\(\int x (a+8 x-8 x^2+4 x^3-x^4) \, dx\) [126]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [A] (verified)
   Fricas [A] (verification not implemented)
   Sympy [A] (verification not implemented)
   Maxima [A] (verification not implemented)
   Giac [A] (verification not implemented)
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 22, antiderivative size = 35 \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right ) \, dx=\frac {a x^2}{2}+\frac {8 x^3}{3}-2 x^4+\frac {4 x^5}{5}-\frac {x^6}{6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {14} \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right ) \, dx=\frac {a x^2}{2}-\frac {x^6}{6}+\frac {4 x^5}{5}-2 x^4+\frac {8 x^3}{3} \]

[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*} \text {integral}& = \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] (verified)

Time = 0.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right ) \, dx=\frac {a x^2}{2}+\frac {8 x^3}{3}-2 x^4+\frac {4 x^5}{5}-\frac {x^6}{6} \]

[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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80

method result size
gosper \(\frac {1}{2} a \,x^{2}+\frac {8}{3} x^{3}-2 x^{4}+\frac {4}{5} x^{5}-\frac {1}{6} x^{6}\) \(28\)
default \(\frac {1}{2} a \,x^{2}+\frac {8}{3} x^{3}-2 x^{4}+\frac {4}{5} x^{5}-\frac {1}{6} x^{6}\) \(28\)
norman \(\frac {1}{2} a \,x^{2}+\frac {8}{3} x^{3}-2 x^{4}+\frac {4}{5} x^{5}-\frac {1}{6} x^{6}\) \(28\)
risch \(\frac {1}{2} a \,x^{2}+\frac {8}{3} x^{3}-2 x^{4}+\frac {4}{5} x^{5}-\frac {1}{6} x^{6}\) \(28\)
parallelrisch \(\frac {1}{2} a \,x^{2}+\frac {8}{3} x^{3}-2 x^{4}+\frac {4}{5} x^{5}-\frac {1}{6} x^{6}\) \(28\)

[In]

int(x*(-x^4+4*x^3-8*x^2+a+8*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right ) \, dx=-\frac {1}{6} \, x^{6} + \frac {4}{5} \, x^{5} - 2 \, x^{4} + \frac {1}{2} \, a x^{2} + \frac {8}{3} \, x^{3} \]

[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 + 1/2*a*x^2 + 8/3*x^3

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right ) \, dx=\frac {a x^{2}}{2} - \frac {x^{6}}{6} + \frac {4 x^{5}}{5} - 2 x^{4} + \frac {8 x^{3}}{3} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right ) \, dx=-\frac {1}{6} \, x^{6} + \frac {4}{5} \, x^{5} - 2 \, x^{4} + \frac {1}{2} \, a x^{2} + \frac {8}{3} \, x^{3} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right ) \, dx=-\frac {1}{6} \, x^{6} + \frac {4}{5} \, x^{5} - 2 \, x^{4} + \frac {1}{2} \, a x^{2} + \frac {8}{3} \, x^{3} \]

[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

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right ) \, dx=-\frac {x^6}{6}+\frac {4\,x^5}{5}-2\,x^4+\frac {8\,x^3}{3}+\frac {a\,x^2}{2} \]

[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