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

Optimal. Leaf size=30 \[ \frac{a^2 x^3}{3}+\frac{2}{7} a b x^7+\frac{b^2 x^{11}}{11} \]

[Out]

(a^2*x^3)/3 + (2*a*b*x^7)/7 + (b^2*x^11)/11

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Rubi [A]  time = 0.0322706, 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 \[ \frac{a^2 x^3}{3}+\frac{2}{7} a b x^7+\frac{b^2 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x^4)^2,x]

[Out]

(a^2*x^3)/3 + (2*a*b*x^7)/7 + (b^2*x^11)/11

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Rubi in Sympy [A]  time = 5.20119, size = 26, normalized size = 0.87 \[ \frac{a^{2} x^{3}}{3} + \frac{2 a b x^{7}}{7} + \frac{b^{2} x^{11}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**3/3 + 2*a*b*x**7/7 + b**2*x**11/11

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Mathematica [A]  time = 0.00155, size = 30, normalized size = 1. \[ \frac{a^2 x^3}{3}+\frac{2}{7} a b x^7+\frac{b^2 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a + b*x^4)^2,x]

[Out]

(a^2*x^3)/3 + (2*a*b*x^7)/7 + (b^2*x^11)/11

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Maple [A]  time = 0.001, size = 25, normalized size = 0.8 \[{\frac{{x}^{3}{a}^{2}}{3}}+{\frac{2\,ab{x}^{7}}{7}}+{\frac{{b}^{2}{x}^{11}}{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^4+a)^2,x)

[Out]

1/3*x^3*a^2+2/7*a*b*x^7+1/11*b^2*x^11

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Maxima [A]  time = 1.42159, size = 32, normalized size = 1.07 \[ \frac{1}{11} \, b^{2} x^{11} + \frac{2}{7} \, a b x^{7} + \frac{1}{3} \, a^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*b^2*x^11 + 2/7*a*b*x^7 + 1/3*a^2*x^3

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Fricas [A]  time = 0.199596, size = 1, normalized size = 0.03 \[ \frac{1}{11} x^{11} b^{2} + \frac{2}{7} x^{7} b a + \frac{1}{3} x^{3} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*x^11*b^2 + 2/7*x^7*b*a + 1/3*x^3*a^2

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Sympy [A]  time = 0.084343, size = 26, normalized size = 0.87 \[ \frac{a^{2} x^{3}}{3} + \frac{2 a b x^{7}}{7} + \frac{b^{2} x^{11}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b*x**4+a)**2,x)

[Out]

a**2*x**3/3 + 2*a*b*x**7/7 + b**2*x**11/11

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GIAC/XCAS [A]  time = 0.217998, size = 32, normalized size = 1.07 \[ \frac{1}{11} \, b^{2} x^{11} + \frac{2}{7} \, a b x^{7} + \frac{1}{3} \, a^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*b^2*x^11 + 2/7*a*b*x^7 + 1/3*a^2*x^3