3.85.21 \(\int \frac {25 x+15 x^2+e^{5/x} (-500+175 x-30 x^2)+e^x (x-4 x^2)}{9765625 x+e^{5 x} x-9765625 x^2+3906250 x^3-781250 x^4+78125 x^5-3125 x^6+e^{20/x} (48828125 x-87890625 x^2+62500000 x^3-21875000 x^4+3750000 x^5-250000 x^6)+e^{10/x} (97656250 x-136718750 x^2+74218750 x^3-19531250 x^4+2500000 x^5-125000 x^6)+e^{5/x} (-48828125 x+58593750 x^2-27343750 x^3+6250000 x^4-703125 x^5+31250 x^6)+e^{25/x} (-9765625 x+19531250 x^2-15625000 x^3+6250000 x^4-1250000 x^5+100000 x^6)+e^{15/x} (-97656250 x+156250000 x^2-97656250 x^3+29687500 x^4-4375000 x^5+250000 x^6)+e^{4 x} (125 x-25 x^2+e^{5/x} (-125 x+50 x^2))+e^{3 x} (6250 x-2500 x^2+250 x^3+e^{5/x} (-12500 x+7500 x^2-1000 x^3)+e^{10/x} (6250 x-5000 x^2+1000 x^3))+e^{2 x} (156250 x-93750 x^2+18750 x^3-1250 x^4+e^{10/x} (468750 x-468750 x^2+150000 x^3-15000 x^4)+e^{5/x} (-468750 x+375000 x^2-93750 x^3+7500 x^4)+e^{15/x} (-156250 x+187500 x^2-75000 x^3+10000 x^4))+e^x (1953125 x-1562500 x^2+468750 x^3-62500 x^4+3125 x^5+e^{15/x} (-7812500 x+10937500 x^2-5625000 x^3+1250000 x^4-100000 x^5)+e^{5/x} (-7812500 x+7812500 x^2-2812500 x^3+437500 x^4-25000 x^5)+e^{20/x} (1953125 x-3125000 x^2+1875000 x^3-500000 x^4+50000 x^5)+e^{10/x} (11718750 x-14062500 x^2+6093750 x^3-1125000 x^4+75000 x^5))} \, dx\)
Optimal. Leaf size=28 \[ \frac {x}{\left (e^x+5 \left (5-x+e^{5/x} (-5+2 x)\right )\right )^4} \]
________________________________________________________________________________________
Rubi [F] time = 5.65, 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 {25 x+15 x^2+e^{5/x} \left (-500+175 x-30 x^2\right )+e^x \left (x-4 x^2\right )}{9765625 x+e^{5 x} x-9765625 x^2+3906250 x^3-781250 x^4+78125 x^5-3125 x^6+e^{20/x} \left (48828125 x-87890625 x^2+62500000 x^3-21875000 x^4+3750000 x^5-250000 x^6\right )+e^{10/x} \left (97656250 x-136718750 x^2+74218750 x^3-19531250 x^4+2500000 x^5-125000 x^6\right )+e^{5/x} \left (-48828125 x+58593750 x^2-27343750 x^3+6250000 x^4-703125 x^5+31250 x^6\right )+e^{25/x} \left (-9765625 x+19531250 x^2-15625000 x^3+6250000 x^4-1250000 x^5+100000 x^6\right )+e^{15/x} \left (-97656250 x+156250000 x^2-97656250 x^3+29687500 x^4-4375000 x^5+250000 x^6\right )+e^{4 x} \left (125 x-25 x^2+e^{5/x} \left (-125 x+50 x^2\right )\right )+e^{3 x} \left (6250 x-2500 x^2+250 x^3+e^{5/x} \left (-12500 x+7500 x^2-1000 x^3\right )+e^{10/x} \left (6250 x-5000 x^2+1000 x^3\right )\right )+e^{2 x} \left (156250 x-93750 x^2+18750 x^3-1250 x^4+e^{10/x} \left (468750 x-468750 x^2+150000 x^3-15000 x^4\right )+e^{5/x} \left (-468750 x+375000 x^2-93750 x^3+7500 x^4\right )+e^{15/x} \left (-156250 x+187500 x^2-75000 x^3+10000 x^4\right )\right )+e^x \left (1953125 x-1562500 x^2+468750 x^3-62500 x^4+3125 x^5+e^{15/x} \left (-7812500 x+10937500 x^2-5625000 x^3+1250000 x^4-100000 x^5\right )+e^{5/x} \left (-7812500 x+7812500 x^2-2812500 x^3+437500 x^4-25000 x^5\right )+e^{20/x} \left (1953125 x-3125000 x^2+1875000 x^3-500000 x^4+50000 x^5\right )+e^{10/x} \left (11718750 x-14062500 x^2+6093750 x^3-1125000 x^4+75000 x^5\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(25*x + 15*x^2 + E^(5/x)*(-500 + 175*x - 30*x^2) + E^x*(x - 4*x^2))/(9765625*x + E^(5*x)*x - 9765625*x^2 +
