\(\int \frac {1}{128} (128+81 x-54 x^2+8 x^3) \, dx\) [6676]

Optimal result

Integrand size = 19, antiderivative size = 23 \[ \int \frac {1}{128} \left (128+81 x-54 x^2+8 x^3\right ) \, dx=-(-2+e)^2+x+\frac {1}{64} \left (-\frac {9}{2}+x\right )^2 x^2 \]

[Out]

x+1/64*(x-9/2)^2*x^2-(exp(1)-2)^2

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {12} \[ \int \frac {1}{128} \left (128+81 x-54 x^2+8 x^3\right ) \, dx=\frac {x^4}{64}-\frac {9 x^3}{64}+\frac {81 x^2}{256}+x \]

[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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{128} \left (128+81 x-54 x^2+8 x^3\right ) \, dx=\frac {1}{128} \left (128 x+\frac {81 x^2}{2}-18 x^3+2 x^4\right ) \]

[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

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 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	\(\frac {1}{64} x^{4}-\frac {9}{64} x^{3}+\frac {81}{256} x^{2}+x\)	\(18\)
parallelrisch	\(\frac {1}{64} x^{4}-\frac {9}{64} x^{3}+\frac {81}{256} x^{2}+x\)	\(18\)
parts	\(\frac {1}{64} x^{4}-\frac {9}{64} x^{3}+\frac {81}{256} x^{2}+x\)	\(18\)
gosper	\(\frac {x \left (4 x^{3}-36 x^{2}+81 x +256\right )}{256}\)	\(19\)

[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

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {1}{128} \left (128+81 x-54 x^2+8 x^3\right ) \, dx=\frac {1}{64} \, x^{4} - \frac {9}{64} \, x^{3} + \frac {81}{256} \, x^{2} + x \]

[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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{128} \left (128+81 x-54 x^2+8 x^3\right ) \, dx=\frac {x^{4}}{64} - \frac {9 x^{3}}{64} + \frac {81 x^{2}}{256} + x \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {1}{128} \left (128+81 x-54 x^2+8 x^3\right ) \, dx=\frac {1}{64} \, x^{4} - \frac {9}{64} \, x^{3} + \frac {81}{256} \, x^{2} + x \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {1}{128} \left (128+81 x-54 x^2+8 x^3\right ) \, dx=\frac {1}{64} \, x^{4} - \frac {9}{64} \, x^{3} + \frac {81}{256} \, x^{2} + x \]

[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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {1}{128} \left (128+81 x-54 x^2+8 x^3\right ) \, dx=\frac {x^4}{64}-\frac {9\,x^3}{64}+\frac {81\,x^2}{256}+x \]

[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