3.26.68 \(\int \frac {663552 x-2097152 x^5}{6561+41472 x^2+107008 x^4+65536 e^{10} x^4+131072 x^6+65536 x^8+e^5 (41472 x^2+131072 x^4+131072 x^6)} \, dx\) [2568]

3.26.68.1 Optimal result

Integrand size = 62, antiderivative size = 19 \[ \int \frac {663552 x-2097152 x^5}{6561+41472 x^2+107008 x^4+65536 e^{10} x^4+131072 x^6+65536 x^8+e^5 \left (41472 x^2+131072 x^4+131072 x^6\right )} \, dx=\frac {16}{1+e^5+\frac {81}{256 x^2}+x^2} \]

output

16/(81/256/x^2+1+x^2+exp(5))
 

3.26.68.2 Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {663552 x-2097152 x^5}{6561+41472 x^2+107008 x^4+65536 e^{10} x^4+131072 x^6+65536 x^8+e^5 \left (41472 x^2+131072 x^4+131072 x^6\right )} \, dx=\frac {8192 x^2}{162+512 \left (1+e^5\right ) x^2+512 x^4} \]

input

Integrate[(663552*x - 2097152*x^5)/(6561 + 41472*x^2 + 107008*x^4 + 65536* 
E^10*x^4 + 131072*x^6 + 65536*x^8 + E^5*(41472*x^2 + 131072*x^4 + 131072*x 
^6)),x]
 

output

(8192*x^2)/(162 + 512*(1 + E^5)*x^2 + 512*x^4)
 

3.26.68.3 Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {6, 2027, 2460, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {663552 x-2097152 x^5}{65536 x^8+131072 x^6+65536 e^{10} x^4+107008 x^4+41472 x^2+e^5 \left (131072 x^6+131072 x^4+41472 x^2\right )+6561} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {663552 x-2097152 x^5}{65536 x^8+131072 x^6+\left (107008+65536 e^{10}\right ) x^4+41472 x^2+e^5 \left (131072 x^6+131072 x^4+41472 x^2\right )+6561}dx\)

\(\Big \downarrow \) 2027

\(\displaystyle \int \frac {x \left (663552-2097152 x^4\right )}{65536 x^8+131072 x^6+\left (107008+65536 e^{10}\right ) x^4+41472 x^2+e^5 \left (131072 x^6+131072 x^4+41472 x^2\right )+6561}dx\)

\(\Big \downarrow \) 2460

\(\displaystyle \int \left (\frac {8192 x}{-256 x^4-256 \left (1+e^5\right ) x^2-81}+\frac {16384 \left (128 \left (1+e^5\right ) x^2+81\right ) x}{\left (256 x^4+256 \left (1+e^5\right ) x^2+81\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4096 x^2}{256 x^4+256 \left (1+e^5\right ) x^2+81}\)

input

Int[(663552*x - 2097152*x^5)/(6561 + 41472*x^2 + 107008*x^4 + 65536*E^10*x 
^4 + 131072*x^6 + 65536*x^8 + E^5*(41472*x^2 + 131072*x^4 + 131072*x^6)),x 
]
 

output

(4096*x^2)/(81 + 256*(1 + E^5)*x^2 + 256*x^4)
 

3.26.68.3.1 Defintions of rubi rules used

rule 6

Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v 
+ (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] &&  !FreeQ[Fx, x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2027

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ 
(p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & 
& PosQ[s - r] &&  !(EqQ[p, 1] && EqQ[u, 1])
 

rule 2460

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px /. x -> Sqrt[x]]}, 
Int[ExpandIntegrand[u*(Qx /. x -> x^2)^p, x], x] /;  !SumQ[NonfreeFactors[Q 
x, x]]] /; PolyQ[Px, x^2] && GtQ[Expon[Px, x], 2] &&  !BinomialQ[Px, x] && 
 !TrinomialQ[Px, x] && ILtQ[p, 0] && RationalFunctionQ[u, x]
 

3.26.68.4 Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16

method	result	size
risch	\(\frac {16 x^{2}}{x^{4}+x^{2} {\mathrm e}^{5}+x^{2}+\frac {81}{256}}\)	\(22\)
gosper	\(\frac {4096 x^{2}}{256 x^{4}+256 x^{2} {\mathrm e}^{5}+256 x^{2}+81}\)	\(27\)
norman	\(\frac {4096 x^{2}}{256 x^{4}+256 x^{2} {\mathrm e}^{5}+256 x^{2}+81}\)	\(27\)
parallelrisch	\(\frac {4096 x^{2}}{256 x^{4}+256 x^{2} {\mathrm e}^{5}+256 x^{2}+81}\)	\(27\)
default	\(8 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (6561+65536 \textit {\_Z}^{4}+\left (131072 \,{\mathrm e}^{5}+131072\right ) \textit {\_Z}^{3}+\left (65536 \,{\mathrm e}^{10}+131072 \,{\mathrm e}^{5}+107008\right ) \textit {\_Z}^{2}+\left (41472 \,{\mathrm e}^{5}+41472\right ) \textit {\_Z} \right )}{\sum }\frac {\left (-256 \textit {\_R}^{2}+81\right ) \ln \left (x^{2}-\textit {\_R} \right )}{81+512 \textit {\_R}^{3}+768 \textit {\_R}^{2} {\mathrm e}^{5}+256 \textit {\_R} \,{\mathrm e}^{10}+768 \textit {\_R}^{2}+512 \textit {\_R} \,{\mathrm e}^{5}+418 \textit {\_R} +81 \,{\mathrm e}^{5}}\right )\)	\(100\)

