3.69.47 \(\int \frac {-244140000-e^{12}-585938000 x-644531700 x^2-429687600 x^3-193359382 x^4-61875000 x^5-14437500 x^6-2475000 x^7-309375 x^8-27500 x^9-1650 x^{10}-60 x^{11}-x^{12}+e^8 (-1875-1500 x-450 x^2-60 x^3-3 x^4)+e^4 (-1171874-1875000 x-1312500 x^2-525000 x^3-131250 x^4-21000 x^5-2100 x^6-120 x^7-3 x^8)}{488280000 x+2 e^{12} x+1171874000 x^2+1289062200 x^3+859374960 x^4+386718748 x^5+123750000 x^6+28875000 x^7+4950000 x^8+618750 x^9+55000 x^{10}+3300 x^{11}+120 x^{12}+2 x^{13}+e^8 (3750 x+3000 x^2+900 x^3+120 x^4+6 x^5)+e^4 (2343748 x+3750000 x^2+2625000 x^3+1050000 x^4+262500 x^5+42000 x^6+4200 x^7+240 x^8+6 x^9)} \, dx\)

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

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Rubi [B]  time = 1.55, antiderivative size = 81, normalized size of antiderivative = 2.89, number of steps used = 6, number of rules used = 3, integrand size = 282, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {6, 2074, 1587} \begin {gather*} -\frac {1}{2} \log \left (x^4+20 x^3+150 x^2+500 x+e^4+624\right )+\log \left (x^4+20 x^3+150 x^2+500 x+e^4+625\right )-\frac {1}{2} \log \left (x^4+20 x^3+150 x^2+500 x+e^4+626\right )-\frac {\log (x)}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-244140000 - E^12 - 585938000*x - 644531700*x^2 - 429687600*x^3 - 193359382*x^4 - 61875000*x^5 - 14437500
*x^6 - 2475000*x^7 - 309375*x^8 - 27500*x^9 - 1650*x^10 - 60*x^11 - x^12 + E^8*(-1875 - 1500*x - 450*x^2 - 60*
x^3 - 3*x^4) + E^4*(-1171874 - 1875000*x - 1312500*x^2 - 525000*x^3 - 131250*x^4 - 21000*x^5 - 2100*x^6 - 120*
x^7 - 3*x^8))/(488280000*x + 2*E^12*x + 1171874000*x^2 + 1289062200*x^3 + 859374960*x^4 + 386718748*x^5 + 1237
50000*x^6 + 28875000*x^7 + 4950000*x^8 + 618750*x^9 + 55000*x^10 + 3300*x^11 + 120*x^12 + 2*x^13 + E^8*(3750*x
+ 3000*x^2 + 900*x^3 + 120*x^4 + 6*x^5) + E^4*(2343748*x + 3750000*x^2 + 2625000*x^3 + 1050000*x^4 + 262500*x
^5 + 42000*x^6 + 4200*x^7 + 240*x^8 + 6*x^9)),x]

[Out]

-1/2*Log[x] - Log[624 + E^4 + 500*x + 150*x^2 + 20*x^3 + x^4]/2 + Log[625 + E^4 + 500*x + 150*x^2 + 20*x^3 + x
^4] - Log[626 + E^4 + 500*x + 150*x^2 + 20*x^3 + x^4]/2