3906250*x^3 - 781250*x^4 + 78125*x^5 - 3125*x^6 + E^(20/x)*(48828125*x - 87890625*x^2 + 62500000*x^3 - 218750
00*x^4 + 3750000*x^5 - 250000*x^6) + E^(10/x)*(97656250*x - 136718750*x^2 + 74218750*x^3 - 19531250*x^4 + 2500
000*x^5 - 125000*x^6) + E^(5/x)*(-48828125*x + 58593750*x^2 - 27343750*x^3 + 6250000*x^4 - 703125*x^5 + 31250*
x^6) + E^(25/x)*(-9765625*x + 19531250*x^2 - 15625000*x^3 + 6250000*x^4 - 1250000*x^5 + 100000*x^6) + E^(15/x)
*(-97656250*x + 156250000*x^2 - 97656250*x^3 + 29687500*x^4 - 4375000*x^5 + 250000*x^6) + E^(4*x)*(125*x - 25*
x^2 + E^(5/x)*(-125*x + 50*x^2)) + E^(3*x)*(6250*x - 2500*x^2 + 250*x^3 + E^(5/x)*(-12500*x + 7500*x^2 - 1000*
x^3) + E^(10/x)*(6250*x - 5000*x^2 + 1000*x^3)) + E^(2*x)*(156250*x - 93750*x^2 + 18750*x^3 - 1250*x^4 + E^(10
/x)*(468750*x - 468750*x^2 + 150000*x^3 - 15000*x^4) + E^(5/x)*(-468750*x + 375000*x^2 - 93750*x^3 + 7500*x^4)
+ E^(15/x)*(-156250*x + 187500*x^2 - 75000*x^3 + 10000*x^4)) + E^x*(1953125*x - 1562500*x^2 + 468750*x^3 - 62
500*x^4 + 3125*x^5 + E^(15/x)*(-7812500*x + 10937500*x^2 - 5625000*x^3 + 1250000*x^4 - 100000*x^5) + E^(5/x)*(
-7812500*x + 7812500*x^2 - 2812500*x^3 + 437500*x^4 - 25000*x^5) + E^(20/x)*(1953125*x - 3125000*x^2 + 1875000
*x^3 - 500000*x^4 + 50000*x^5) + E^(10/x)*(11718750*x - 14062500*x^2 + 6093750*x^3 - 1125000*x^4 + 75000*x^5))
),x]
[Out]
200*Defer[Int][E^(5/x)/(25 - 25*E^(5/x) + E^x - 5*x + 10*E^(5/x)*x)^5, x] - 500*Defer[Int][E^(5/x)/(x*(25 - 25
*E^(5/x) + E^x - 5*x + 10*E^(5/x)*x)^5), x] + 120*Defer[Int][x/(25 - 25*E^(5/x) + E^x - 5*x + 10*E^(5/x)*x)^5,
x] - 140*Defer[Int][(E^(5/x)*x)/(25 - 25*E^(5/x) + E^x - 5*x + 10*E^(5/x)*x)^5, x] - 20*Defer[Int][x^2/(25 -
25*E^(5/x) + E^x - 5*x + 10*E^(5/x)*x)^5, x] + 40*Defer[Int][(E^(5/x)*x^2)/(25 - 25*E^(5/x) + E^x - 5*x + 10*E
^(5/x)*x)^5, x] + Defer[Int][(25 - 25*E^(5/x) + E^x - 5*x + 10*E^(5/x)*x)^(-4), x] - 4*Defer[Int][x/(25 - 25*E
^(5/x) + E^x - 5*x + 10*E^(5/x)*x)^4, x]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 x (5+3 x)+e^x \left (x-4 x^2\right )-5 e^{5/x} \left (100-35 x+6 x^2\right )}{x \left (e^x-5 (-5+x)+5 e^{5/x} (-5+2 x)\right )^5} \, dx\\ &=\int \left (-\frac {-1+4 x}{\left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^4}+\frac {20 \left (-25 e^{5/x}+10 e^{5/x} x+6 x^2-7 e^{5/x} x^2-x^3+2 e^{5/x} x^3\right )}{x \left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^5}\right ) \, dx\\ &=20 \int \frac {-25 e^{5/x}+10 e^{5/x} x+6 x^2-7 e^{5/x} x^2-x^3+2 e^{5/x} x^3}{x \left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^5} \, dx-\int \frac {-1+4 x}{\left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^4} \, dx\\ &=20 \int \frac {-\left ((-6+x) x^2\right )+e^{5/x} \left (-25+10 x-7 x^2+2 x^3\right )}{x \left (e^x-5 (-5+x)+5 e^{5/x} (-5+2 x)\right )^5} \, dx-\int \left (-\frac {1}{\left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^4}+\frac {4 x}{\left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^4}\right ) \, dx\\ &=-\left (4 \int \frac {x}{\left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^4} \, dx\right )+20 \int \left (\frac {10 e^{5/x}}{\left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^5}-\frac {25 e^{5/x}}{x \left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^5}+\frac {6 x}{\left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^5}-\frac {7 e^{5/x} x}{\left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^5}-\frac {x^2}{\left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^5}+\frac {2 e^{5/x} x^2}{\left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^5}\right ) \, dx+\int \frac {1}{\left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^4} \, dx\\ &=-\left (4 \int \frac {x}{\left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^4} \, dx\right )-20 \int \frac {x^2}{\left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^5} \, dx+40 \int \frac {e^{5/x} x^2}{\left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^5} \, dx+120 \int \frac {x}{\left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^5} \, dx-140 \int \frac {e^{5/x} x}{\left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^5} \, dx+200 \int \frac {e^{5/x}}{\left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^5} \, dx-500 \int \frac {e^{5/x}}{x \left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^5} \, dx+\int \frac {1}{\left (25-25 e^{5/x}+e^x-5 x+10 e^{5/x} x\right )^4} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 1.03, size = 27, normalized size = 0.96 \begin {gather*} \frac {x}{\left (e^x-5 (-5+x)+5 e^{5/x} (-5+2 x)\right )^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(25*x + 15*x^2 + E^(5/x)*(-500 + 175*x - 30*x^2) + E^x*(x - 4*x^2))/(9765625*x + E^(5*x)*x - 9765625
*x^2 + 3906250*x^3 - 781250*x^4 + 78125*x^5 - 3125*x^6 + E^(20/x)*(48828125*x - 87890625*x^2 + 62500000*x^3 -
21875000*x^4 + 3750000*x^5 - 250000*x^6) + E^(10/x)*(97656250*x - 136718750*x^2 + 74218750*x^3 - 19531250*x^4
+ 2500000*x^5 - 125000*x^6) + E^(5/x)*(-48828125*x + 58593750*x^2 - 27343750*x^3 + 6250000*x^4 - 703125*x^5 +
31250*x^6) + E^(25/x)*(-9765625*x + 19531250*x^2 - 15625000*x^3 + 6250000*x^4 - 1250000*x^5 + 100000*x^6) + E^
(15/x)*(-97656250*x + 156250000*x^2 - 97656250*x^3 + 29687500*x^4 - 4375000*x^5 + 250000*x^6) + E^(4*x)*(125*x
- 25*x^2 + E^(5/x)*(-125*x + 50*x^2)) + E^(3*x)*(6250*x - 2500*x^2 + 250*x^3 + E^(5/x)*(-12500*x + 7500*x^2 -
1000*x^3) + E^(10/x)*(6250*x - 5000*x^2 + 1000*x^3)) + E^(2*x)*(156250*x - 93750*x^2 + 18750*x^3 - 1250*x^4 +
E^(10/x)*(468750*x - 468750*x^2 + 150000*x^3 - 15000*x^4) + E^(5/x)*(-468750*x + 375000*x^2 - 93750*x^3 + 750
0*x^4) + E^(15/x)*(-156250*x + 187500*x^2 - 75000*x^3 + 10000*x^4)) + E^x*(1953125*x - 1562500*x^2 + 468750*x^
3 - 62500*x^4 + 3125*x^5 + E^(15/x)*(-7812500*x + 10937500*x^2 - 5625000*x^3 + 1250000*x^4 - 100000*x^5) + E^(
5/x)*(-7812500*x + 7812500*x^2 - 2812500*x^3 + 437500*x^4 - 25000*x^5) + E^(20/x)*(1953125*x - 3125000*x^2 + 1
875000*x^3 - 500000*x^4 + 50000*x^5) + E^(10/x)*(11718750*x - 14062500*x^2 + 6093750*x^3 - 1125000*x^4 + 75000
*x^5))),x]
[Out]
x/(E^x - 5*(-5 + x) + 5*E^(5/x)*(-5 + 2*x))^4
