3.58.17 \(\int \frac {-864+4752 x-5904 x^2+3176 x^3-810 x^4+81 x^5}{-2592+6480 x-6480 x^2+3240 x^3-810 x^4+81 x^5} \, dx\)

Optimal. Leaf size=28 \[ 1-5 e^{1+e^3}+x-\left (1-\frac {x}{3 (-2+x)}\right )^4 \]

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 46, normalized size of antiderivative = 1.64, number of steps used = 2, number of rules used = 1, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {2074} \begin {gather*} x-\frac {64}{81 (2-x)}-\frac {32}{27 (2-x)^2}-\frac {64}{81 (2-x)^3}-\frac {16}{81 (2-x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-864 + 4752*x - 5904*x^2 + 3176*x^3 - 810*x^4 + 81*x^5)/(-2592 + 6480*x - 6480*x^2 + 3240*x^3 - 810*x^4 +
81*x^5),x]

[Out]

-16/(81*(2 - x)^4) - 64/(81*(2 - x)^3) - 32/(27*(2 - x)^2) - 64/(81*(2 - x)) + x

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {64}{81 (-2+x)^5}-\frac {64}{27 (-2+x)^4}+\frac {64}{27 (-2+x)^3}-\frac {64}{81 (-2+x)^2}\right ) \, dx\\ &=-\frac {16}{81 (2-x)^4}-\frac {64}{81 (2-x)^3}-\frac {32}{27 (2-x)^2}-\frac {64}{81 (2-x)}+x\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 38, normalized size = 1.36 \begin {gather*} \frac {1}{81} \left (-\frac {16}{(-2+x)^4}+\frac {64}{(-2+x)^3}-\frac {96}{(-2+x)^2}+\frac {64}{-2+x}+81 (-2+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-864 + 4752*x - 5904*x^2 + 3176*x^3 - 810*x^4 + 81*x^5)/(-2592 + 6480*x - 6480*x^2 + 3240*x^3 - 810
*x^4 + 81*x^5),x]

[Out]

(-16/(-2 + x)^4 + 64/(-2 + x)^3 - 96/(-2 + x)^2 + 64/(-2 + x) + 81*(-2 + x))/81

________________________________________________________________________________________

fricas [B]  time = 0.68, size = 47, normalized size = 1.68 \begin {gather*} \frac {81 \, x^{5} - 648 \, x^{4} + 2008 \, x^{3} - 3072 \, x^{2} + 2512 \, x - 1040}{81 \, {\left (x^{4} - 8 \, x^{3} + 24 \, x^{2} - 32 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((81*x^5-810*x^4+3176*x^3-5904*x^2+4752*x-864)/(81*x^5-810*x^4+3240*x^3-6480*x^2+6480*x-2592),x, algo
rithm="fricas")

[Out]

1/81*(81*x^5 - 648*x^4 + 2008*x^3 - 3072*x^2 + 2512*x - 1040)/(x^4 - 8*x^3 + 24*x^2 - 32*x + 16)

________________________________________________________________________________________

giac [A]  time = 0.14, size = 24, normalized size = 0.86 \begin {gather*} x + \frac {16 \, {\left (4 \, x^{3} - 30 \, x^{2} + 76 \, x - 65\right )}}{81 \, {\left (x - 2\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((81*x^5-810*x^4+3176*x^3-5904*x^2+4752*x-864)/(81*x^5-810*x^4+3240*x^3-6480*x^2+6480*x-2592),x, algo
rithm="giac")

[Out]

x + 16/81*(4*x^3 - 30*x^2 + 76*x - 65)/(x - 2)^4

________________________________________________________________________________________

maple [A]  time = 0.04, size = 25, normalized size = 0.89

method	result	size
norman	\(\frac {x^{5}-\frac {18224}{81} x -\frac {3176}{81} x^{3}+\frac {4160}{27} x^{2}+\frac {9328}{81}}{\left (x -2\right )^{4}}\)	\(25\)
default	\(x +\frac {64}{81 \left (x -2\right )}+\frac {64}{81 \left (x -2\right )^{3}}-\frac {16}{81 \left (x -2\right )^{4}}-\frac {32}{27 \left (x -2\right )^{2}}\)	\(31\)
risch	\(x +\frac {\frac {64}{81} x^{3}-\frac {160}{27} x^{2}+\frac {1216}{81} x -\frac {1040}{81}}{x^{4}-8 x^{3}+24 x^{2}-32 x +16}\)	\(39\)
gosper	\(\frac {81 x^{5}-3176 x^{3}+12480 x^{2}-18224 x +9328}{81 x^{4}-648 x^{3}+1944 x^{2}-2592 x +1296}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((81*x^5-810*x^4+3176*x^3-5904*x^2+4752*x-864)/(81*x^5-810*x^4+3240*x^3-6480*x^2+6480*x-2592),x,method=_RET
URNVERBOSE)

[Out]

(x^5-18224/81*x-3176/81*x^3+4160/27*x^2+9328/81)/(x-2)^4

________________________________________________________________________________________

maxima [A]  time = 0.35, size = 39, normalized size = 1.39 \begin {gather*} x + \frac {16 \, {\left (4 \, x^{3} - 30 \, x^{2} + 76 \, x - 65\right )}}{81 \, {\left (x^{4} - 8 \, x^{3} + 24 \, x^{2} - 32 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((81*x^5-810*x^4+3176*x^3-5904*x^2+4752*x-864)/(81*x^5-810*x^4+3240*x^3-6480*x^2+6480*x-2592),x, algo
rithm="maxima")

[Out]

x + 16/81*(4*x^3 - 30*x^2 + 76*x - 65)/(x^4 - 8*x^3 + 24*x^2 - 32*x + 16)

________________________________________________________________________________________

mupad [B]  time = 3.69, size = 23, normalized size = 0.82 \begin {gather*} x+\frac {\frac {64\,x^3}{81}-\frac {160\,x^2}{27}+\frac {1216\,x}{81}-\frac {1040}{81}}{{\left (x-2\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4752*x - 5904*x^2 + 3176*x^3 - 810*x^4 + 81*x^5 - 864)/(6480*x - 6480*x^2 + 3240*x^3 - 810*x^4 + 81*x^5 -
2592),x)

[Out]

x + ((1216*x)/81 - (160*x^2)/27 + (64*x^3)/81 - 1040/81)/(x - 2)^4

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 36, normalized size = 1.29 \begin {gather*} x + \frac {64 x^{3} - 480 x^{2} + 1216 x - 1040}{81 x^{4} - 648 x^{3} + 1944 x^{2} - 2592 x + 1296} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((81*x**5-810*x**4+3176*x**3-5904*x**2+4752*x-864)/(81*x**5-810*x**4+3240*x**3-6480*x**2+6480*x-2592)
,x)

[Out]

x + (64*x**3 - 480*x**2 + 1216*x - 1040)/(81*x**4 - 648*x**3 + 1944*x**2 - 2592*x + 1296)

________________________________________________________________________________________