Rule 6

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

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, 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 \frac {-244140000-e^{12}-585938000 x-644531700 x^2-429687600 x^3-193359382 x^4-61875000 x^5-14437500 x^6-2475000 x^7-309375 x^8-27500 x^9-1650 x^{10}-60 x^{11}-x^{12}+e^8 \left (-1875-1500 x-450 x^2-60 x^3-3 x^4\right )+e^4 \left (-1171874-1875000 x-1312500 x^2-525000 x^3-131250 x^4-21000 x^5-2100 x^6-120 x^7-3 x^8\right )}{\left (488280000+2 e^{12}\right ) x+1171874000 x^2+1289062200 x^3+859374960 x^4+386718748 x^5+123750000 x^6+28875000 x^7+4950000 x^8+618750 x^9+55000 x^{10}+3300 x^{11}+120 x^{12}+2 x^{13}+e^8 \left (3750 x+3000 x^2+900 x^3+120 x^4+6 x^5\right )+e^4 \left (2343748 x+3750000 x^2+2625000 x^3+1050000 x^4+262500 x^5+42000 x^6+4200 x^7+240 x^8+6 x^9\right )} \, dx\\ &=\int \left (-\frac {1}{2 x}-\frac {2 (5+x)^3}{624+e^4+500 x+150 x^2+20 x^3+x^4}+\frac {4 (5+x)^3}{625+e^4+500 x+150 x^2+20 x^3+x^4}-\frac {2 (5+x)^3}{626+e^4+500 x+150 x^2+20 x^3+x^4}\right ) \, dx\\ &=-\frac {\log (x)}{2}-2 \int \frac {(5+x)^3}{624+e^4+500 x+150 x^2+20 x^3+x^4} \, dx-2 \int \frac {(5+x)^3}{626+e^4+500 x+150 x^2+20 x^3+x^4} \, dx+4 \int \frac {(5+x)^3}{625+e^4+500 x+150 x^2+20 x^3+x^4} \, dx\\ &=-\frac {\log (x)}{2}-\frac {1}{2} \log \left (624+e^4+500 x+150 x^2+20 x^3+x^4\right )+\log \left (625+e^4+500 x+150 x^2+20 x^3+x^4\right )-\frac {1}{2} \log \left (626+e^4+500 x+150 x^2+20 x^3+x^4\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.27, size = 112, normalized size = 4.00 \begin {gather*} \frac {1}{2} \left (-\log (x)+2 \log \left (625+e^4+500 x+150 x^2+20 x^3+x^4\right )-\log \left (390624+1250 e^4+e^8+625000 x+1000 e^4 x+437500 x^2+300 e^4 x^2+175000 x^3+40 e^4 x^3+43750 x^4+2 e^4 x^4+7000 x^5+700 x^6+40 x^7+x^8\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-244140000 - E^12 - 585938000*x - 644531700*x^2 - 429687600*x^3 - 193359382*x^4 - 61875000*x^5 - 14
437500*x^6 - 2475000*x^7 - 309375*x^8 - 27500*x^9 - 1650*x^10 - 60*x^11 - x^12 + E^8*(-1875 - 1500*x - 450*x^2
- 60*x^3 - 3*x^4) + E^4*(-1171874 - 1875000*x - 1312500*x^2 - 525000*x^3 - 131250*x^4 - 21000*x^5 - 2100*x^6
- 120*x^7 - 3*x^8))/(488280000*x + 2*E^12*x + 1171874000*x^2 + 1289062200*x^3 + 859374960*x^4 + 386718748*x^5
+ 123750000*x^6 + 28875000*x^7 + 4950000*x^8 + 618750*x^9 + 55000*x^10 + 3300*x^11 + 120*x^12 + 2*x^13 + E^8*(
3750*x + 3000*x^2 + 900*x^3 + 120*x^4 + 6*x^5) + E^4*(2343748*x + 3750000*x^2 + 2625000*x^3 + 1050000*x^4 + 26
2500*x^5 + 42000*x^6 + 4200*x^7 + 240*x^8 + 6*x^9)),x]

[Out]

(-Log[x] + 2*Log[625 + E^4 + 500*x + 150*x^2 + 20*x^3 + x^4] - Log[390624 + 1250*E^4 + E^8 + 625000*x + 1000*E
^4*x + 437500*x^2 + 300*E^4*x^2 + 175000*x^3 + 40*E^4*x^3 + 43750*x^4 + 2*E^4*x^4 + 7000*x^5 + 700*x^6 + 40*x^
7 + x^8])/2

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fricas [B]  time = 0.82, size = 97, normalized size = 3.46 \begin {gather*} -\frac {1}{2} \, \log \left (x^{9} + 40 \, x^{8} + 700 \, x^{7} + 7000 \, x^{6} + 43750 \, x^{5} + 175000 \, x^{4} + 437500 \, x^{3} + 625000 \, x^{2} + x e^{8} + 2 \, {\left (x^{5} + 20 \, x^{4} + 150 \, x^{3} + 500 \, x^{2} + 625 \, x\right )} e^{4} + 390624 \, x\right ) + \log \left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + e^{4} + 625\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(4)^3+(-3*x^4-60*x^3-450*x^2-1500*x-1875)*exp(4)^2+(-3*x^8-120*x^7-2100*x^6-21000*x^5-131250*x^
4-525000*x^3-1312500*x^2-1875000*x-1171874)*exp(4)-x^12-60*x^11-1650*x^10-27500*x^9-309375*x^8-2475000*x^7-144
37500*x^6-61875000*x^5-193359382*x^4-429687600*x^3-644531700*x^2-585938000*x-244140000)/(2*x*exp(4)^3+(6*x^5+1
20*x^4+900*x^3+3000*x^2+3750*x)*exp(4)^2+(6*x^9+240*x^8+4200*x^7+42000*x^6+262500*x^5+1050000*x^4+2625000*x^3+
3750000*x^2+2343748*x)*exp(4)+2*x^13+120*x^12+3300*x^11+55000*x^10+618750*x^9+4950000*x^8+28875000*x^7+1237500
00*x^6+386718748*x^5+859374960*x^4+1289062200*x^3+1171874000*x^2+488280000*x),x, algorithm="fricas")