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fricas [B] time = 0.71, size = 298, normalized size = 10.64 \begin {gather*} \frac {x}{625 \, x^{4} - 12500 \, x^{3} + 93750 \, x^{2} + 20 \, {\left ({\left (2 \, x - 5\right )} e^{\frac {5}{x}} - x + 5\right )} e^{\left (3 \, x\right )} + 150 \, {\left (x^{2} + {\left (4 \, x^{2} - 20 \, x + 25\right )} e^{\frac {10}{x}} - 2 \, {\left (2 \, x^{2} - 15 \, x + 25\right )} e^{\frac {5}{x}} - 10 \, x + 25\right )} e^{\left (2 \, x\right )} - 500 \, {\left (x^{3} - 15 \, x^{2} - {\left (8 \, x^{3} - 60 \, x^{2} + 150 \, x - 125\right )} e^{\frac {15}{x}} + 3 \, {\left (4 \, x^{3} - 40 \, x^{2} + 125 \, x - 125\right )} e^{\frac {10}{x}} - 3 \, {\left (2 \, x^{3} - 25 \, x^{2} + 100 \, x - 125\right )} e^{\frac {5}{x}} + 75 \, x - 125\right )} e^{x} + 625 \, {\left (16 \, x^{4} - 160 \, x^{3} + 600 \, x^{2} - 1000 \, x + 625\right )} e^{\frac {20}{x}} - 2500 \, {\left (8 \, x^{4} - 100 \, x^{3} + 450 \, x^{2} - 875 \, x + 625\right )} e^{\frac {15}{x}} + 3750 \, {\left (4 \, x^{4} - 60 \, x^{3} + 325 \, x^{2} - 750 \, x + 625\right )} e^{\frac {10}{x}} - 2500 \, {\left (2 \, x^{4} - 35 \, x^{3} + 225 \, x^{2} - 625 \, x + 625\right )} e^{\frac {5}{x}} - 312500 \, x + e^{\left (4 \, x\right )} + 390625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-4*x^2+x)*exp(x)+(-30*x^2+175*x-500)*exp(5/x)+15*x^2+25*x)/(x*exp(x)^5+((50*x^2-125*x)*exp(5/x)-25
*x^2+125*x)*exp(x)^4+((1000*x^3-5000*x^2+6250*x)*exp(5/x)^2+(-1000*x^3+7500*x^2-12500*x)*exp(5/x)+250*x^3-2500
*x^2+6250*x)*exp(x)^3+((10000*x^4-75000*x^3+187500*x^2-156250*x)*exp(5/x)^3+(-15000*x^4+150000*x^3-468750*x^2+
468750*x)*exp(5/x)^2+(7500*x^4-93750*x^3+375000*x^2-468750*x)*exp(5/x)-1250*x^4+18750*x^3-93750*x^2+156250*x)*
exp(x)^2+((50000*x^5-500000*x^4+1875000*x^3-3125000*x^2+1953125*x)*exp(5/x)^4+(-100000*x^5+1250000*x^4-5625000
*x^3+10937500*x^2-7812500*x)*exp(5/x)^3+(75000*x^5-1125000*x^4+6093750*x^3-14062500*x^2+11718750*x)*exp(5/x)^2
+(-25000*x^5+437500*x^4-2812500*x^3+7812500*x^2-7812500*x)*exp(5/x)+3125*x^5-62500*x^4+468750*x^3-1562500*x^2+
1953125*x)*exp(x)+(100000*x^6-1250000*x^5+6250000*x^4-15625000*x^3+19531250*x^2-9765625*x)*exp(5/x)^5+(-250000
*x^6+3750000*x^5-21875000*x^4+62500000*x^3-87890625*x^2+48828125*x)*exp(5/x)^4+(250000*x^6-4375000*x^5+2968750
0*x^4-97656250*x^3+156250000*x^2-97656250*x)*exp(5/x)^3+(-125000*x^6+2500000*x^5-19531250*x^4+74218750*x^3-136
718750*x^2+97656250*x)*exp(5/x)^2+(31250*x^6-703125*x^5+6250000*x^4-27343750*x^3+58593750*x^2-48828125*x)*exp(
5/x)-3125*x^6+78125*x^5-781250*x^4+3906250*x^3-9765625*x^2+9765625*x),x, algorithm="fricas")
[Out]
x/(625*x^4 - 12500*x^3 + 93750*x^2 + 20*((2*x - 5)*e^(5/x) - x + 5)*e^(3*x) + 150*(x^2 + (4*x^2 - 20*x + 25)*e
^(10/x) - 2*(2*x^2 - 15*x + 25)*e^(5/x) - 10*x + 25)*e^(2*x) - 500*(x^3 - 15*x^2 - (8*x^3 - 60*x^2 + 150*x - 1
25)*e^(15/x) + 3*(4*x^3 - 40*x^2 + 125*x - 125)*e^(10/x) - 3*(2*x^3 - 25*x^2 + 100*x - 125)*e^(5/x) + 75*x - 1
25)*e^x + 625*(16*x^4 - 160*x^3 + 600*x^2 - 1000*x + 625)*e^(20/x) - 2500*(8*x^4 - 100*x^3 + 450*x^2 - 875*x +
625)*e^(15/x) + 3750*(4*x^4 - 60*x^3 + 325*x^2 - 750*x + 625)*e^(10/x) - 2500*(2*x^4 - 35*x^3 + 225*x^2 - 625
