3.53.9 \(\int \frac {1-5 x-9 x^2-2 x^3+2 x^4+e^{2 x} (1-6 x-2 x^2+2 x^3)+e^x (2-10 x-12 x^2+2 x^3+2 x^4)+(-4 x-6 x^2-3 x^3+e^{2 x} (-3 x-2 x^2)+e^x (-6 x-8 x^2-2 x^3)) \log (x)}{5 x+40 x^2+110 x^3+130 x^4+85 x^5+30 x^6+5 x^7+e^{4 x} (5 x+10 x^2+5 x^3)+e^{3 x} (20 x+60 x^2+60 x^3+20 x^4)+e^{2 x} (30 x+130 x^2+190 x^3+120 x^4+30 x^5)+e^x (20 x+120 x^2+240 x^3+220 x^4+100 x^5+20 x^6)} \, dx\)
Optimal. Leaf size=29 \[ 5+\frac {2-x+\log (x)}{5 \left (x+(1+x) \left (1+e^x+x\right )^2\right )} \]
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Rubi [F] time = 49.83, antiderivative size = 0, normalized size of antiderivative = 0.00,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used =
{} \begin {gather*} \int \frac {1-5 x-9 x^2-2 x^3+2 x^4+e^{2 x} \left (1-6 x-2 x^2+2 x^3\right )+e^x \left (2-10 x-12 x^2+2 x^3+2 x^4\right )+\left (-4 x-6 x^2-3 x^3+e^{2 x} \left (-3 x-2 x^2\right )+e^x \left (-6 x-8 x^2-2 x^3\right )\right ) \log (x)}{5 x+40 x^2+110 x^3+130 x^4+85 x^5+30 x^6+5 x^7+e^{4 x} \left (5 x+10 x^2+5 x^3\right )+e^{3 x} \left (20 x+60 x^2+60 x^3+20 x^4\right )+e^{2 x} \left (30 x+130 x^2+190 x^3+120 x^4+30 x^5\right )+e^x \left (20 x+120 x^2+240 x^3+220 x^4+100 x^5+20 x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(1 - 5*x - 9*x^2 - 2*x^3 + 2*x^4 + E^(2*x)*(1 - 6*x - 2*x^2 + 2*x^3) + E^x*(2 - 10*x - 12*x^2 + 2*x^3 + 2*
x^4) + (-4*x - 6*x^2 - 3*x^3 + E^(2*x)*(-3*x - 2*x^2) + E^x*(-6*x - 8*x^2 - 2*x^3))*Log[x])/(5*x + 40*x^2 + 11
0*x^3 + 130*x^4 + 85*x^5 + 30*x^6 + 5*x^7 + E^(4*x)*(5*x + 10*x^2 + 5*x^3) + E^(3*x)*(20*x + 60*x^2 + 60*x^3 +
20*x^4) + E^(2*x)*(30*x + 130*x^2 + 190*x^3 + 120*x^4 + 30*x^5) + E^x*(20*x + 120*x^2 + 240*x^3 + 220*x^4 + 1
00*x^5 + 20*x^6)),x]
[Out]
Defer[Int][(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)^(-2), x]/5 + (8*Defer[Int][x/(1 + 4*x +
3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)^2, x])/5 + (4*Log[x]*Defer[Int][x/(1 + 4*x + 3*x^2 + x^3 + E
^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)^2, x])/5 + (4*Defer[Int][(E^x*x)/(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) +
2*E^x*(1 + x)^2)^2, x])/5 + (2*Log[x]*Defer[Int][(E^x*x)/(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 +
x)^2)^2, x])/5 + (4*Defer[Int][x^2/(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)^2, x])/5 + (4*
Log[x]*Defer[Int][x^2/(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)^2, x])/5 + (2*Defer[Int][(E^
x*x^2)/(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)^2, x])/5 + (2*Log[x]*Defer[Int][(E^x*x^2)/(
