3.35.67 \(\int \frac {65536+256 e^2-98304 x^4+65536 x^6-12288 x^8+e (-8192+2048 x^4)+e^2 (-16384 x+24576 x^3-12288 x^5+2048 x^7+e (1024 x-512 x^3))}{e^6-24 e^4 x+192 e^2 x^2-512 x^3} \, dx\)

Optimal. Leaf size=32 \[ \frac {4 \left (\frac {e}{4}-\left (2-x^2\right )^2\right )^2}{\left (-\frac {e^2}{8}+x\right )^2} \]

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Rubi [B]  time = 0.20, antiderivative size = 153, normalized size of antiderivative = 4.78, number of steps used = 2, number of rules used = 1, integrand size = 91, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {2074} \begin {gather*} 4 x^6+e^2 x^5-\frac {1}{16} \left (512-3 e^4\right ) x^4-\frac {1}{32} e^2 \left (256-e^4\right ) x^3+\frac {\left (98304-2048 e-1536 e^4+5 e^8\right ) x^2}{1024}+\frac {e^2 \left (98304-2048 e-1024 e^4+3 e^8\right ) x}{4096}+\frac {e^2 \left (128-e^4\right ) \left (16384-1024 e-256 e^4+e^8\right )}{8192 \left (e^2-8 x\right )}+\frac {\left (16384-1024 e-256 e^4+e^8\right )^2}{65536 \left (e^2-8 x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(65536 + 256*E^2 - 98304*x^4 + 65536*x^6 - 12288*x^8 + E*(-8192 + 2048*x^4) + E^2*(-16384*x + 24576*x^3 -
12288*x^5 + 2048*x^7 + E*(1024*x - 512*x^3)))/(E^6 - 24*E^4*x + 192*E^2*x^2 - 512*x^3),x]

[Out]

(16384 - 1024*E - 256*E^4 + E^8)^2/(65536*(E^2 - 8*x)^2) + (E^2*(128 - E^4)*(16384 - 1024*E - 256*E^4 + E^8))/
(8192*(E^2 - 8*x)) + (E^2*(98304 - 2048*E - 1024*E^4 + 3*E^8)*x)/4096 + ((98304 - 2048*E - 1536*E^4 + 5*E^8)*x
^2)/1024 - (E^2*(256 - E^4)*x^3)/32 - ((512 - 3*E^4)*x^4)/16 + E^2*x^5 + 4*x^6

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 (\frac {e^2 \left (98304-2048 e-1024 e^4+3 e^8\right )}{4096}+\frac {\left (16384-1024 e-256 e^4+e^8\right )^2}{4096 \left (e^2-8 x\right )^3}+\frac {e^2 \left (128-e^4\right ) \left (16384-1024 e-256 e^4+e^8\right )}{1024 \left (e^2-8 x\right )^2}+\frac {1}{512} \left (98304-2048 e-1536 e^4+5 e^8\right ) x+\frac {3}{32} e^2 \left (-256+e^4\right ) x^2+\frac {1}{4} \left (-512+3 e^4\right ) x^3+5 e^2 x^4+24 x^5\right ) \, dx\\ &=\frac {\left (16384-1024 e-256 e^4+e^8\right )^2}{65536 \left (e^2-8 x\right )^2}+\frac {e^2 \left (128-e^4\right ) \left (16384-1024 e-256 e^4+e^8\right )}{8192 \left (e^2-8 x\right )}+\frac {e^2 \left (98304-2048 e-1024 e^4+3 e^8\right ) x}{4096}+\frac {\left (98304-2048 e-1536 e^4+5 e^8\right ) x^2}{1024}-\frac {1}{32} e^2 \left (256-e^4\right ) x^3-\frac {1}{16} \left (512-3 e^4\right ) x^4+e^2 x^5+4 x^6\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.06, size = 137, normalized size = 4.28 \begin {gather*} -\frac {-3072 e^9+7 e^{16}-2097152 e^3 x-2359296 e^6 x+49152 e^7 x+30720 e^{10} x-112 e^{14} x+262144 e^2 (-1+128 x)-65536 e^5 \left (-2+3 x^2\right )-24576 e^8 \left (-6+5 x^2\right )+64 e^{12} \left (-30+7 x^2\right )+1048576 e^4 \left (-2+9 x^2\right )+2097152 e \left (4+x^4\right )-4194304 \left (16+24 x^4-8 x^6+x^8\right )}{16384 \left (e^2-8 x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(65536 + 256*E^2 - 98304*x^4 + 65536*x^6 - 12288*x^8 + E*(-8192 + 2048*x^4) + E^2*(-16384*x + 24576*
x^3 - 12288*x^5 + 2048*x^7 + E*(1024*x - 512*x^3)))/(E^6 - 24*E^4*x + 192*E^2*x^2 - 512*x^3),x]

