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

Optimal. Leaf size=23 \[ -1-3 \left (-3+\left (\frac {1}{3}-x\right )^2\right ) x+(-6-x) x \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 14, normalized size of antiderivative = 0.61, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {12} \begin {gather*} -3 x^3+x^2+\frac {8 x}{3} \end {gather*}

Antiderivative was successfully verified.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 14, normalized size = 0.61 \begin {gather*} \frac {8 x}{3}+x^2-3 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 + 6*x - 27*x^2)/3,x]

[Out]

(8*x)/3 + x^2 - 3*x^3

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 12, normalized size = 0.52 \begin {gather*} -3 \, x^{3} + x^{2} + \frac {8}{3} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*x^3 + x^2 + 8/3*x

________________________________________________________________________________________

giac [A]  time = 0.18, size = 12, normalized size = 0.52 \begin {gather*} -3 \, x^{3} + x^{2} + \frac {8}{3} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*x^3 + x^2 + 8/3*x

________________________________________________________________________________________

maple [A]  time = 0.01, size = 13, normalized size = 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	\(-3 x^{3}+x^{2}+\frac {8}{3} x\)	\(13\)
gosper	\(-\frac {x \left (9 x^{2}-3 x -8\right )}{3}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.39, size = 12, normalized size = 0.52 \begin {gather*} -3 \, x^{3} + x^{2} + \frac {8}{3} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*x^3 + x^2 + 8/3*x

________________________________________________________________________________________

mupad [B]  time = 0.02, size = 13, normalized size = 0.57 \begin {gather*} \frac {x\,\left (-9\,x^2+3\,x+8\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x - 9*x^2 + 8/3,x)

[Out]

(x*(3*x - 9*x^2 + 8))/3

________________________________________________________________________________________

sympy [A]  time = 0.05, size = 12, normalized size = 0.52 \begin {gather*} - 3 x^{3} + x^{2} + \frac {8 x}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*x**3 + x**2 + 8*x/3

________________________________________________________________________________________