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3.12.82 \(\int \frac {e^5 (-65537-262144 x-344064 x^2-229376 x^3-89600 x^4-21504 x^5-3136 x^6-256 x^7-9 x^8)}{25-655370 x+4293787649 x^2+17178984448 x^3+32211910656 x^4+37580899328 x^5+30534533120 x^6+18320722560 x^7+8396997184 x^8+2998927414 x^9+843448322 x^{10}+187432960 x^{11}+32800768 x^{12}+4472832 x^{13}+465920 x^{14}+35840 x^{15}+1920 x^{16}+64 x^{17}+x^{18}} \, dx\)

Optimal. Leaf size=16 \[ \frac {e^5}{-5+x+x (4+x)^8} \]

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Rubi [B]  time = 3.00, antiderivative size = 52, normalized size of antiderivative = 3.25, number of steps used = 6, number of rules used = 4, integrand size = 134, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {12, 2074, 2101, 6742} \begin {gather*} -\frac {e^5}{-x^9-32 x^8-448 x^7-3584 x^6-17920 x^5-57344 x^4-114688 x^3-131072 x^2-65537 x+5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^5*(-65537 - 262144*x - 344064*x^2 - 229376*x^3 - 89600*x^4 - 21504*x^5 - 3136*x^6 - 256*x^7 - 9*x^8))/(
25 - 655370*x + 4293787649*x^2 + 17178984448*x^3 + 32211910656*x^4 + 37580899328*x^5 + 30534533120*x^6 + 18320
722560*x^7 + 8396997184*x^8 + 2998927414*x^9 + 843448322*x^10 + 187432960*x^11 + 32800768*x^12 + 4472832*x^13
+ 465920*x^14 + 35840*x^15 + 1920*x^16 + 64*x^17 + x^18),x]

[Out]

-(E^5/(5 - 65537*x - 131072*x^2 - 114688*x^3 - 57344*x^4 - 17920*x^5 - 3584*x^6 - 448*x^7 - 32*x^8 - x^9))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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]

Rule 2101

Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[(Coeff[Pm, x, m]*Qn^(p + 1)
)/(n*(p + 1)*Coeff[Qn, x, n]), x] + Dist[1/(n*Coeff[Qn, x, n]), Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[P
m, x, m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm, x] && PolyQ[Qn, x] && NeQ[p,
-1]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^5 \int \frac {-65537-262144 x-344064 x^2-229376 x^3-89600 x^4-21504 x^5-3136 x^6-256 x^7-9 x^8}{25-655370 x+4293787649 x^2+17178984448 x^3+32211910656 x^4+37580899328 x^5+30534533120 x^6+18320722560 x^7+8396997184 x^8+2998927414 x^9+843448322 x^{10}+187432960 x^{11}+32800768 x^{12}+4472832 x^{13}+465920 x^{14}+35840 x^{15}+1920 x^{16}+64 x^{17}+x^{18}} \, dx\\ &=e^5 \int \left (-\frac {65537}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2}-\frac {262144 x}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2}-\frac {344064 x^2}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2}-\frac {229376 x^3}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2}-\frac {89600 x^4}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2}-\frac {21504 x^5}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2}-\frac {3136 x^6}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2}-\frac {256 x^7}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2}-\frac {9 x^8}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2}\right ) \, dx\\ &=-\left (\left (9 e^5\right ) \int \frac {x^8}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2} \, dx\right )-\left (256 e^5\right ) \int \frac {x^7}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2} \, dx-\left (3136 e^5\right ) \int \frac {x^6}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2} \, dx-\left (21504 e^5\right ) \int \frac {x^5}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2} \, dx-\left (65537 e^5\right ) \int \frac {1}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2} \, dx-\left (89600 e^5\right ) \int \frac {x^4}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2} \, dx-\left (229376 e^5\right ) \int \frac {x^3}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2} \, dx-\left (262144 e^5\right ) \int \frac {x}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2} \, dx-\left (344064 e^5\right ) \int \frac {x^2}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2} \, dx\\ &=-\frac {e^5}{5-65537 x-131072 x^2-114688 x^3-57344 x^4-17920 x^5-3584 x^6-448 x^7-32 x^8-x^9}-e^5 \int \frac {-65537-262144 x-344064 x^2-229376 x^3-89600 x^4-21504 x^5-3136 x^6-256 x^7}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2} \, dx-\left (256 e^5\right ) \int \frac {x^7}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2} \, dx-\left (3136 e^5\right ) \int \frac {x^6}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2} \, dx-\left (21504 e^5\right ) \int \frac {x^5}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2} \, dx-\left (65537 e^5\right ) \int \frac {1}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2} \, dx-\left (89600 e^5\right ) \int \frac {x^4}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2} \, dx-\left (229376 e^5\right ) \int \frac {x^3}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2} \, dx-\left (262144 e^5\right ) \int \frac {x}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2} \, dx-\left (344064 e^5\right ) \int \frac {x^2}{\left (-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.02, size = 49, normalized size = 3.06 \begin {gather*} \frac {e^5}{-5+65537 x+131072 x^2+114688 x^3+57344 x^4+17920 x^5+3584 x^6+448 x^7+32 x^8+x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^5*(-65537 - 262144*x - 344064*x^2 - 229376*x^3 - 89600*x^4 - 21504*x^5 - 3136*x^6 - 256*x^7 - 9*x
^8))/(25 - 655370*x + 4293787649*x^2 + 17178984448*x^3 + 32211910656*x^4 + 37580899328*x^5 + 30534533120*x^6 +
 18320722560*x^7 + 8396997184*x^8 + 2998927414*x^9 + 843448322*x^10 + 187432960*x^11 + 32800768*x^12 + 4472832
*x^13 + 465920*x^14 + 35840*x^15 + 1920*x^16 + 64*x^17 + x^18),x]

