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

Optimal. Leaf size=30 \[ \frac{a^2 x^3}{3}+\frac{2}{5} a b x^5+\frac{b^2 x^7}{7} \]

[Out]

(a^2*x^3)/3 + (2*a*b*x^5)/5 + (b^2*x^7)/7

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Rubi [A]  time = 0.0253667, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{a^2 x^3}{3}+\frac{2}{5} a b x^5+\frac{b^2 x^7}{7} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*x^3)/3 + (2*a*b*x^5)/5 + (b^2*x^7)/7

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Rubi in Sympy [A]  time = 12.0253, size = 26, normalized size = 0.87 \[ \frac{a^{2} x^{3}}{3} + \frac{2 a b x^{5}}{5} + \frac{b^{2} x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**3/3 + 2*a*b*x**5/5 + b**2*x**7/7

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Mathematica [A]  time = 0.00170071, size = 30, normalized size = 1. \[ \frac{a^2 x^3}{3}+\frac{2}{5} a b x^5+\frac{b^2 x^7}{7} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*x^3)/3 + (2*a*b*x^5)/5 + (b^2*x^7)/7

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Maple [A]  time = 0.001, size = 25, normalized size = 0.8 \[{\frac{{a}^{2}{x}^{3}}{3}}+{\frac{2\,ab{x}^{5}}{5}}+{\frac{{b}^{2}{x}^{7}}{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a^2*x^3+2/5*a*b*x^5+1/7*b^2*x^7

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Maxima [A]  time = 0.683371, size = 32, normalized size = 1.07 \[ \frac{1}{7} \, b^{2} x^{7} + \frac{2}{5} \, a b x^{5} + \frac{1}{3} \, a^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*b^2*x^7 + 2/5*a*b*x^5 + 1/3*a^2*x^3

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Fricas [A]  time = 0.240613, size = 1, normalized size = 0.03 \[ \frac{1}{7} x^{7} b^{2} + \frac{2}{5} x^{5} b a + \frac{1}{3} x^{3} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*x^7*b^2 + 2/5*x^5*b*a + 1/3*x^3*a^2

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Sympy [A]  time = 0.071444, size = 26, normalized size = 0.87 \[ \frac{a^{2} x^{3}}{3} + \frac{2 a b x^{5}}{5} + \frac{b^{2} x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**3/3 + 2*a*b*x**5/5 + b**2*x**7/7

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GIAC/XCAS [A]  time = 0.267457, size = 32, normalized size = 1.07 \[ \frac{1}{7} \, b^{2} x^{7} + \frac{2}{5} \, a b x^{5} + \frac{1}{3} \, a^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*b^2*x^7 + 2/5*a*b*x^5 + 1/3*a^2*x^3