input

int((-2097152*x^5+663552*x)/(65536*x^4*exp(5)^2+(131072*x^6+131072*x^4+414 
72*x^2)*exp(5)+65536*x^8+131072*x^6+107008*x^4+41472*x^2+6561),x,method=_R 
ETURNVERBOSE)
 

output

16*x^2/(x^4+x^2*exp(5)+x^2+81/256)
 

3.26.68.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.37 \[ \int \frac {663552 x-2097152 x^5}{6561+41472 x^2+107008 x^4+65536 e^{10} x^4+131072 x^6+65536 x^8+e^5 \left (41472 x^2+131072 x^4+131072 x^6\right )} \, dx=\frac {4096 \, x^{2}}{256 \, x^{4} + 256 \, x^{2} e^{5} + 256 \, x^{2} + 81} \]

input

integrate((-2097152*x^5+663552*x)/(65536*x^4*exp(5)^2+(131072*x^6+131072*x 
^4+41472*x^2)*exp(5)+65536*x^8+131072*x^6+107008*x^4+41472*x^2+6561),x, al 
gorithm=\
 

output

4096*x^2/(256*x^4 + 256*x^2*e^5 + 256*x^2 + 81)
 

3.26.68.6 Sympy [A] (verification not implemented)

Time = 0.63 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {663552 x-2097152 x^5}{6561+41472 x^2+107008 x^4+65536 e^{10} x^4+131072 x^6+65536 x^8+e^5 \left (41472 x^2+131072 x^4+131072 x^6\right )} \, dx=\frac {4096 x^{2}}{256 x^{4} + x^{2} \cdot \left (256 + 256 e^{5}\right ) + 81} \]

input

integrate((-2097152*x**5+663552*x)/(65536*x**4*exp(5)**2+(131072*x**6+1310 
72*x**4+41472*x**2)*exp(5)+65536*x**8+131072*x**6+107008*x**4+41472*x**2+6 
561),x)
 

output

4096*x**2/(256*x**4 + x**2*(256 + 256*exp(5)) + 81)
 

3.26.68.7 Maxima [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {663552 x-2097152 x^5}{6561+41472 x^2+107008 x^4+65536 e^{10} x^4+131072 x^6+65536 x^8+e^5 \left (41472 x^2+131072 x^4+131072 x^6\right )} \, dx=\frac {4096 \, x^{2}}{256 \, x^{4} + 256 \, x^{2} {\left (e^{5} + 1\right )} + 81} \]

input

integrate((-2097152*x^5+663552*x)/(65536*x^4*exp(5)^2+(131072*x^6+131072*x 
^4+41472*x^2)*exp(5)+65536*x^8+131072*x^6+107008*x^4+41472*x^2+6561),x, al 
gorithm=\
 

output

4096*x^2/(256*x^4 + 256*x^2*(e^5 + 1) + 81)
 

3.26.68.8 Giac [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.37 \[ \int \frac {663552 x-2097152 x^5}{6561+41472 x^2+107008 x^4+65536 e^{10} x^4+131072 x^6+65536 x^8+e^5 \left (41472 x^2+131072 x^4+131072 x^6\right )} \, dx=\frac {4096 \, x^{2}}{256 \, x^{4} + 256 \, x^{2} e^{5} + 256 \, x^{2} + 81} \]

input

integrate((-2097152*x^5+663552*x)/(65536*x^4*exp(5)^2+(131072*x^6+131072*x 
^4+41472*x^2)*exp(5)+65536*x^8+131072*x^6+107008*x^4+41472*x^2+6561),x, al 
gorithm=\
 

output

4096*x^2/(256*x^4 + 256*x^2*e^5 + 256*x^2 + 81)
 

3.26.68.9 Mupad [B] (verification not implemented)

Time = 11.59 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {663552 x-2097152 x^5}{6561+41472 x^2+107008 x^4+65536 e^{10} x^4+131072 x^6+65536 x^8+e^5 \left (41472 x^2+131072 x^4+131072 x^6\right )} \, dx=\frac {4096\,x^2}{256\,x^4+\left (256\,{\mathrm {e}}^5+256\right )\,x^2+81} \]

input

int((663552*x - 2097152*x^5)/(65536*x^4*exp(10) + exp(5)*(41472*x^2 + 1310 
72*x^4 + 131072*x^6) + 41472*x^2 + 107008*x^4 + 131072*x^6 + 65536*x^8 + 6 
561),x)
 

output

(4096*x^2)/(x^2*(256*exp(5) + 256) + 256*x^4 + 81)