^{2} x^{13}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(b*x**4+a)**2,x)

[Out]

a**2*x**5/5 + 2*a*b*x**9/9 + b**2*x**13/13

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GIAC/XCAS [A]  time = 0.220052, size = 32, normalized size = 1.07 \[ \frac{1}{13} \, b^{2} x^{13} + \frac{2}{9} \, a b x^{9} + \frac{1}{5} \, a^{2} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/13*b^2*x^13 + 2/9*a*b*x^9 + 1/5*a^2*x^5