[Out]

-1/2*log(x^9 + 40*x^8 + 700*x^7 + 7000*x^6 + 43750*x^5 + 175000*x^4 + 437500*x^3 + 625000*x^2 + x*e^8 + 2*(x^5
+ 20*x^4 + 150*x^3 + 500*x^2 + 625*x)*e^4 + 390624*x) + log(x^4 + 20*x^3 + 150*x^2 + 500*x + e^4 + 625)

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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((-exp(4)^3+(-3*x^4-60*x^3-450*x^2-1500*x-1875)*exp(4)^2+(-3*x^8-120*x^7-2100*x^6-21000*x^5-131250*x^
4-525000*x^3-1312500*x^2-1875000*x-1171874)*exp(4)-x^12-60*x^11-1650*x^10-27500*x^9-309375*x^8-2475000*x^7-144
37500*x^6-61875000*x^5-193359382*x^4-429687600*x^3-644531700*x^2-585938000*x-244140000)/(2*x*exp(4)^3+(6*x^5+1
20*x^4+900*x^3+3000*x^2+3750*x)*exp(4)^2+(6*x^9+240*x^8+4200*x^7+42000*x^6+262500*x^5+1050000*x^4+2625000*x^3+
3750000*x^2+2343748*x)*exp(4)+2*x^13+120*x^12+3300*x^11+55000*x^10+618750*x^9+4950000*x^8+28875000*x^7+1237500
00*x^6+386718748*x^5+859374960*x^4+1289062200*x^3+1171874000*x^2+488280000*x),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 1.34, size = 73, normalized size = 2.61

method	result	size
norman	\(-\frac {\ln \relax (x )}{2}-\frac {\ln \left (x^{4}+20 x^{3}+150 x^{2}+{\mathrm e}^{4}+500 x +624\right )}{2}-\frac {\ln \left (x^{4}+20 x^{3}+150 x^{2}+{\mathrm e}^{4}+500 x +626\right )}{2}+\ln \left (x^{4}+20 x^{3}+150 x^{2}+{\mathrm e}^{4}+500 x +625\right )\)	\(73\)
risch	\(\ln \left (-x^{4}-20 x^{3}-150 x^{2}-{\mathrm e}^{4}-500 x -625\right )-\frac {\ln \left (x^{9}+40 x^{8}+700 x^{7}+7000 x^{6}+\left (2 \,{\mathrm e}^{4}+43750\right ) x^{5}+\left (40 \,{\mathrm e}^{4}+175000\right ) x^{4}+\left (300 \,{\mathrm e}^{4}+437500\right ) x^{3}+\left (1000 \,{\mathrm e}^{4}+625000\right ) x^{2}+\left ({\mathrm e}^{8}+1250 \,{\mathrm e}^{4}+390624\right ) x \right )}{2}\)	\(99\)
default	\(-\frac {\ln \relax (x )}{2}+\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{12}+60 \textit {\_Z}^{11}+1650 \textit {\_Z}^{10}+27500 \textit {\_Z}^{9}+\left (3 \,{\mathrm e}^{4}+309375\right ) \textit {\_Z}^{8}+\left (120 \,{\mathrm e}^{4}+2475000\right ) \textit {\_Z}^{7}+\left (2100 \,{\mathrm e}^{4}[phone]\right ) \textit {\_Z}^{6}+\left (21000 \,{\mathrm e}^{4}[phone]\right ) \textit {\_Z}^{5}+\left (3 \,{\mathrm e}^{8}+131250 \,{\mathrm e}^{4}[phone]\right ) \textit {\_Z}^{4}+\left (60 \,{\mathrm e}^{8}+525000 \,{\mathrm e}^{4}[phone]\right ) \textit {\_Z}^{3}+\left (450 \,{\mathrm e}^{8}+1312500 \,{\mathrm e}^{4}[phone]\right ) \textit {\_Z}^{2}+\left (1500 \,{\mathrm e}^{8}+1875000 \,{\mathrm e}^{4}[phone]\right ) \textit {\_Z} +{\mathrm e}^{12}+1875 \,{\mathrm e}^{8}+1171874 \,{\mathrm e}^{4}[phone]\right )}{\sum }\frac {\left (-\textit {\_R}^{3}-15 \textit {\_R}^{2}-75 \textit {\_R} -125\right ) \ln \left (x -\textit {\_R} \right )}{146484250+375 \,{\mathrm e}^{8}+225 \textit {\_R} \,{\mathrm e}^{8}+468750 \,{\mathrm e}^{4}+165 \textit {\_R}^{10}+3 \textit {\_R}^{11}+4331250 \textit {\_R}^{6}+618750 \textit {\_R}^{7}+4125 \textit {\_R}^{9}+61875 \textit {\_R}^{8}[phone] \textit {\_R}^{5}[phone] \textit {\_R}^{4}[phone] \textit {\_R}^{3}[phone] \textit {\_R}^{2}[phone] \textit {\_R} +210 \textit {\_R}^{6} {\mathrm e}^{4}+6 \textit {\_R}^{7} {\mathrm e}^{4}+393750 \textit {\_R}^{2} {\mathrm e}^{4}+656250 \textit {\_R} \,{\mathrm e}^{4}+3 \textit {\_R}^{3} {\mathrm e}^{8}+45 \textit {\_R}^{2} {\mathrm e}^{8}+131250 \textit {\_R}^{3} {\mathrm e}^{4}+26250 \textit {\_R}^{4} {\mathrm e}^{4}+3150 \textit {\_R}^{5} {\mathrm e}^{4}}\right )\)	\(287\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-exp(4)^3+(-3*x^4-60*x^3-450*x^2-1500*x-1875)*exp(4)^2+(-3*x^8-120*x^7-2100*x^6-21000*x^5-131250*x^4-5250
00*x^3-1312500*x^2-1875000*x-1171874)*exp(4)-x^12-60*x^11-1650*x^10-27500*x^9-309375*x^8-2475000*x^7-14437500*
x^6-61875000*x^5-193359382*x^4-429687600*x^3-644531700*x^2-585938000*x-244140000)/(2*x*exp(4)^3+(6*x^5+120*x^4
+900*x^3+3000*x^2+3750*x)*exp(4)^2+(6*x^9+240*x^8+4200*x^7+42000*x^6+262500*x^5+1050000*x^4+2625000*x^3+375000
0*x^2+2343748*x)*exp(4)+2*x^13+120*x^12+3300*x^11+55000*x^10+618750*x^9+4950000*x^8+28875000*x^7+123750000*x^6
[phone]*x^5+859374960*x^4+1289062200*x^3+1171874000*x^2+488280000*x),x,method=_RETURNVERBOSE)