[Out]

-1/16384*(-3072*E^9 + 7*E^16 - 2097152*E^3*x - 2359296*E^6*x + 49152*E^7*x + 30720*E^10*x - 112*E^14*x + 26214
4*E^2*(-1 + 128*x) - 65536*E^5*(-2 + 3*x^2) - 24576*E^8*(-6 + 5*x^2) + 64*E^12*(-30 + 7*x^2) + 1048576*E^4*(-2
+ 9*x^2) + 2097152*E*(4 + x^4) - 4194304*(16 + 24*x^4 - 8*x^6 + x^8))/(E^2 - 8*x)^2

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fricas [B]  time = 0.59, size = 129, normalized size = 4.03 \begin {gather*} \frac {16777216 \, x^{8} - 134217728 \, x^{6} + 402653184 \, x^{4} + 112 \, x e^{14} - 64 \, {\left (7 \, x^{2} - 40\right )} e^{12} - 40960 \, x e^{10} + 32768 \, {\left (5 \, x^{2} - 9\right )} e^{8} - 98304 \, x e^{7} + 4718592 \, x e^{6} + 131072 \, {\left (3 \, x^{2} - 4\right )} e^{5} - 2097152 \, {\left (9 \, x^{2} - 4\right )} e^{4} + 8388608 \, x e^{3} - 1048576 \, {\left (128 \, x - 1\right )} e^{2} - 8388608 \, {\left (x^{4} + 4\right )} e - 7 \, e^{16} + 6144 \, e^{9} + 268435456}{65536 \, {\left (64 \, x^{2} - 16 \, x e^{2} + e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-512*x^3+1024*x)*exp(1)+2048*x^7-12288*x^5+24576*x^3-16384*x)*exp(2)+256*exp(1)^2+(2048*x^4-8192)
*exp(1)-12288*x^8+65536*x^6-98304*x^4+65536)/(exp(2)^3-24*x*exp(2)^2+192*x^2*exp(2)-512*x^3),x, algorithm="fri
cas")

[Out]

1/65536*(16777216*x^8 - 134217728*x^6 + 402653184*x^4 + 112*x*e^14 - 64*(7*x^2 - 40)*e^12 - 40960*x*e^10 + 327
68*(5*x^2 - 9)*e^8 - 98304*x*e^7 + 4718592*x*e^6 + 131072*(3*x^2 - 4)*e^5 - 2097152*(9*x^2 - 4)*e^4 + 8388608*
x*e^3 - 1048576*(128*x - 1)*e^2 - 8388608*(x^4 + 4)*e - 7*e^16 + 6144*e^9 + 268435456)/(64*x^2 - 16*x*e^2 + e^
4)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-512*x^3+1024*x)*exp(1)+2048*x^7-12288*x^5+24576*x^3-16384*x)*exp(2)+256*exp(1)^2+(2048*x^4-8192)
*exp(1)-12288*x^8+65536*x^6-98304*x^4+65536)/(exp(2)^3-24*x*exp(2)^2+192*x^2*exp(2)-512*x^3),x, algorithm="gia
c")

[Out]

