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

Optimal. Leaf size=30 \[ \frac{a^2 x^4}{4}+\frac{1}{3} a b x^6+\frac{b^2 x^8}{8} \]

[Out]

(a^2*x^4)/4 + (a*b*x^6)/3 + (b^2*x^8)/8

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Rubi [A]  time = 0.0267797, 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 \[ \frac{a^2 x^4}{4}+\frac{1}{3} a b x^6+\frac{b^2 x^8}{8} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*x^4)/4 + (a*b*x^6)/3 + (b^2*x^8)/8

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Rubi in Sympy [A]  time = 14.1419, size = 27, normalized size = 0.9 \[ - \frac{a \left (a + b x^{2}\right )^{3}}{6 b^{2}} + \frac{\left (a + b x^{2}\right )^{4}}{8 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*(a + b*x**2)**3/(6*b**2) + (a + b*x**2)**4/(8*b**2)

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Mathematica [A]  time = 0.00253875, size = 30, normalized size = 1. \[ \frac{a^2 x^4}{4}+\frac{1}{3} a b x^6+\frac{b^2 x^8}{8} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*x^4)/4 + (a*b*x^6)/3 + (b^2*x^8)/8

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Maple [A]  time = 0.001, size = 25, normalized size = 0.8 \[{\frac{{a}^{2}{x}^{4}}{4}}+{\frac{ab{x}^{6}}{3}}+{\frac{{b}^{2}{x}^{8}}{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*a^2*x^4+1/3*a*b*x^6+1/8*b^2*x^8

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Maxima [A]  time = 0.69491, size = 32, normalized size = 1.07 \[ \frac{1}{8} \, b^{2} x^{8} + \frac{1}{3} \, a b x^{6} + \frac{1}{4} \, a^{2} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*b^2*x^8 + 1/3*a*b*x^6 + 1/4*a^2*x^4

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Fricas [A]  time = 0.237872, size = 1, normalized size = 0.03 \[ \frac{1}{8} x^{8} b^{2} + \frac{1}{3} x^{6} b a + \frac{1}{4} x^{4} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*x^8*b^2 + 1/3*x^6*b*a + 1/4*x^4*a^2

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Sympy [A]  time = 0.076611, size = 24, normalized size = 0.8 \[ \frac{a^{2} x^{4}}{4} + \frac{a b x^{6}}{3} + \frac{b^{2} x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**4/4 + a*b*x**6/3 + b**2*x**8/8

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GIAC/XCAS [A]  time = 0.268267, size = 32, normalized size = 1.07 \[ \frac{1}{8} \, b^{2} x^{8} + \frac{1}{3} \, a b x^{6} + \frac{1}{4} \, a^{2} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*b^2*x^8 + 1/3*a*b*x^6 + 1/4*a^2*x^4