\(\int \frac {1}{120} (-4+x) (-3+x) (-2+x) (-1+x) x \, dx\) [283]

Optimal result

Integrand size = 17, antiderivative size = 36 \[ \int \frac {1}{120} (-4+x) (-3+x) (-2+x) (-1+x) x \, dx=\frac {x^2}{10}-\frac {5 x^3}{36}+\frac {7 x^4}{96}-\frac {x^5}{60}+\frac {x^6}{720} \]

[Out]

1/10*x^2-5/36*x^3+7/96*x^4-1/60*x^5+1/720*x^6

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {12, 1620} \[ \int \frac {1}{120} (-4+x) (-3+x) (-2+x) (-1+x) x \, dx=\frac {x^6}{720}-\frac {x^5}{60}+\frac {7 x^4}{96}-\frac {5 x^3}{36}+\frac {x^2}{10} \]

[In]

Int[((-4 + x)*(-3 + x)*(-2 + x)*(-1 + x)*x)/120,x]

[Out]

x^2/10 - (5*x^3)/36 + (7*x^4)/96 - x^5/60 + x^6/720

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.)*((g_.) + (h_.)*(x_)
)^(q_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x] /; FreeQ[
{a, b, c, d, e, f, g, h, m, n, p, q}, x] && PolyQ[Px, x] && IntegersQ[m, n]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{120} \int (-4+x) (-3+x) (-2+x) (-1+x) x \, dx \\ & = \frac {1}{120} \int \left (24 x-50 x^2+35 x^3-10 x^4+x^5\right ) \, dx \\ & = \frac {x^2}{10}-\frac {5 x^3}{36}+\frac {7 x^4}{96}-\frac {x^5}{60}+\frac {x^6}{720} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {1}{120} (-4+x) (-3+x) (-2+x) (-1+x) x \, dx=\frac {1}{120} \left (12 x^2-\frac {50 x^3}{3}+\frac {35 x^4}{4}-2 x^5+\frac {x^6}{6}\right ) \]

[In]

Integrate[((-4 + x)*(-3 + x)*(-2 + x)*(-1 + x)*x)/120,x]

[Out]

(12*x^2 - (50*x^3)/3 + (35*x^4)/4 - 2*x^5 + x^6/6)/120

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75

method	result	size
gosper	\(\frac {1}{10} x^{2}-\frac {5}{36} x^{3}+\frac {7}{96} x^{4}-\frac {1}{60} x^{5}+\frac {1}{720} x^{6}\)	\(27\)
default	\(\frac {1}{10} x^{2}-\frac {5}{36} x^{3}+\frac {7}{96} x^{4}-\frac {1}{60} x^{5}+\frac {1}{720} x^{6}\)	\(27\)
norman	\(\frac {1}{10} x^{2}-\frac {5}{36} x^{3}+\frac {7}{96} x^{4}-\frac {1}{60} x^{5}+\frac {1}{720} x^{6}\)	\(27\)
risch	\(\frac {1}{10} x^{2}-\frac {5}{36} x^{3}+\frac {7}{96} x^{4}-\frac {1}{60} x^{5}+\frac {1}{720} x^{6}\)	\(27\)
parallelrisch	\(\frac {1}{10} x^{2}-\frac {5}{36} x^{3}+\frac {7}{96} x^{4}-\frac {1}{60} x^{5}+\frac {1}{720} x^{6}\)	\(27\)

[In]

int(1/120*(x-4)*(-3+x)*(-2+x)*(-1+x)*x,x,method=_RETURNVERBOSE)

[Out]

1/10*x^2-5/36*x^3+7/96*x^4-1/60*x^5+1/720*x^6

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {1}{120} (-4+x) (-3+x) (-2+x) (-1+x) x \, dx=\frac {1}{720} \, x^{6} - \frac {1}{60} \, x^{5} + \frac {7}{96} \, x^{4} - \frac {5}{36} \, x^{3} + \frac {1}{10} \, x^{2} \]

[In]

integrate(1/120*(x-4)*(-3+x)*(-2+x)*(-1+x)*x,x, algorithm="fricas")

[Out]

1/720*x^6 - 1/60*x^5 + 7/96*x^4 - 5/36*x^3 + 1/10*x^2

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \frac {1}{120} (-4+x) (-3+x) (-2+x) (-1+x) x \, dx=\frac {x^{6}}{720} - \frac {x^{5}}{60} + \frac {7 x^{4}}{96} - \frac {5 x^{3}}{36} + \frac {x^{2}}{10} \]

[In]

integrate(1/120*(x-4)*(-3+x)*(-2+x)*(-1+x)*x,x)

[Out]

x**6/720 - x**5/60 + 7*x**4/96 - 5*x**3/36 + x**2/10

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {1}{120} (-4+x) (-3+x) (-2+x) (-1+x) x \, dx=\frac {1}{720} \, x^{6} - \frac {1}{60} \, x^{5} + \frac {7}{96} \, x^{4} - \frac {5}{36} \, x^{3} + \frac {1}{10} \, x^{2} \]

[In]

integrate(1/120*(x-4)*(-3+x)*(-2+x)*(-1+x)*x,x, algorithm="maxima")

[Out]

1/720*x^6 - 1/60*x^5 + 7/96*x^4 - 5/36*x^3 + 1/10*x^2

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.64 \[ \int \frac {1}{120} (-4+x) (-3+x) (-2+x) (-1+x) x \, dx=\frac {1}{720} \, {\left (x^{2} - 4 \, x\right )}^{3} + \frac {1}{160} \, {\left (x^{2} - 4 \, x\right )}^{2} \]

[In]

integrate(1/120*(x-4)*(-3+x)*(-2+x)*(-1+x)*x,x, algorithm="giac")

[Out]

1/720*(x^2 - 4*x)^3 + 1/160*(x^2 - 4*x)^2

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {1}{120} (-4+x) (-3+x) (-2+x) (-1+x) x \, dx=\frac {x^6}{720}-\frac {x^5}{60}+\frac {7\,x^4}{96}-\frac {5\,x^3}{36}+\frac {x^2}{10} \]

[In]

int((x*(x - 1)*(x - 2)*(x - 3)*(x - 4))/120,x)

[Out]

x^2/10 - (5*x^3)/36 + (7*x^4)/96 - x^5/60 + x^6/720