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

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

[Out]

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

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Rubi [A]  time = 0.0245305, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{a^2 x^2}{2}+\frac{1}{2} a b x^4+\frac{b^2 x^6}{6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.91427, size = 10, normalized size = 0.33 \[ \frac{\left (a + b x^{2}\right )^{3}}{6 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x**2)**3/(6*b)

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Mathematica [A]  time = 0.00339342, size = 16, normalized size = 0.53 \[ \frac{\left (a+b x^2\right )^3}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x^2)^3/(6*b)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.677782, size = 32, normalized size = 1.07 \[ \frac{1}{6} \, b^{2} x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{2} \, a^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.24067, size = 1, normalized size = 0.03 \[ \frac{1}{6} x^{6} b^{2} + \frac{1}{2} x^{4} b a + \frac{1}{2} x^{2} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.074715, size = 24, normalized size = 0.8 \[ \frac{a^{2} x^{2}}{2} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.267847, size = 32, normalized size = 1.07 \[ \frac{1}{6} \, b^{2} x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{2} \, a^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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