Timed out

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maple [B]  time = 0.11, size = 69, normalized size = 2.16

method	result	size
norman	\(\frac {\left (6144-128 \,{\mathrm e}\right ) x^{4}+\left (128 \,{\mathrm e} \,{\mathrm e}^{2}-2048 \,{\mathrm e}^{2}\right ) x -2048 x^{6}+256 x^{8}+4096-8 \,{\mathrm e} \,{\mathrm e}^{4}+128 \,{\mathrm e}^{4}+16 \,{\mathrm e}^{2}-512 \,{\mathrm e}}{\left ({\mathrm e}^{2}-8 x \right )^{2}}\)	\(69\)
gosper	\(-\frac {8 \left (-32 x^{8}+256 x^{6}+16 x^{4} {\mathrm e}-768 x^{4}+{\mathrm e} \,{\mathrm e}^{4}-16 x \,{\mathrm e} \,{\mathrm e}^{2}-16 \,{\mathrm e}^{4}+256 \,{\mathrm e}^{2} x -2 \,{\mathrm e}^{2}+64 \,{\mathrm e}-512\right )}{{\mathrm e}^{4}-16 \,{\mathrm e}^{2} x +64 x^{2}}\)	\(79\)
risch	\(\frac {3 x \,{\mathrm e}^{10}}{4096}+\frac {5 x^{2} {\mathrm e}^{8}}{1024}+\frac {x^{3} {\mathrm e}^{6}}{32}+\frac {3 x^{4} {\mathrm e}^{4}}{16}-\frac {x \,{\mathrm e}^{6}}{4}+{\mathrm e}^{2} x^{5}-\frac {3 x^{2} {\mathrm e}^{4}}{2}+4 x^{6}-8 x^{3} {\mathrm e}^{2}-\frac {x \,{\mathrm e}^{3}}{2}-32 x^{4}+24 \,{\mathrm e}^{2} x -2 x^{2} {\mathrm e}+96 x^{2}+\frac {\frac {\left (4 \,{\mathrm e}^{14}-1536 \,{\mathrm e}^{10}-4096 \,{\mathrm e}^{7}+196608 \,{\mathrm e}^{6}+524288 \,{\mathrm e}^{3}-8388608 \,{\mathrm e}^{2}\right ) x}{4096}-\frac {7 \,{\mathrm e}^{16}}{65536}+\frac {5 \,{\mathrm e}^{12}}{128}+\frac {3 \,{\mathrm e}^{9}}{32}-\frac {9 \,{\mathrm e}^{8}}{2}-8 \,{\mathrm e}^{5}+128 \,{\mathrm e}^{4}+16 \,{\mathrm e}^{2}-512 \,{\mathrm e}+4096}{{\mathrm e}^{4}-16 \,{\mathrm e}^{2} x +64 x^{2}}\)	\(163\)
default	\(24 \,{\mathrm e}^{2} x -\frac {3 x \,{\mathrm e} \,{\mathrm e}^{2}}{2}+{\mathrm e}^{2} x^{5}+\frac {405 x \,{\mathrm e}^{10}}{4096}+\frac {x^{3} {\mathrm e}^{6}}{4}+\frac {81 x^{2} {\mathrm e}^{8}}{512}+4 x^{6}-32 x^{4}+96 x^{2}-\frac {171 x^{2} \left ({\mathrm e}^{4}\right )^{2}}{1024}-\frac {3 x^{2} {\mathrm e}^{4}}{2}-\frac {29 x \,{\mathrm e}^{6}}{8}-8 x^{3} {\mathrm e}^{2}+x \,{\mathrm e}^{3}-2 x^{2} {\mathrm e}+\frac {3 x^{4} {\mathrm e}^{4}}{16}-\frac {609 \,{\mathrm e}^{6} {\mathrm e}^{4} x}{4096}-\frac {7 x^{3} {\mathrm e}^{2} {\mathrm e}^{4}}{32}+\frac {7 x^{2} {\mathrm e}^{2} {\mathrm e}^{6}}{512}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (-{\mathrm e}^{6}+24 \textit {\_Z} \,{\mathrm e}^{4}-192 \textit {\_Z}^{2} {\mathrm e}^{2}+512 \textit {\_Z}^{3}\right )}{\sum }\frac {\left (-67108864 \,{\mathrm e}^{2} \textit {\_R} +4194304 \textit {\_R} \,{\mathrm e}^{3}+1572864 \textit {\_R} \,{\mathrm e}^{6}+32 \textit {\_R} \,{\mathrm e}^{14}-12288 \textit {\_R} \,{\mathrm e}^{10}-32768 \textit {\_R} \,{\mathrm e}^{7}+1048576 \,{\mathrm e}^{2}-33554432 \,{\mathrm e}-3 \,{\mathrm e}^{16}+1024 \,{\mathrm e}^{12}+2048 \,{\mathrm e}^{9}-98304 \,{\mathrm e}^{8}[phone]\right ) \ln \left (x -\textit {\_R} \right )}{{\mathrm e}^{4}-16 \,{\mathrm e}^{2} \textit {\_R} +64 \textit {\_R}^{2}}\right )}{98304}+\frac {207 \,{\mathrm e}^{2} {\mathrm e}^{8} x}{4096}+\frac {27 \,{\mathrm e}^{2} {\mathrm e}^{4} x}{8}\)	\(314\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-512*x^3+1024*x)*exp(1)+2048*x^7-12288*x^5+24576*x^3-16384*x)*exp(2)+256*exp(1)^2+(2048*x^4-8192)*exp(1
)-12288*x^8+65536*x^6-98304*x^4+65536)/(exp(2)^3-24*x*exp(2)^2+192*x^2*exp(2)-512*x^3),x,method=_RETURNVERBOSE
)