1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)^2, x])/5 + (2*Log[x]*Defer[Int][x^3/(1 + 4*x + 3*x^
2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)^2, x])/5 - (2*Defer[Int][(E^x*x^3)/(1 + 4*x + 3*x^2 + x^3 + E^(2*
x)*(1 + x) + 2*E^x*(1 + x)^2)^2, x])/5 - (2*Defer[Int][x^4/(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1
+ x)^2)^2, x])/5 - (3*Defer[Int][1/((1 + x)*(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)^2), x
])/5 - (Log[x]*Defer[Int][1/((1 + x)*(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)^2), x])/5 - (
4*Defer[Int][(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)^(-1), x])/5 - (2*Log[x]*Defer[Int][(1
+ 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)^(-1), x])/5 + Defer[Int][1/(x*(1 + 4*x + 3*x^2 + x^3
+ E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)), x]/5 + (2*Defer[Int][x/(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^
x*(1 + x)^2), x])/5 - (3*Defer[Int][1/((1 + x)*(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)), x
])/5 - (Log[x]*Defer[Int][1/((1 + x)*(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)), x])/5 - (4*
Defer[Int][Defer[Int][x/(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)^2, x]/x, x])/5 - (2*Defer[
Int][Defer[Int][(E^x*x)/(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)^2, x]/x, x])/5 - (4*Defer[
Int][Defer[Int][x^2/(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)^2, x]/x, x])/5 - (2*Defer[Int]
[Defer[Int][(E^x*x^2)/(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)^2, x]/x, x])/5 - (2*Defer[In
t][Defer[Int][x^3/(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)^2, x]/x, x])/5 + Defer[Int][Defe
r[Int][1/((1 + x)*(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)^2), x]/x, x]/5 + (2*Defer[Int][D
efer[Int][(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)^(-1), x]/x, x])/5 + Defer[Int][Defer[Int
][1/((1 + x)*(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2)), x]/x, x]/5
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1-5 x-9 x^2-2 x^3+2 x^4+e^{2 x} \left (1-6 x-2 x^2+2 x^3\right )+2 e^x \left (1-5 x-6 x^2+x^3+x^4\right )-x \left (4+6 x+3 x^2+e^{2 x} (3+2 x)+2 e^x \left (3+4 x+x^2\right )\right ) \log (x)}{5 x \left (1+4 x+3 x^2+x^3+e^{2 x} (1+x)+2 e^x (1+x)^2\right )^2} \, dx\\ &=\frac {1}{5} \int \frac {1-5 x-9 x^2-2 x^3+2 x^4+e^{2 x} \left (1-6 x-2 x^2+2 x^3\right )+2 e^x \left (1-5 x-6 x^2+x^3+x^4\right )-x \left (4+6 x+3 x^2+e^{2 x} (3+2 x)+2 e^x \left (3+4 x+x^2\right )\right ) \log (x)}{x \left (1+4 x+3 x^2+x^3+e^{2 