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]

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

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 35, normalized size = 1.00 \[ \frac {a x^2}{2}-\frac {x^6}{6}+\frac {4 x^5}{5}-2 x^4+\frac {8 x^3}{3} \]

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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fricas [A]  time = 0.40, size = 27, normalized size = 0.77 \[ -\frac {1}{6} x^{6} + \frac {4}{5} x^{5} - 2 x^{4} + \frac {8}{3} x^{3} + \frac {1}{2} x^{2} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-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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giac [A]  time = 0.36, size = 27, normalized size = 0.77 \[ -\frac {1}{6} \, x^{6} + \frac {4}{5} \, x^{5} - 2 \, x^{4} + \frac {1}{2} \, a x^{2} + \frac {8}{3} \, x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-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

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maple [A]  time = 0.00, size = 28, normalized size = 0.80 \[ -\frac {1}{6} x^{6}+\frac {4}{5} x^{5}-2 x^{4}+\frac {1}{2} a \,x^{2}+\frac {8}{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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 = 0.61, size = 27, normalized size = 0.77 \[ -\frac {1}{6} \, x^{6} + \frac {4}{5} \, x^{5} - 2 \, x^{4} + \frac {1}{2} \, a x^{2} + \frac {8}{3} \, x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-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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mupad [B]  time = 0.02, size = 27, normalized size = 0.77 \[ -\frac {x^6}{6}+\frac {4\,x^5}{5}-2\,x^4+\frac {8\,x^3}{3}+\frac {a\,x^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[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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sympy [A]  time = 0.06, 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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