*x + 625)*e^(5/x) - 312500*x + e^(4*x) + 390625)
________________________________________________________________________________________
giac [B] time = 1.40, size = 849, normalized size = 30.32 result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-4*x^2+x)*exp(x)+(-30*x^2+175*x-500)*exp(5/x)+15*x^2+25*x)/(x*exp(x)^5+((50*x^2-125*x)*exp(5/x)-25
*x^2+125*x)*exp(x)^4+((1000*x^3-5000*x^2+6250*x)*exp(5/x)^2+(-1000*x^3+7500*x^2-12500*x)*exp(5/x)+250*x^3-2500
*x^2+6250*x)*exp(x)^3+((10000*x^4-75000*x^3+187500*x^2-156250*x)*exp(5/x)^3+(-15000*x^4+150000*x^3-468750*x^2+
468750*x)*exp(5/x)^2+(7500*x^4-93750*x^3+375000*x^2-468750*x)*exp(5/x)-1250*x^4+18750*x^3-93750*x^2+156250*x)*
exp(x)^2+((50000*x^5-500000*x^4+1875000*x^3-3125000*x^2+1953125*x)*exp(5/x)^4+(-100000*x^5+1250000*x^4-5625000
*x^3+10937500*x^2-7812500*x)*exp(5/x)^3+(75000*x^5-1125000*x^4+6093750*x^3-14062500*x^2+11718750*x)*exp(5/x)^2
+(-25000*x^5+437500*x^4-2812500*x^3+7812500*x^2-7812500*x)*exp(5/x)+3125*x^5-62500*x^4+468750*x^3-1562500*x^2+
1953125*x)*exp(x)+(100000*x^6-1250000*x^5+6250000*x^4-15625000*x^3+19531250*x^2-9765625*x)*exp(5/x)^5+(-250000
*x^6+3750000*x^5-21875000*x^4+62500000*x^3-87890625*x^2+48828125*x)*exp(5/x)^4+(250000*x^6-4375000*x^5+2968750
0*x^4-97656250*x^3+156250000*x^2-97656250*x)*exp(5/x)^3+(-125000*x^6+2500000*x^5-19531250*x^4+74218750*x^3-136
718750*x^2+97656250*x)*exp(5/x)^2+(31250*x^6-703125*x^5+6250000*x^4-27343750*x^3+58593750*x^2-48828125*x)*exp(
5/x)-3125*x^6+78125*x^5-781250*x^4+3906250*x^3-9765625*x^2+9765625*x),x, algorithm="giac")
[Out]
x*e^(40/x)/(10000*x^4*e^(60/x) - 20000*x^4*e^(55/x) + 15000*x^4*e^(50/x) - 5000*x^4*e^(45/x) + 625*x^4*e^(40/x
) + 4000*x^3*e^((x^2 + 10)/x + 45/x) - 6000*x^3*e^((x^2 + 10)/x + 40/x) + 3000*x^3*e^((x^2 + 10)/x + 35/x) - 5
00*x^3*e^((x^2 + 10)/x + 30/x) - 100000*x^3*e^(60/x) + 250000*x^3*e^(55/x) - 225000*x^3*e^(50/x) + 87500*x^3*e
^(45/x) - 12500*x^3*e^(40/x) + 600*x^2*e^(2*(x^2 + 10)/x + 30/x) - 600*x^2*e^(2*(x^2 + 10)/x + 25/x) + 150*x^2
*e^(2*(x^2 + 10)/x + 20/x) - 30000*x^2*e^((x^2 + 10)/x + 45/x) + 60000*x^2*e^((x^2 + 10)/x + 40/x) - 37500*x^2
*e^((x^2 + 10)/x + 35/x) + 7500*x^2*e^((x^2 + 10)/x + 30/x) + 375000*x^2*e^(60/x) - 1125000*x^2*e^(55/x) + 121
8750*x^2*e^(50/x) - 562500*x^2*e^(45/x) + 93750*x^2*e^(40/x) + 40*x*e^(3*(x^2 + 10)/x + 15/x) - 20*x*e^(3*(x^2
+ 10)/x + 10/x) - 3000*x*e^(2*(x^2 + 10)/x + 30/x) + 4500*x*e^(2*(x^2 + 10)/x + 25/x) - 1500*x*e^(2*(x^2 + 10
)/x + 20/x) + 75000*x*e^((x^2 + 10)/x + 45/x) - 187500*x*e^((x^2 + 10)/x + 40/x) + 150000*x*e^((x^2 + 10)/x +
35/x) - 37500*x*e^((x^2 + 10)/x + 30/x) - 625000*x*e^(60/x) + 2187500*x*e^(55/x) - 2812500*x*e^(50/x) + 156250
0*x*e^(45/x) - 312500*x*e^(40/x) + e^(4*(x^2 + 10)/x) - 100*e^(3*(x^2 + 10)/x + 15/x) + 100*e^(3*(x^2 + 10)/x
+ 10/x) + 3750*e^(2*(x^2 + 10)/x + 30/x) - 7500*e^(2*(x^2 + 10)/x + 25/x) + 3750*e^(2*(x^2 + 10)/x + 20/x) - 6
2500*e^((x^2 + 10)/x + 45/x) + 187500*e^((x^2 + 10)/x + 40/x) - 187500*e^((x^2 + 10)/x + 35/x) + 62500*e^((x^2
