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

Optimal. Leaf size=16 \[ \frac{\left (a+b x^4\right )^3}{12 b} \]

[Out]

(a + b*x^4)^3/(12*b)

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Rubi [A]  time = 0.0110391, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{\left (a+b x^4\right )^3}{12 b} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x^4)^2,x]

[Out]

(a + b*x^4)^3/(12*b)

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Rubi in Sympy [A]  time = 2.17803, size = 10, normalized size = 0.62 \[ \frac{\left (a + b x^{4}\right )^{3}}{12 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x**4)**3/(12*b)

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Mathematica [A]  time = 0.00148984, size = 30, normalized size = 1.88 \[ \frac{a^2 x^4}{4}+\frac{1}{4} a b x^8+\frac{b^2 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + b*x^4)^2,x]

[Out]

(a^2*x^4)/4 + (a*b*x^8)/4 + (b^2*x^12)/12

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Maple [A]  time = 0.001, size = 25, normalized size = 1.6 \[{\frac{{b}^{2}{x}^{12}}{12}}+{\frac{ab{x}^{8}}{4}}+{\frac{{x}^{4}{a}^{2}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^4+a)^2,x)

[Out]

1/12*b^2*x^12+1/4*a*b*x^8+1/4*x^4*a^2

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Maxima [A]  time = 1.42769, size = 19, normalized size = 1.19 \[ \frac{{\left (b x^{4} + a\right )}^{3}}{12 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(b*x^4 + a)^3/b

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Fricas [A]  time = 0.201558, size = 1, normalized size = 0.06 \[ \frac{1}{12} x^{12} b^{2} + \frac{1}{4} x^{8} b a + \frac{1}{4} x^{4} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*x^12*b^2 + 1/4*x^8*b*a + 1/4*x^4*a^2

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Sympy [A]  time = 0.090557, size = 24, normalized size = 1.5 \[ \frac{a^{2} x^{4}}{4} + \frac{a b x^{8}}{4} + \frac{b^{2} x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x**4+a)**2,x)

[Out]

a**2*x**4/4 + a*b*x**8/4 + b**2*x**12/12

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GIAC/XCAS [A]  time = 0.220883, size = 19, normalized size = 1.19 \[ \frac{{\left (b x^{4} + a\right )}^{3}}{12 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(b*x^4 + a)^3/b