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

Optimal. Leaf size=16 \[ \frac{\left (a+b x^2\right )^3}{6 b} \]

[Out]

(a + b*x^2)^3/(6*b)

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Rubi [A]  time = 0.0122781, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\left (a+b x^2\right )^3}{6 b} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x^2)^2,x]

[Out]

(a + b*x^2)^3/(6*b)

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Rubi in Sympy [A]  time = 2.16477, size = 10, normalized size = 0.62 \[ \frac{\left (a + b x^{2}\right )^{3}}{6 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x**2)**3/(6*b)

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Mathematica [A]  time = 0.0037182, size = 16, normalized size = 1. \[ \frac{\left (a+b x^2\right )^3}{6 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x^2)^2,x]

[Out]

(a + b*x^2)^3/(6*b)

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Maple [A]  time = 0., size = 25, normalized size = 1.6 \[{\frac{{b}^{2}{x}^{6}}{6}}+{\frac{ab{x}^{4}}{2}}+{\frac{{a}^{2}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^2+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.32701, size = 19, normalized size = 1.19 \[ \frac{{\left (b x^{2} + a\right )}^{3}}{6 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(b*x^2 + a)^3/b

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Fricas [A]  time = 0.193589, size = 1, normalized size = 0.06 \[ \frac{1}{6} x^{6} b^{2} + \frac{1}{2} x^{4} b a + \frac{1}{2} x^{2} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*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 = 0.085306, size = 24, normalized size = 1.5 \[ \frac{a^{2} x^{2}}{2} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*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 = 0.207605, size = 19, normalized size = 1.19 \[ \frac{{\left (b x^{2} + a\right )}^{3}}{6 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(b*x^2 + a)^3/b