x} (1+x)+2 e^x (1+x)^2\right )^2} \, dx\\ &=\frac {1}{5} \int \left (-\frac {\left (-1+4 x+2 e^x x+8 x^2+4 e^x x^2+6 x^3+2 e^x x^3+2 x^4\right ) (-2+x-\log (x))}{(1+x) \left (1+2 e^x+e^{2 x}+4 x+4 e^x x+e^{2 x} x+3 x^2+2 e^x x^2+x^3\right )^2}+\frac {1-6 x-2 x^2+2 x^3-3 x \log (x)-2 x^2 \log (x)}{x (1+x) \left (1+2 e^x+e^{2 x}+4 x+4 e^x x+e^{2 x} x+3 x^2+2 e^x x^2+x^3\right )}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {\left (-1+4 x+2 e^x x+8 x^2+4 e^x x^2+6 x^3+2 e^x x^3+2 x^4\right ) (-2+x-\log (x))}{(1+x) \left (1+2 e^x+e^{2 x}+4 x+4 e^x x+e^{2 x} x+3 x^2+2 e^x x^2+x^3\right )^2} \, dx\right )+\frac {1}{5} \int \frac {1-6 x-2 x^2+2 x^3-3 x \log (x)-2 x^2 \log (x)}{x (1+x) \left (1+2 e^x+e^{2 x}+4 x+4 e^x x+e^{2 x} x+3 x^2+2 e^x x^2+x^3\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.16, size = 45, normalized size = 1.55 \begin {gather*} \frac {2-x+\log (x)}{5 \left (1+4 x+3 x^2+x^3+e^{2 x} (1+x)+2 e^x (1+x)^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(1 - 5*x - 9*x^2 - 2*x^3 + 2*x^4 + E^(2*x)*(1 - 6*x - 2*x^2 + 2*x^3) + E^x*(2 - 10*x - 12*x^2 + 2*x^
3 + 2*x^4) + (-4*x - 6*x^2 - 3*x^3 + E^(2*x)*(-3*x - 2*x^2) + E^x*(-6*x - 8*x^2 - 2*x^3))*Log[x])/(5*x + 40*x^
2 + 110*x^3 + 130*x^4 + 85*x^5 + 30*x^6 + 5*x^7 + E^(4*x)*(5*x + 10*x^2 + 5*x^3) + E^(3*x)*(20*x + 60*x^2 + 60
*x^3 + 20*x^4) + E^(2*x)*(30*x + 130*x^2 + 190*x^3 + 120*x^4 + 30*x^5) + E^x*(20*x + 120*x^2 + 240*x^3 + 220*x
^4 + 100*x^5 + 20*x^6)),x]
[Out]
(2 - x + Log[x])/(5*(1 + 4*x + 3*x^2 + x^3 + E^(2*x)*(1 + x) + 2*E^x*(1 + x)^2))
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fricas [A] time = 0.63, size = 44, normalized size = 1.52 \begin {gather*} -\frac {x - \log \relax (x) - 2}{5 \, {\left (x^{3} + 3 \, x^{2} + {\left (x + 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{2} + 2 \, x + 1\right )} e^{x} + 4 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-2*x^2-3*x)*exp(x)^2+(-2*x^3-8*x^2-6*x)*exp(x)-3*x^3-6*x^2-4*x)*log(x)+(2*x^3-2*x^2-6*x+1)*exp(x)
^2+(2*x^4+2*x^3-12*x^2-10*x+2)*exp(x)+2*x^4-2*x^3-9*x^2-5*x+1)/((5*x^3+10*x^2+5*x)*exp(x)^4+(20*x^4+60*x^3+60*
x^2+20*x)*exp(x)^3+(30*x^5+120*x^4+190*x^3+130*x^2+30*x)*exp(x)^2+(20*x^6+100*x^5+220*x^4+240*x^3+120*x^2+20*x
)*exp(x)+5*x^7+30*x^6+85*x^5+130*x^4+110*x^3+40*x^2+5*x),x, algorithm="fricas")
[Out]
-1/5*(x - log(x) - 2)/(x^3 + 3*x^2 + (x + 1)*e^(2*x) + 2*(x^2 + 2*x + 1)*e^x + 4*x + 1)
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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((((-2*x^2-3*x)*exp(x)^2+(-2*x^3-8*x^2-6*x)*exp(x)-3*x^3-6*x^2-4*x)*log(x)+(2*x^3-2*x^2-6*x+1)*exp(x)
