3.67.76 \(\int \frac {1}{128} (128+81 x-54 x^2+8 x^3) \, dx\)

Optimal. Leaf size=23 \[ -(-2+e)^2+x+\frac {1}{64} \left (-\frac {9}{2}+x\right )^2 x^2 \]

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Rubi [A]  time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {12} \begin {gather*} \frac {x^4}{64}-\frac {9 x^3}{64}+\frac {81 x^2}{256}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(128 + 81*x - 54*x^2 + 8*x^3)/128,x]

[Out]

x + (81*x^2)/256 - (9*x^3)/64 + x^4/64

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}{128} \int \left (128+81 x-54 x^2+8 x^3\right ) \, dx\\ &=x+\frac {81 x^2}{256}-\frac {9 x^3}{64}+\frac {x^4}{64}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 25, normalized size = 1.09 \begin {gather*} \frac {1}{128} \left (128 x+\frac {81 x^2}{2}-18 x^3+2 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(128 + 81*x - 54*x^2 + 8*x^3)/128,x]

[Out]

(128*x + (81*x^2)/2 - 18*x^3 + 2*x^4)/128

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fricas [A]  time = 0.57, size = 17, normalized size = 0.74 \begin {gather*} \frac {1}{64} \, x^{4} - \frac {9}{64} \, x^{3} + \frac {81}{256} \, x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*x^3-27/64*x^2+81/128*x+1,x, algorithm="fricas")

[Out]

1/64*x^4 - 9/64*x^3 + 81/256*x^2 + x

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giac [A]  time = 0.13, size = 17, normalized size = 0.74 \begin {gather*} \frac {1}{64} \, x^{4} - \frac {9}{64} \, x^{3} + \frac {81}{256} \, x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*x^3-27/64*x^2+81/128*x+1,x, algorithm="giac")

[Out]

1/64*x^4 - 9/64*x^3 + 81/256*x^2 + x

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maple [A]  time = 0.01, size = 18, normalized size = 0.78

method	result	size
default	\(\frac {1}{64} x^{4}-\frac {9}{64} x^{3}+\frac {81}{256} x^{2}+x\)	\(18\)
norman	\(\frac {1}{64} x^{4}-\frac {9}{64} x^{3}+\frac {81}{256} x^{2}+x\)	\(18\)
risch	\(\frac {1}{64} x^{4}-\frac {9}{64} x^{3}+\frac {81}{256} x^{2}+x\)	\(18\)
gosper	\(\frac {x \left (4 x^{3}-36 x^{2}+81 x +256\right )}{256}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/16*x^3-27/64*x^2+81/128*x+1,x,method=_RETURNVERBOSE)

[Out]

1/64*x^4-9/64*x^3+81/256*x^2+x

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maxima [A]  time = 0.36, size = 17, normalized size = 0.74 \begin {gather*} \frac {1}{64} \, x^{4} - \frac {9}{64} \, x^{3} + \frac {81}{256} \, x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*x^3-27/64*x^2+81/128*x+1,x, algorithm="maxima")

[Out]

1/64*x^4 - 9/64*x^3 + 81/256*x^2 + x

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mupad [B]  time = 0.03, size = 17, normalized size = 0.74 \begin {gather*} \frac {x^4}{64}-\frac {9\,x^3}{64}+\frac {81\,x^2}{256}+x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((81*x)/128 - (27*x^2)/64 + x^3/16 + 1,x)

[Out]

x + (81*x^2)/256 - (9*x^3)/64 + x^4/64

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sympy [A]  time = 0.06, size = 19, normalized size = 0.83 \begin {gather*} \frac {x^{4}}{64} - \frac {9 x^{3}}{64} + \frac {81 x^{2}}{256} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*x**3-27/64*x**2+81/128*x+1,x)

[Out]

x**4/64 - 9*x**3/64 + 81*x**2/256 + x

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