[Out]

((6144-128*exp(1))*x^4+(128*exp(1)*exp(2)-2048*exp(2))*x-2048*x^6+256*x^8+4096-8*exp(2)^2*exp(1)+128*exp(2)^2+
16*exp(1)^2-512*exp(1))/(exp(2)-8*x)^2

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maxima [B]  time = 0.50, size = 151, normalized size = 4.72 \begin {gather*} 4 \, x^{6} + x^{5} e^{2} + \frac {1}{16} \, x^{4} {\left (3 \, e^{4} - 512\right )} + \frac {1}{32} \, x^{3} {\left (e^{6} - 256 \, e^{2}\right )} + \frac {1}{1024} \, x^{2} {\left (5 \, e^{8} - 1536 \, e^{4} - 2048 \, e + 98304\right )} + \frac {1}{4096} \, x {\left (3 \, e^{10} - 1024 \, e^{6} - 2048 \, e^{3} + 98304 \, e^{2}\right )} + \frac {64 \, x {\left (e^{14} - 384 \, e^{10} - 1024 \, e^{7} + 49152 \, e^{6} + 131072 \, e^{3} - 2097152 \, e^{2}\right )} - 7 \, e^{16} + 2560 \, e^{12} + 6144 \, e^{9} - 294912 \, e^{8} - 524288 \, e^{5} + 8388608 \, e^{4} + 1048576 \, e^{2} - 33554432 \, e + 268435456}{65536 \, {\left (64 \, x^{2} - 16 \, x e^{2} + e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-512*x^3+1024*x)*exp(1)+2048*x^7-12288*x^5+24576*x^3-16384*x)*exp(2)+256*exp(1)^2+(2048*x^4-8192)
*exp(1)-12288*x^8+65536*x^6-98304*x^4+65536)/(exp(2)^3-24*x*exp(2)^2+192*x^2*exp(2)-512*x^3),x, algorithm="max
ima")

[Out]