^2+(2*x^4+2*x^3-12*x^2-10*x+2)*exp(x)+2*x^4-2*x^3-9*x^2-5*x+1)/((5*x^3+10*x^2+5*x)*exp(x)^4+(20*x^4+60*x^3+60*
x^2+20*x)*exp(x)^3+(30*x^5+120*x^4+190*x^3+130*x^2+30*x)*exp(x)^2+(20*x^6+100*x^5+220*x^4+240*x^3+120*x^2+20*x
)*exp(x)+5*x^7+30*x^6+85*x^5+130*x^4+110*x^3+40*x^2+5*x),x, algorithm="giac")
[Out]
Timed out
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maple [B] time = 0.07, size = 93, normalized size = 3.21
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size |
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| risch |
\(\frac {\ln \relax (x )}{5 x \,{\mathrm e}^{2 x}+10 \,{\mathrm e}^{x} x^{2}+5 x^{3}+5 \,{\mathrm e}^{2 x}+20 \,{\mathrm e}^{x} x +15 x^{2}+10 \,{\mathrm e}^{x}+20 x +5}-\frac {x -2}{5 \left (x \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} x^{2}+x^{3}+{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x} x +3 x^{2}+2 \,{\mathrm e}^{x}+4 x +1\right )}\) |
\(93\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((((-2*x^2-3*x)*exp(x)^2+(-2*x^3-8*x^2-6*x)*exp(x)-3*x^3-6*x^2-4*x)*ln(x)+(2*x^3-2*x^2-6*x+1)*exp(x)^2+(2*x
^4+2*x^3-12*x^2-10*x+2)*exp(x)+2*x^4-2*x^3-9*x^2-5*x+1)/((5*x^3+10*x^2+5*x)*exp(x)^4+(20*x^4+60*x^3+60*x^2+20*
x)*exp(x)^3+(30*x^5+120*x^4+190*x^3+130*x^2+30*x)*exp(x)^2+(20*x^6+100*x^5+220*x^4+240*x^3+120*x^2+20*x)*exp(x
)+5*x^7+30*x^6+85*x^5+130*x^4+110*x^3+40*x^2+5*x),x,method=_RETURNVERBOSE)
[Out]
1/5/(x*exp(2*x)+2*exp(x)*x^2+x^3+exp(2*x)+4*exp(x)*x+3*x^2+2*exp(x)+4*x+1)*ln(x)-1/5*(x-2)/(x*exp(2*x)+2*exp(x
)*x^2+x^3+exp(2*x)+4*exp(x)*x+3*x^2+2*exp(x)+4*x+1)
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maxima [A] time = 0.54, size = 44, normalized size = 1.52 \begin {gather*} -\frac {x - \log \relax (x) - 2}{5 \, {\left (x^{3} + 3 \, x^{2} + {\left (x + 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{2} + 2 \, x + 1\right )} e^{x} + 4 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-2*x^2-3*x)*exp(x)^2+(-2*x^3-8*x^2-6*x)*exp(x)-3*x^3-6*x^2-4*x)*log(x)+(2*x^3-2*x^2-6*x+1)*exp(x)
^2+(2*x^4+2*x^3-12*x^2-10*x+2)*exp(x)+2*x^4-2*x^3-9*x^2-5*x+1)/((5*x^3+10*x^2+5*x)*exp(x)^4+(20*x^4+60*x^3+60*
x^2+20*x)*exp(x)^3+(30*x^5+120*x^4+190*x^3+130*x^2+30*x)*exp(x)^2+(20*x^6+100*x^5+220*x^4+240*x^3+120*x^2+20*x
)*exp(x)+5*x^7+30*x^6+85*x^5+130*x^4+110*x^3+40*x^2+5*x),x, algorithm="maxima")
[Out]
-1/5*(x - log(x) - 2)/(x^3 + 3*x^2 + (x + 1)*e^(2*x) + 2*(x^2 + 2*x + 1)*e^x + 4*x + 1)