[Out]

-1/2*ln(x)-1/2*ln(x^4+20*x^3+150*x^2+exp(4)+500*x+624)-1/2*ln(x^4+20*x^3+150*x^2+exp(4)+500*x+626)+ln(x^4+20*x
^3+150*x^2+exp(4)+500*x+625)

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maxima [B]  time = 0.37, size = 72, normalized size = 2.57 \begin {gather*} -\frac {1}{2} \, \log \left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + e^{4} + 626\right ) + \log \left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + e^{4} + 625\right ) - \frac {1}{2} \, \log \left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + e^{4} + 624\right ) - \frac {1}{2} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(4)^3+(-3*x^4-60*x^3-450*x^2-1500*x-1875)*exp(4)^2+(-3*x^8-120*x^7-2100*x^6-21000*x^5-131250*x^
4-525000*x^3-1312500*x^2-1875000*x-1171874)*exp(4)-x^12-60*x^11-1650*x^10-27500*x^9-309375*x^8-2475000*x^7-144
37500*x^6-61875000*x^5-193359382*x^4-429687600*x^3-644531700*x^2-585938000*x-244140000)/(2*x*exp(4)^3+(6*x^5+1
20*x^4+900*x^3+3000*x^2+3750*x)*exp(4)^2+(6*x^9+240*x^8+4200*x^7+42000*x^6+262500*x^5+1050000*x^4+2625000*x^3+
3750000*x^2+2343748*x)*exp(4)+2*x^13+120*x^12+3300*x^11+55000*x^10+618750*x^9+4950000*x^8+28875000*x^7+1237500
00*x^6+386718748*x^5+859374960*x^4+1289062200*x^3+1171874000*x^2+488280000*x),x, algorithm="maxima")

[Out]

-1/2*log(x^4 + 20*x^3 + 150*x^2 + 500*x + e^4 + 626) + log(x^4 + 20*x^3 + 150*x^2 + 500*x + e^4 + 625) - 1/2*l
og(x^4 + 20*x^3 + 150*x^2 + 500*x + e^4 + 624) - 1/2*log(x)