4*x^6 + x^5*e^2 + 1/16*x^4*(3*e^4 - 512) + 1/32*x^3*(e^6 - 256*e^2) + 1/1024*x^2*(5*e^8 - 1536*e^4 - 2048*e +
98304) + 1/4096*x*(3*e^10 - 1024*e^6 - 2048*e^3 + 98304*e^2) + 1/65536*(64*x*(e^14 - 384*e^10 - 1024*e^7 + 491
52*e^6 + 131072*e^3 - 2097152*e^2) - 7*e^16 + 2560*e^12 + 6144*e^9 - 294912*e^8 - 524288*e^5 + 8388608*e^4 + 1
048576*e^2 - 33554432*e + 268435456)/(64*x^2 - 16*x*e^2 + e^4)

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mupad [B]  time = 2.19, size = 262, normalized size = 8.19 \begin {gather*} x^3\,\left (8\,{\mathrm {e}}^2-\frac {{\mathrm {e}}^6}{16}+\frac {{\mathrm {e}}^2\,\left (\frac {3\,{\mathrm {e}}^4}{4}-128\right )}{8}\right )+x^4\,\left (\frac {3\,{\mathrm {e}}^4}{16}-32\right )+x\,\left (\frac {3\,{\mathrm {e}}^2\,\left (\frac {5\,{\mathrm {e}}^8}{512}-4\,\mathrm {e}+\frac {3\,{\mathrm {e}}^2\,\left (24\,{\mathrm {e}}^2-\frac {3\,{\mathrm {e}}^6}{16}+\frac {3\,{\mathrm {e}}^2\,\left (\frac {3\,{\mathrm {e}}^4}{4}-128\right )}{8}\right )}{8}-\frac {3\,{\mathrm {e}}^4\,\left (\frac {3\,{\mathrm {e}}^4}{4}-128\right )}{64}+192\right )}{8}+{\mathrm {e}}^2\,\left (\mathrm {e}-48\right )-\frac {3\,{\mathrm {e}}^4\,\left (24\,{\mathrm {e}}^2-\frac {3\,{\mathrm {e}}^6}{16}+\frac {3\,{\mathrm {e}}^2\,\left (\frac {3\,{\mathrm {e}}^4}{4}-128\right )}{8}\right )}{64}+\frac {{\mathrm {e}}^6\,\left (\frac {3\,{\mathrm {e}}^4}{4}-128\right )}{512}\right )+x^5\,{\mathrm {e}}^2+x^2\,\left (\frac {5\,{\mathrm {e}}^8}{1024}-2\,\mathrm {e}+\frac {3\,{\mathrm {e}}^2\,\left (24\,{\mathrm {e}}^2-\frac {3\,{\mathrm {e}}^6}{16}+\frac {3\,{\mathrm {e}}^2\,\left (\frac {3\,{\mathrm {e}}^4}{4}-128\right )}{8}\right )}{16}-\frac {3\,{\mathrm {e}}^4\,\left (\frac {3\,{\mathrm {e}}^4}{4}-128\right )}{128}+96\right )-\frac {2097152\,\mathrm {e}-65536\,{\mathrm {e}}^2-524288\,{\mathrm {e}}^4+32768\,{\mathrm {e}}^5+18432\,{\mathrm {e}}^8-384\,{\mathrm {e}}^9-160\,{\mathrm {e}}^{12}+\frac {7\,{\mathrm {e}}^{16}}{16}+x\,\left (8388608\,{\mathrm {e}}^2-524288\,{\mathrm {e}}^3-196608\,{\mathrm {e}}^6+4096\,{\mathrm {e}}^7+1536\,{\mathrm {e}}^{10}-4\,{\mathrm {e}}^{14}\right )-16777216}{262144\,x^2-65536\,{\mathrm {e}}^2\,x+4096\,{\mathrm {e}}^4}+4\,x^6 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((256*exp(2) + exp(1)*(2048*x^4 - 8192) + exp(2)*(exp(1)*(1024*x - 512*x^3) - 16384*x + 24576*x^3 - 12288*x
^5 + 2048*x^7) - 98304*x^4 + 65536*x^6 - 12288*x^8 + 65536)/(exp(6) - 24*x*exp(4) + 192*x^2*exp(2) - 512*x^3),
x)

