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

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

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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} $$\begin{array}{c}-3x^3+x^2+\frac{8x}{3}\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{1}{3}\int (8+6x-27x^2)dx\\ & =\frac{8x}{3}+x^2-3x^3\end{array}\end{array}$$

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

Antiderivative was successfully verified.

[In]

Integrate[(8 + 6*x - 27*x^2)/3,x]

[Out]

(8*x)/3 + x^2 - 3*x^3

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

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

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

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

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maple [A]  time = 0.01, size = 13, normalized size = 0.57

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

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

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

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

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

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

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

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

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