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mupad [B]  time = 5.49, size = 97, normalized size = 3.46 \begin {gather*} \ln \left (x^4+20\,x^3+150\,x^2+500\,x+{\mathrm {e}}^4+625\right )-\frac {\ln \left (x\,\left (625000\,x+1250\,{\mathrm {e}}^4+{\mathrm {e}}^8+1000\,x\,{\mathrm {e}}^4+300\,x^2\,{\mathrm {e}}^4+40\,x^3\,{\mathrm {e}}^4+2\,x^4\,{\mathrm {e}}^4+437500\,x^2+175000\,x^3+43750\,x^4+7000\,x^5+700\,x^6+40\,x^7+x^8+390624\right )\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(585938000*x + exp(12) + exp(4)*(1875000*x + 1312500*x^2 + 525000*x^3 + 131250*x^4 + 21000*x^5 + 2100*x^6
+ 120*x^7 + 3*x^8 + 1171874) + exp(8)*(1500*x + 450*x^2 + 60*x^3 + 3*x^4 + 1875) + 644531700*x^2 + 429687600*
x^3 + 193359382*x^4 + 61875000*x^5 + 14437500*x^6 + 2475000*x^7 + 309375*x^8 + 27500*x^9 + 1650*x^10 + 60*x^11
+ x^12 + 244140000)/(488280000*x + 2*x*exp(12) + exp(4)*(2343748*x + 3750000*x^2 + 2625000*x^3 + 1050000*x^4
+ 262500*x^5 + 42000*x^6 + 4200*x^7 + 240*x^8 + 6*x^9) + exp(8)*(3750*x + 3000*x^2 + 900*x^3 + 120*x^4 + 6*x^5
) + 1171874000*x^2 + 1289062200*x^3 + 859374960*x^4 + 386718748*x^5 + 123750000*x^6 + 28875000*x^7 + 4950000*x
^8 + 618750*x^9 + 55000*x^10 + 3300*x^11 + 120*x^12 + 2*x^13),x)

[Out]

log(500*x + exp(4) + 150*x^2 + 20*x^3 + x^4 + 625) - log(x*(625000*x + 1250*exp(4) + exp(8) + 1000*x*exp(4) +
300*x^2*exp(4) + 40*x^3*exp(4) + 2*x^4*exp(4) + 437500*x^2 + 175000*x^3 + 43750*x^4 + 7000*x^5 + 700*x^6 + 40*
x^7 + x^8 + 390624))/2

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sympy [B]  time = 41.52, size = 97, normalized size = 3.46 \begin {gather*} \log {\left (x^{4} + 20 x^{3} + 150 x^{2} + 500 x + e^{4} + 625 \right )} - \frac {\log {\left (x^{9} + 40 x^{8} + 700 x^{7} + 7000 x^{6} + x^{5} \left (2 e^{4} + 43750\right ) + x^{4} \left (40 e^{4} + 175000\right ) + x^{3} \left (300 e^{4} + 437500\right ) + x^{2} \left (1000 e^{4} + 625000\right ) + x \left (e^{8} + 1250 e^{4} + 390624\right ) \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(4)**3+(-3*x**4-60*x**3-450*x**2-1500*x-1875)*exp(4)**2+(-3*x**8-120*x**7-2100*x**6-21000*x**5-
131250*x**4-525000*x**3-1312500*x**2-1875000*x-1171874)*exp(4)-x**12-60*x**11-1650*x**10-27500*x**9-309375*x**
8-2475000*x**7-14437500*x**6-61875000*x**5-193359382*x**4-429687600*x**3-644531700*x**2-585938000*x-244140000)
/(2*x*exp(4)**3+(6*x**5+120*x**4+900*x**3+3000*x**2+3750*x)*exp(4)**2+(6*x**9+240*x**8+4200*x**7+42000*x**6+26
2500*x**5+1050000*x**4+2625000*x**3+3750000*x**2+2343748*x)*exp(4)+2*x**13+120*x**12+3300*x**11+55000*x**10+61
8750*x**9+4950000*x**8+28875000*x**7+123750000*x**6+386718748*x**5+859374960*x**4+1289062200*x**3+1171874000*x
**2+488280000*x),x)

[Out]

log(x**4 + 20*x**3 + 150*x**2 + 500*x + exp(4) + 625) - log(x**9 + 40*x**8 + 700*x**7 + 7000*x**6 + x**5*(2*ex
p(4) + 43750) + x**4*(40*exp(4) + 175000) + x**3*(300*exp(4) + 437500) + x**2*(1000*exp(4) + 625000) + x*(exp(
8) + 1250*exp(4) + 390624))/2

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