3.126 \(\int x \left (a+8 x-8 x^2+4 x^3-x^4\right ) \, dx\)

Optimal. Leaf size=35 \[ \frac{a x^2}{2}-\frac{x^6}{6}+\frac{4 x^5}{5}-2 x^4+\frac{8 x^3}{3} \]

[Out]

(a*x^2)/2 + (8*x^3)/3 - 2*x^4 + (4*x^5)/5 - x^6/6

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Rubi [A]  time = 0.0268965, antiderivative size = 35, 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 \[ \frac{a x^2}{2}-\frac{x^6}{6}+\frac{4 x^5}{5}-2 x^4+\frac{8 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + 8*x - 8*x^2 + 4*x^3 - x^4),x]

[Out]

(a*x^2)/2 + (8*x^3)/3 - 2*x^4 + (4*x^5)/5 - x^6/6

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x \left (a - x^{4} + 4 x^{3} - 8 x^{2} + 8 x\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(-x**4+4*x**3-8*x**2+a+8*x),x)

[Out]

Integral(x*(a - x**4 + 4*x**3 - 8*x**2 + 8*x), x)

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Mathematica [A]  time = 0.00213301, size = 35, normalized size = 1. \[ \frac{a x^2}{2}-\frac{x^6}{6}+\frac{4 x^5}{5}-2 x^4+\frac{8 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + 8*x - 8*x^2 + 4*x^3 - x^4),x]

[Out]

(a*x^2)/2 + (8*x^3)/3 - 2*x^4 + (4*x^5)/5 - x^6/6

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Maple [A]  time = 0.001, size = 28, normalized size = 0.8 \[{\frac{a{x}^{2}}{2}}+{\frac{8\,{x}^{3}}{3}}-2\,{x}^{4}+{\frac{4\,{x}^{5}}{5}}-{\frac{{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(-x^4+4*x^3-8*x^2+a+8*x),x)

[Out]

1/2*a*x^2+8/3*x^3-2*x^4+4/5*x^5-1/6*x^6

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Maxima [A]  time = 0.808736, size = 36, normalized size = 1.03 \[ -\frac{1}{6} \, x^{6} + \frac{4}{5} \, x^{5} - 2 \, x^{4} + \frac{1}{2} \, a x^{2} + \frac{8}{3} \, x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^4 - 4*x^3 + 8*x^2 - a - 8*x)*x,x, algorithm="maxima")

[Out]

-1/6*x^6 + 4/5*x^5 - 2*x^4 + 1/2*a*x^2 + 8/3*x^3

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Fricas [A]  time = 0.2392, size = 1, normalized size = 0.03 \[ -\frac{1}{6} x^{6} + \frac{4}{5} x^{5} - 2 x^{4} + \frac{8}{3} x^{3} + \frac{1}{2} x^{2} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^4 - 4*x^3 + 8*x^2 - a - 8*x)*x,x, algorithm="fricas")

[Out]

-1/6*x^6 + 4/5*x^5 - 2*x^4 + 8/3*x^3 + 1/2*x^2*a

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Sympy [A]  time = 0.064578, size = 29, normalized size = 0.83 \[ \frac{a x^{2}}{2} - \frac{x^{6}}{6} + \frac{4 x^{5}}{5} - 2 x^{4} + \frac{8 x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(-x**4+4*x**3-8*x**2+a+8*x),x)

[Out]

a*x**2/2 - x**6/6 + 4*x**5/5 - 2*x**4 + 8*x**3/3

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GIAC/XCAS [A]  time = 0.261089, size = 36, normalized size = 1.03 \[ -\frac{1}{6} \, x^{6} + \frac{4}{5} \, x^{5} - 2 \, x^{4} + \frac{1}{2} \, a x^{2} + \frac{8}{3} \, x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^4 - 4*x^3 + 8*x^2 - a - 8*x)*x,x, algorithm="giac")

[Out]

-1/6*x^6 + 4/5*x^5 - 2*x^4 + 1/2*a*x^2 + 8/3*x^3