+ 10)/x + 30/x) + 390625*e^(60/x) - 1562500*e^(55/x) + 2343750*e^(50/x) - 1562500*e^(45/x) + 390625*e^(40/x))
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maple [A] time = 0.71, size = 29, normalized size = 1.04
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\(\frac {x}{\left (10 x \,{\mathrm e}^{\frac {5}{x}}+{\mathrm e}^{x}-25 \,{\mathrm e}^{\frac {5}{x}}-5 x +25\right )^{4}}\) |
\(29\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((-4*x^2+x)*exp(x)+(-30*x^2+175*x-500)*exp(5/x)+15*x^2+25*x)/(x*exp(x)^5+((50*x^2-125*x)*exp(5/x)-25*x^2+1
25*x)*exp(x)^4+((1000*x^3-5000*x^2+6250*x)*exp(5/x)^2+(-1000*x^3+7500*x^2-12500*x)*exp(5/x)+250*x^3-2500*x^2+6
250*x)*exp(x)^3+((10000*x^4-75000*x^3+187500*x^2-156250*x)*exp(5/x)^3+(-15000*x^4+150000*x^3-468750*x^2+468750
*x)*exp(5/x)^2+(7500*x^4-93750*x^3+375000*x^2-468750*x)*exp(5/x)-1250*x^4+18750*x^3-93750*x^2+156250*x)*exp(x)
^2+((50000*x^5-500000*x^4+1875000*x^3-3125000*x^2+1953125*x)*exp(5/x)^4+(-100000*x^5+1250000*x^4-5625000*x^3+1
0937500*x^2-7812500*x)*exp(5/x)^3+(75000*x^5-1125000*x^4+6093750*x^3-14062500*x^2+11718750*x)*exp(5/x)^2+(-250
00*x^5+437500*x^4-2812500*x^3+7812500*x^2-7812500*x)*exp(5/x)+3125*x^5-62500*x^4+468750*x^3-1562500*x^2+195312
5*x)*exp(x)+(100000*x^6-1250000*x^5+6250000*x^4-15625000*x^3+19531250*x^2-9765625*x)*exp(5/x)^5+(-250000*x^6+3
750000*x^5-21875000*x^4+62500000*x^3-87890625*x^2+48828125*x)*exp(5/x)^4+(250000*x^6-4375000*x^5+29687500*x^4-
97656250*x^3+156250000*x^2-97656250*x)*exp(5/x)^3+(-125000*x^6+2500000*x^5-19531250*x^4+74218750*x^3-136718750
*x^2+97656250*x)*exp(5/x)^2+(31250*x^6-703125*x^5+6250000*x^4-27343750*x^3+58593750*x^2-48828125*x)*exp(5/x)-3
125*x^6+78125*x^5-781250*x^4+3906250*x^3-9765625*x^2+9765625*x),x,method=_RETURNVERBOSE)
[Out]
x/(10*x*exp(5/x)+exp(x)-25*exp(5/x)-5*x+25)^4
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maxima [B] time = 2.04, size = 279, normalized size = 9.96 \begin {gather*} \frac {x}{625 \, x^{4} - 12500 \, x^{3} + 93750 \, x^{2} - 20 \, {\left (x - 5\right )} e^{\left (3 \, x\right )} + 150 \, {\left (x^{2} - 10 \, x + 25\right )} e^{\left (2 \, x\right )} - 500 \, {\left (x^{3} - 15 \, x^{2} + 75 \, x - 125\right )} e^{x} + 625 \, {\left (16 \, x^{4} - 160 \, x^{3} + 600 \, x^{2} - 1000 \, x + 625\right )} e^{\frac {20}{x}} - 500 \, {\left (40 \, x^{4} - 500 \, x^{3} + 2250 \, x^{2} - {\left (8 \, x^{3} - 60 \, x^{2} + 150 \, x - 125\right )} e^{x} - 4375 \, x + 3125\right )} e^{\frac {15}{x}} + 150 \, {\left (100 \, x^{4} - 1500 \, x^{3} + 8125 \, x^{2} + {\left (4 \, x^{2} - 20 \, x + 25\right )} e^{\left (2 \, x\right )} - 10 \, {\left (4 \, x^{3} - 40 \, x^{2} + 125 \, x - 125\right )} e^{x} - 18750 \, x + 15625\right )} e^{\frac {10}{x}} - 20 \, {\left (250 \, x^{4} - 4375 \, x^{3} + 28125 \, x^{2} - {\left (2 \, x - 5\right )} e^{\left (3 \, x\right )} + 15 \, {\left (2 \, x^{2} - 15 \, x + 25\right )} e^{\left (2 \, x\right )} - 75 \, {\left (2 \, x^{3} - 25 \, x^{2} + 100 \, x - 125\right )} e^{x} - 78125 \, x + 78125\right )} e^{\frac {5}{x}} - 312500 \, x + e^{\left (4 \, x\right )} + 390625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-4*x^2+x)*exp(x)+(-30*x^2+175*x-500)*exp(5/x)+15*x^2+25*x)/(x*exp(x)^5+((50*x^2-125*x)*exp(5/x)-25