[Out]

x^3*(8*exp(2) - exp(6)/16 + (exp(2)*((3*exp(4))/4 - 128))/8) + x^4*((3*exp(4))/16 - 32) + x*((3*exp(2)*((5*exp
(8))/512 - 4*exp(1) + (3*exp(2)*(24*exp(2) - (3*exp(6))/16 + (3*exp(2)*((3*exp(4))/4 - 128))/8))/8 - (3*exp(4)
*((3*exp(4))/4 - 128))/64 + 192))/8 + exp(2)*(exp(1) - 48) - (3*exp(4)*(24*exp(2) - (3*exp(6))/16 + (3*exp(2)*
((3*exp(4))/4 - 128))/8))/64 + (exp(6)*((3*exp(4))/4 - 128))/512) + x^5*exp(2) + x^2*((5*exp(8))/1024 - 2*exp(
1) + (3*exp(2)*(24*exp(2) - (3*exp(6))/16 + (3*exp(2)*((3*exp(4))/4 - 128))/8))/16 - (3*exp(4)*((3*exp(4))/4 -
128))/128 + 96) - (2097152*exp(1) - 65536*exp(2) - 524288*exp(4) + 32768*exp(5) + 18432*exp(8) - 384*exp(9) -
160*exp(12) + (7*exp(16))/16 + x*(8388608*exp(2) - 524288*exp(3) - 196608*exp(6) + 4096*exp(7) + 1536*exp(10)
- 4*exp(14)) - 16777216)/(4096*exp(4) - 65536*x*exp(2) + 262144*x^2) + 4*x^6

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sympy [B]  time = 1.73, size = 175, normalized size = 5.47 \begin {gather*} 4 x^{6} + x^{5} e^{2} + x^{4} \left (-32 + \frac {3 e^{4}}{16}\right ) + x^{3} \left (- 8 e^{2} + \frac {e^{6}}{32}\right ) + x^{2} \left (- \frac {3 e^{4}}{2} - 2 e + \frac {5 e^{8}}{1024} + 96\right ) + x \left (- \frac {e^{6}}{4} - \frac {e^{3}}{2} + \frac {3 e^{10}}{4096} + 24 e^{2}\right ) + \frac {x \left (- 134217728 e^{2} - 24576 e^{10} - 65536 e^{7} + 64 e^{14} + 8388608 e^{3} + 3145728 e^{6}\right ) - 294912 e^{8} - 33554432 e - 524288 e^{5} - 7 e^{16} + 1048576 e^{2} + 6144 e^{9} + 268435456 + 2560 e^{12} + 8388608 e^{4}}{4194304 x^{2} - 1048576 x e^{2} + 65536 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-512*x**3+1024*x)*exp(1)+2048*x**7-12288*x**5+24576*x**3-16384*x)*exp(2)+256*exp(1)**2+(2048*x**4
-8192)*exp(1)-12288*x**8+65536*x**6-98304*x**4+65536)/(exp(2)**3-24*x*exp(2)**2+192*x**2*exp(2)-512*x**3),x)

[Out]

4*x**6 + x**5*exp(2) + x**4*(-32 + 3*exp(4)/16) + x**3*(-8*exp(2) + exp(6)/32) + x**2*(-3*exp(4)/2 - 2*E + 5*e
xp(8)/1024 + 96) + x*(-exp(6)/4 - exp(3)/2 + 3*exp(10)/4096 + 24*exp(2)) + (x*(-134217728*exp(2) - 24576*exp(1
0) - 65536*exp(7) + 64*exp(14) + 8388608*exp(3) + 3145728*exp(6)) - 294912*exp(8) - 33554432*E - 524288*exp(5)
- 7*exp(16) + 1048576*exp(2) + 6144*exp(9) + 268435456 + 2560*exp(12) + 8388608*exp(4))/(4194304*x**2 - 10485
76*x*exp(2) + 65536*exp(4))

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