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

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 = 0.0099643, 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, Rules used = {14} \[ \frac{a x^2}{2}-\frac{x^6}{6}+\frac{4 x^5}{5}-2 x^4+\frac{8 x^3}{3} \]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int x \left (a+8 x-8 x^2+4 x^3-x^4\right ) \, dx &=\int \left (a x+8 x^2-8 x^3+4 x^4-x^5\right ) \, dx\\ &=\frac{a x^2}{2}+\frac{8 x^3}{3}-2 x^4+\frac{4 x^5}{5}-\frac{x^6}{6}\\ \end{align*}

Mathematica [A]  time = 0.0013309, 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 \begin{align*}{\frac{a{x}^{2}}{2}}+{\frac{8\,{x}^{3}}{3}}-2\,{x}^{4}+{\frac{4\,{x}^{5}}{5}}-{\frac{{x}^{6}}{6}} \end{align*}

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 = 1.08354, size = 36, normalized size = 1.03 \begin{align*} -\frac{1}{6} \, x^{6} + \frac{4}{5} \, x^{5} - 2 \, x^{4} + \frac{1}{2} \, a x^{2} + \frac{8}{3} \, x^{3} \end{align*}

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, 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 = 1.2997, size = 68, normalized size = 1.94 \begin{align*} -\frac{1}{6} x^{6} + \frac{4}{5} x^{5} - 2 x^{4} + \frac{8}{3} x^{3} + \frac{1}{2} x^{2} a \end{align*}

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, 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 = 0.057822, size = 29, normalized size = 0.83 \begin{align*} \frac{a x^{2}}{2} - \frac{x^{6}}{6} + \frac{4 x^{5}}{5} - 2 x^{4} + \frac{8 x^{3}}{3} \end{align*}

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 [A]  time = 1.13136, size = 36, normalized size = 1.03 \begin{align*} -\frac{1}{6} \, x^{6} + \frac{4}{5} \, x^{5} - 2 \, x^{4} + \frac{1}{2} \, a x^{2} + \frac{8}{3} \, x^{3} \end{align*}

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, algorithm="giac")

[Out]

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