[Out]

E^5/(-5 + 65537*x + 131072*x^2 + 114688*x^3 + 57344*x^4 + 17920*x^5 + 3584*x^6 + 448*x^7 + 32*x^8 + x^9)

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fricas [B]  time = 0.73, size = 48, normalized size = 3.00 \begin {gather*} \frac {e^{5}}{x^{9} + 32 \, x^{8} + 448 \, x^{7} + 3584 \, x^{6} + 17920 \, x^{5} + 57344 \, x^{4} + 114688 \, x^{3} + 131072 \, x^{2} + 65537 \, x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x^8-256*x^7-3136*x^6-21504*x^5-89600*x^4-229376*x^3-344064*x^2-262144*x-65537)*exp(5)/(x^18+64*x
^17+1920*x^16+35840*x^15+465920*x^14+4472832*x^13+32800768*x^12+187432960*x^11+843448322*x^10+2998927414*x^9+8
396997184*x^8+18320722560*x^7+30534533120*x^6+37580899328*x^5+32211910656*x^4+17178984448*x^3+4293787649*x^2-6
55370*x+25),x, algorithm="fricas")

[Out]

e^5/(x^9 + 32*x^8 + 448*x^7 + 3584*x^6 + 17920*x^5 + 57344*x^4 + 114688*x^3 + 131072*x^2 + 65537*x - 5)

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giac [B]  time = 0.66, size = 48, normalized size = 3.00 \begin {gather*} \frac {e^{5}}{x^{9} + 32 \, x^{8} + 448 \, x^{7} + 3584 \, x^{6} + 17920 \, x^{5} + 57344 \, x^{4} + 114688 \, x^{3} + 131072 \, x^{2} + 65537 \, x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x^8-256*x^7-3136*x^6-21504*x^5-89600*x^4-229376*x^3-344064*x^2-262144*x-65537)*exp(5)/(x^18+64*x
^17+1920*x^16+35840*x^15+465920*x^14+4472832*x^13+32800768*x^12+187432960*x^11+843448322*x^10+2998927414*x^9+8
396997184*x^8+18320722560*x^7+30534533120*x^6+37580899328*x^5+32211910656*x^4+17178984448*x^3+4293787649*x^2-6
55370*x+25),x, algorithm="giac")

[Out]

e^5/(x^9 + 32*x^8 + 448*x^7 + 3584*x^6 + 17920*x^5 + 57344*x^4 + 114688*x^3 + 131072*x^2 + 65537*x - 5)

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maple [B]  time = 0.09, size = 49, normalized size = 3.06

