\(\int \frac {1}{3} (8+6 x-27 x^2) \, dx\) [3246]

   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 = 14, antiderivative size = 23 \[ \int \frac {1}{3} \left (8+6 x-27 x^2\right ) \, dx=-1-3 \left (-3+\left (\frac {1}{3}-x\right )^2\right ) x+(-6-x) x \]

[Out]

(-x-6)*x-x*(3*(1/3-x)^2-9)-1

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {12} \[ \int \frac {1}{3} \left (8+6 x-27 x^2\right ) \, dx=-3 x^3+x^2+\frac {8 x}{3} \]

[In]

Int[(8 + 6*x - 27*x^2)/3,x]

[Out]

(8*x)/3 + x^2 - 3*x^3

Rule 12

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \left (8+6 x-27 x^2\right ) \, dx \\ & = \frac {8 x}{3}+x^2-3 x^3 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61 \[ \int \frac {1}{3} \left (8+6 x-27 x^2\right ) \, dx=\frac {8 x}{3}+x^2-3 x^3 \]

[In]

Integrate[(8 + 6*x - 27*x^2)/3,x]

[Out]

(8*x)/3 + x^2 - 3*x^3

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.57

method result size
default \(-3 x^{3}+x^{2}+\frac {8}{3} x\) \(13\)
norman \(-3 x^{3}+x^{2}+\frac {8}{3} x\) \(13\)
risch \(-3 x^{3}+x^{2}+\frac {8}{3} x\) \(13\)
parallelrisch \(-3 x^{3}+x^{2}+\frac {8}{3} x\) \(13\)
parts \(-3 x^{3}+x^{2}+\frac {8}{3} x\) \(13\)
gosper \(-\frac {x \left (9 x^{2}-3 x -8\right )}{3}\) \(14\)

[In]

int(-9*x^2+2*x+8/3,x,method=_RETURNVERBOSE)

[Out]

-3*x^3+x^2+8/3*x

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52 \[ \int \frac {1}{3} \left (8+6 x-27 x^2\right ) \, dx=-3 \, x^{3} + x^{2} + \frac {8}{3} \, x \]

[In]

integrate(-9*x^2+2*x+8/3,x, algorithm="fricas")

[Out]

-3*x^3 + x^2 + 8/3*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52 \[ \int \frac {1}{3} \left (8+6 x-27 x^2\right ) \, dx=- 3 x^{3} + x^{2} + \frac {8 x}{3} \]

[In]

integrate(-9*x**2+2*x+8/3,x)

[Out]

-3*x**3 + x**2 + 8*x/3

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52 \[ \int \frac {1}{3} \left (8+6 x-27 x^2\right ) \, dx=-3 \, x^{3} + x^{2} + \frac {8}{3} \, x \]

[In]

integrate(-9*x^2+2*x+8/3,x, algorithm="maxima")

[Out]

-3*x^3 + x^2 + 8/3*x

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52 \[ \int \frac {1}{3} \left (8+6 x-27 x^2\right ) \, dx=-3 \, x^{3} + x^{2} + \frac {8}{3} \, x \]

[In]

integrate(-9*x^2+2*x+8/3,x, algorithm="giac")

[Out]

-3*x^3 + x^2 + 8/3*x

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.57 \[ \int \frac {1}{3} \left (8+6 x-27 x^2\right ) \, dx=\frac {x\,\left (-9\,x^2+3\,x+8\right )}{3} \]

[In]

int(2*x - 9*x^2 + 8/3,x)

[Out]

(x*(3*x - 9*x^2 + 8))/3