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {5\,x-{\mathrm {e}}^x\,\left (2\,x^4+2\,x^3-12\,x^2-10\,x+2\right )+{\mathrm {e}}^{2\,x}\,\left (-2\,x^3+2\,x^2+6\,x-1\right )+\ln \relax (x)\,\left (4\,x+{\mathrm {e}}^{2\,x}\,\left (2\,x^2+3\,x\right )+6\,x^2+3\,x^3+{\mathrm {e}}^x\,\left (2\,x^3+8\,x^2+6\,x\right )\right )+9\,x^2+2\,x^3-2\,x^4-1}{5\,x+{\mathrm {e}}^{4\,x}\,\left (5\,x^3+10\,x^2+5\,x\right )+{\mathrm {e}}^{3\,x}\,\left (20\,x^4+60\,x^3+60\,x^2+20\,x\right )+{\mathrm {e}}^x\,\left (20\,x^6+100\,x^5+220\,x^4+240\,x^3+120\,x^2+20\,x\right )+{\mathrm {e}}^{2\,x}\,\left (30\,x^5+120\,x^4+190\,x^3+130\,x^2+30\,x\right )+40\,x^2+110\,x^3+130\,x^4+85\,x^5+30\,x^6+5\,x^7} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(5*x - exp(x)*(2*x^3 - 12*x^2 - 10*x + 2*x^4 + 2) + exp(2*x)*(6*x + 2*x^2 - 2*x^3 - 1) + log(x)*(4*x + ex
p(2*x)*(3*x + 2*x^2) + 6*x^2 + 3*x^3 + exp(x)*(6*x + 8*x^2 + 2*x^3)) + 9*x^2 + 2*x^3 - 2*x^4 - 1)/(5*x + exp(4
*x)*(5*x + 10*x^2 + 5*x^3) + exp(3*x)*(20*x + 60*x^2 + 60*x^3 + 20*x^4) + exp(x)*(20*x + 120*x^2 + 240*x^3 + 2
20*x^4 + 100*x^5 + 20*x^6) + exp(2*x)*(30*x + 130*x^2 + 190*x^3 + 120*x^4 + 30*x^5) + 40*x^2 + 110*x^3 + 130*x
^4 + 85*x^5 + 30*x^6 + 5*x^7),x)
[Out]
int(-(5*x - exp(x)*(2*x^3 - 12*x^2 - 10*x + 2*x^4 + 2) + exp(2*x)*(6*x + 2*x^2 - 2*x^3 - 1) + log(x)*(4*x + ex
p(2*x)*(3*x + 2*x^2) + 6*x^2 + 3*x^3 + exp(x)*(6*x + 8*x^2 + 2*x^3)) + 9*x^2 + 2*x^3 - 2*x^4 - 1)/(5*x + exp(4
*x)*(5*x + 10*x^2 + 5*x^3) + exp(3*x)*(20*x + 60*x^2 + 60*x^3 + 20*x^4) + exp(x)*(20*x + 120*x^2 + 240*x^3 + 2
20*x^4 + 100*x^5 + 20*x^6) + exp(2*x)*(30*x + 130*x^2 + 190*x^3 + 120*x^4 + 30*x^5) + 40*x^2 + 110*x^3 + 130*x
^4 + 85*x^5 + 30*x^6 + 5*x^7), x)
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sympy [A] time = 0.64, size = 44, normalized size = 1.52 \begin {gather*} \frac {- x + \log {\relax (x )} + 2}{5 x^{3} + 15 x^{2} + 20 x + \left (5 x + 5\right ) e^{2 x} + \left (10 x^{2} + 20 x + 10\right ) e^{x} + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-2*x**2-3*x)*exp(x)**2+(-2*x**3-8*x**2-6*x)*exp(x)-3*x**3-6*x**2-4*x)*ln(x)+(2*x**3-2*x**2-6*x+1)
*exp(x)**2+(2*x**4+2*x**3-12*x**2-10*x+2)*exp(x)+2*x**4-2*x**3-9*x**2-5*x+1)/((5*x**3+10*x**2+5*x)*exp(x)**4+(
20*x**4+60*x**3+60*x**2+20*x)*exp(x)**3+(30*x**5+120*x**4+190*x**3+130*x**2+30*x)*exp(x)**2+(20*x**6+100*x**5+
220*x**4+240*x**3+120*x**2+20*x)*exp(x)+5*x**7+30*x**6+85*x**5+130*x**4+110*x**3+40*x**2+5*x),x)
[Out]
(-x + log(x) + 2)/(5*x**3 + 15*x**2 + 20*x + (5*x + 5)*exp(2*x) + (10*x**2 + 20*x + 10)*exp(x) + 5)
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