method result size
gosper \(\frac {{\mathrm e}^{5}}{x^{9}+32 x^{8}+448 x^{7}+3584 x^{6}+17920 x^{5}+57344 x^{4}+114688 x^{3}+131072 x^{2}+65537 x -5}\) \(49\)
default \(\frac {{\mathrm e}^{5}}{x^{9}+32 x^{8}+448 x^{7}+3584 x^{6}+17920 x^{5}+57344 x^{4}+114688 x^{3}+131072 x^{2}+65537 x -5}\) \(49\)
norman \(\frac {{\mathrm e}^{5}}{x^{9}+32 x^{8}+448 x^{7}+3584 x^{6}+17920 x^{5}+57344 x^{4}+114688 x^{3}+131072 x^{2}+65537 x -5}\) \(49\)
risch \(\frac {{\mathrm e}^{5}}{x^{9}+32 x^{8}+448 x^{7}+3584 x^{6}+17920 x^{5}+57344 x^{4}+114688 x^{3}+131072 x^{2}+65537 x -5}\) \(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-9*x^8-256*x^7-3136*x^6-21504*x^5-89600*x^4-229376*x^3-344064*x^2-262144*x-65537)*exp(5)/(x^18+64*x^17+19
20*x^16+35840*x^15+465920*x^14+4472832*x^13+32800768*x^12+187432960*x^11+843448322*x^10+2998927414*x^9+8396997
184*x^8+18320722560*x^7+30534533120*x^6+37580899328*x^5+32211910656*x^4+17178984448*x^3+4293787649*x^2-655370*
x+25),x,method=_RETURNVERBOSE)

[Out]

exp(5)/(x^9+32*x^8+448*x^7+3584*x^6+17920*x^5+57344*x^4+114688*x^3+131072*x^2+65537*x-5)

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maxima [B]  time = 0.36, size = 48, normalized size = 3.00 \begin {gather*} \frac {e^{5}}{x^{9} + 32 \, x^{8} + 448 \, x^{7} + 3584 \, x^{6} + 17920 \, x^{5} + 57344 \, x^{4} + 114688 \, x^{3} + 131072 \, x^{2} + 65537 \, x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x^8-256*x^7-3136*x^6-21504*x^5-89600*x^4-229376*x^3-344064*x^2-262144*x-65537)*exp(5)/(x^18+64*x
^17+1920*x^16+35840*x^15+465920*x^14+4472832*x^13+32800768*x^12+187432960*x^11+843448322*x^10+2998927414*x^9+8
396997184*x^8+18320722560*x^7+30534533120*x^6+37580899328*x^5+32211910656*x^4+17178984448*x^3+4293787649*x^2-6
55370*x+25),x, algorithm="maxima")

[Out]

e^5/(x^9 + 32*x^8 + 448*x^7 + 3584*x^6 + 17920*x^5 + 57344*x^4 + 114688*x^3 + 131072*x^2 + 65537*x - 5)

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mupad [B]  time = 0.25, size = 48, normalized size = 3.00 \begin {gather*} \frac {{\mathrm {e}}^5}{x^9+32\,x^8+448\,x^7+3584\,x^6+17920\,x^5+57344\,x^4+114688\,x^3+131072\,x^2+65537\,x-5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(5)*(262144*x + 344064*x^2 + 229376*x^3 + 89600*x^4 + 21504*x^5 + 3136*x^6 + 256*x^7 + 9*x^8 + 65537)
)/(4293787649*x^2 - 655370*x + 17178984448*x^3 + 32211910656*x^4 + 37580899328*x^5 + 30534533120*x^6 + 1832072
2560*x^7 + 8396997184*x^8 + 2998927414*x^9 + 843448322*x^10 + 187432960*x^11 + 32800768*x^12 + 4472832*x^13 +
465920*x^14 + 35840*x^15 + 1920*x^16 + 64*x^17 + x^18 + 25),x)

[Out]

exp(5)/(65537*x + 131072*x^2 + 114688*x^3 + 57344*x^4 + 17920*x^5 + 3584*x^6 + 448*x^7 + 32*x^8 + x^9 - 5)

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sympy [B]  time = 1.24, size = 46, normalized size = 2.88 \begin {gather*} \frac {e^{5}}{x^{9} + 32 x^{8} + 448 x^{7} + 3584 x^{6} + 17920 x^{5} + 57344 x^{4} + 114688 x^{3} + 131072 x^{2} + 65537 x - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x**8-256*x**7-3136*x**6-21504*x**5-89600*x**4-229376*x**3-344064*x**2-262144*x-65537)*exp(5)/(x*
*18+64*x**17+1920*x**16+35840*x**15+465920*x**14+4472832*x**13+32800768*x**12+187432960*x**11+843448322*x**10+
2998927414*x**9+8396997184*x**8+18320722560*x**7+30534533120*x**6+37580899328*x**5+32211910656*x**4+1717898444
8*x**3+4293787649*x**2-655370*x+25),x)

[Out]

exp(5)/(x**9 + 32*x**8 + 448*x**7 + 3584*x**6 + 17920*x**5 + 57344*x**4 + 114688*x**3 + 131072*x**2 + 65537*x
- 5)

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