*x^2+125*x)*exp(x)^4+((1000*x^3-5000*x^2+6250*x)*exp(5/x)^2+(-1000*x^3+7500*x^2-12500*x)*exp(5/x)+250*x^3-2500
*x^2+6250*x)*exp(x)^3+((10000*x^4-75000*x^3+187500*x^2-156250*x)*exp(5/x)^3+(-15000*x^4+150000*x^3-468750*x^2+
468750*x)*exp(5/x)^2+(7500*x^4-93750*x^3+375000*x^2-468750*x)*exp(5/x)-1250*x^4+18750*x^3-93750*x^2+156250*x)*
exp(x)^2+((50000*x^5-500000*x^4+1875000*x^3-3125000*x^2+1953125*x)*exp(5/x)^4+(-100000*x^5+1250000*x^4-5625000
*x^3+10937500*x^2-7812500*x)*exp(5/x)^3+(75000*x^5-1125000*x^4+6093750*x^3-14062500*x^2+11718750*x)*exp(5/x)^2
+(-25000*x^5+437500*x^4-2812500*x^3+7812500*x^2-7812500*x)*exp(5/x)+3125*x^5-62500*x^4+468750*x^3-1562500*x^2+
1953125*x)*exp(x)+(100000*x^6-1250000*x^5+6250000*x^4-15625000*x^3+19531250*x^2-9765625*x)*exp(5/x)^5+(-250000
*x^6+3750000*x^5-21875000*x^4+62500000*x^3-87890625*x^2+48828125*x)*exp(5/x)^4+(250000*x^6-4375000*x^5+2968750
0*x^4-97656250*x^3+156250000*x^2-97656250*x)*exp(5/x)^3+(-125000*x^6+2500000*x^5-19531250*x^4+74218750*x^3-136
718750*x^2+97656250*x)*exp(5/x)^2+(31250*x^6-703125*x^5+6250000*x^4-27343750*x^3+58593750*x^2-48828125*x)*exp(
5/x)-3125*x^6+78125*x^5-781250*x^4+3906250*x^3-9765625*x^2+9765625*x),x, algorithm="maxima")
[Out]
x/(625*x^4 - 12500*x^3 + 93750*x^2 - 20*(x - 5)*e^(3*x) + 150*(x^2 - 10*x + 25)*e^(2*x) - 500*(x^3 - 15*x^2 +
75*x - 125)*e^x + 625*(16*x^4 - 160*x^3 + 600*x^2 - 1000*x + 625)*e^(20/x) - 500*(40*x^4 - 500*x^3 + 2250*x^2
- (8*x^3 - 60*x^2 + 150*x - 125)*e^x - 4375*x + 3125)*e^(15/x) + 150*(100*x^4 - 1500*x^3 + 8125*x^2 + (4*x^2 -
20*x + 25)*e^(2*x) - 10*(4*x^3 - 40*x^2 + 125*x - 125)*e^x - 18750*x + 15625)*e^(10/x) - 20*(250*x^4 - 4375*x
^3 + 28125*x^2 - (2*x - 5)*e^(3*x) + 15*(2*x^2 - 15*x + 25)*e^(2*x) - 75*(2*x^3 - 25*x^2 + 100*x - 125)*e^x -
78125*x + 78125)*e^(5/x) - 312500*x + e^(4*x) + 390625)
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {25\,x+{\mathrm {e}}^x\,\left (x-4\,x^2\right )-{\mathrm {e}}^{5/x}\,\left (30\,x^2-175\,x+500\right )+15\,x^2}{9765625\,x+x\,{\mathrm {e}}^{5\,x}-{\mathrm {e}}^{25/x}\,\left (-100000\,x^6+1250000\,x^5-6250000\,x^4+15625000\,x^3-19531250\,x^2+9765625\,x\right )-{\mathrm {e}}^{5/x}\,\left (-31250\,x^6+703125\,x^5-6250000\,x^4+27343750\,x^3-58593750\,x^2+48828125\,x\right )+{\mathrm {e}}^{20/x}\,\left (-250000\,x^6+3750000\,x^5-21875000\,x^4+62500000\,x^3-87890625\,x^2+48828125\,x\right )+{\mathrm {e}}^{10/x}\,\left (-125000\,x^6+2500000\,x^5-19531250\,x^4+74218750\,x^3-136718750\,x^2+97656250\,x\right )-{\mathrm {e}}^{15/x}\,\left (-250000\,x^6+4375000\,x^5-29687500\,x^4+97656250\,x^3-156250000\,x^2+97656250\,x\right )-{\mathrm {e}}^{4\,x}\,\left ({\mathrm {e}}^{5/x}\,\left (125\,x-50\,x^2\right )-125\,x+25\,x^2\right )+{\mathrm {e}}^{3\,x}\,\left (6250\,x+{\mathrm {e}}^{10/x}\,\left (1000\,x^3-5000\,x^2+6250\,x\right )-{\mathrm {e}}^{5/x}\,\left (1000\,x^3-7500\,x^2+12500\,x\right )-2500\,x^2+250\,x^3\right )+{\mathrm {e}}^x\,\left (1953125\,x+{\mathrm {e}}^{20/x}\,\left (50000\,x^5-500000\,x^4+1875000\,x^3-3125000\,x^2+1953125\,x\right )-{\mathrm {e}}^{5/x}\,\left (25000\,x^5-437500\,x^4+2812500\,x^3-7812500\,x^2+7812500\,x\right )-{\mathrm {e}}^{15/x}\,\left (100000\,x^5-1250000\,x^4+5625000\,x^3-10937500\,x^2+7812500\,x\right )+{\mathrm {e}}^{10/x}\,\left (75000\,x^5-1125000\,x^4+6093750\,x^3-14062500\,x^2+11718750\,x\right )-1562500\,x^2+468750\,x^3-62500\,x^4+3125\,x^5\right )-9765625\,x^2+3906250\,x^3-781250\,x^4+78125\,x^5-3125\,x^6-{\mathrm {e}}^{2\,x}\,\left ({\mathrm {e}}^{15/x}\,\left (-10000\,x^4+75000\,x^3-187500\,x^2+156250\,x\right )-156250\,x+{\mathrm {e}}^{5/x}\,\left (-7500\,x^4+93750\,x^3-375000\,x^2+468750\,x\right )-{\mathrm {e}}^{10/x}\,\left (-15000\,x^4+150000\,x^3-468750\,x^2+468750\,x\right )+93750\,x^2-18750\,x^3+1250\,x^4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((25*x + exp(x)*(x - 4*x^2) - exp(5/x)*(30*x^2 - 175*x + 500) + 15*x^2)/(9765625*x + x*exp(5*x) - exp(25/x)
*(9765625*x - 19531250*x^2 + 15625000*x^3 - 6250000*x^4 + 1250000*x^5 - 100000*x^6) - exp(5/x)*(48828125*x - 5
8593750*x^2 + 27343750*x^3 - 6250000*x^4 + 703125*x^5 - 31250*x^6) + exp(20/x)*(48828125*x - 87890625*x^2 + 62
500000*x^3 - 21875000*x^4 + 3750000*x^5 - 250000*x^6) + exp(10/x)*(97656250*x - 136718750*x^2 + 74218750*x^3 -
19531250*x^4 + 2500000*x^5 - 125000*x^6) - exp(15/x)*(97656250*x - 156250000*x^2 + 97656250*x^3 - 29687500*x^
4 + 4375000*x^5 - 250000*x^6) - exp(4*x)*(exp(5/x)*(125*x - 50*x^2) - 125*x + 25*x^2) + exp(3*x)*(6250*x + exp
(10/x)*(6250*x - 5000*x^2 + 1000*x^3) - exp(5/x)*(12500*x - 7500*x^2 + 1000*x^3) - 2500*x^2 + 250*x^3) + exp(x
)*(1953125*x + exp(20/x)*(1953125*x - 3125000*x^2 + 1875000*x^3 - 500000*x^4 + 50000*x^5) - exp(5/x)*(7812500*
x - 7812500*x^2 + 2812500*x^3 - 437500*x^4 + 25000*x^5) - exp(15/x)*(7812500*x - 10937500*x^2 + 5625000*x^3 -
1250000*x^4 + 100000*x^5) + exp(10/x)*(11718750*x - 14062500*x^2 + 6093750*x^3 - 1125000*x^4 + 75000*x^5) - 15
62500*x^2 + 468750*x^3 - 62500*x^4 + 3125*x^5) - 9765625*x^2 + 3906250*x^3 - 781250*x^4 + 78125*x^5 - 3125*x^6
- exp(2*x)*(exp(15/x)*(156250*x - 187500*x^2 + 75000*x^3 - 10000*x^4) - 156250*x + exp(5/x)*(468750*x - 37500
0*x^2 + 93750*x^3 - 7500*x^4) - exp(10/x)*(468750*x - 468750*x^2 + 150000*x^3 - 15000*x^4) + 93750*x^2 - 18750
*x^3 + 1250*x^4)),x)
[Out]
int((25*x + exp(x)*(x - 4*x^2) - exp(5/x)*(30*x^2 - 175*x + 500) + 15*x^2)/(9765625*x + x*exp(5*x) - exp(25/x)
*(9765625*x - 19531250*x^2 + 15625000*x^3 - 6250000*x^4 + 1250000*x^5 - 100000*x^6) - exp(5/x)*(48828125*x - 5
8593750*x^2 + 27343750*x^3 - 6250000*x^4 + 703125*x^5 - 31250*x^6) + exp(20/x)*(48828125*x - 87890625*x^2 + 62
500000*x^3 - 21875000*x^4 + 3750000*x^5 - 250000*x^6) + exp(10/x)*(97656250*x - 136718750*x^2 + 74218750*x^3 -
19531250*x^4 + 2500000*x^5 - 125000*x^6) - exp(15/x)*(97656250*x - 156250000*x^2 + 97656250*x^3 - 29687500*x^
4 + 4375000*x^5 - 250000*x^6) - exp(4*x)*(exp(5/x)*(125*x - 50*x^2) - 125*x + 25*x^2) + exp(3*x)*(6250*x + exp
(10/x)*(6250*x - 5000*x^2 + 1000*x^3) - exp(5/x)*(12500*x - 7500*x^2 + 1000*x^3) - 2500*x^2 + 250*x^3) + exp(x
)*(1953125*x + exp(20/x)*(1953125*x - 3125000*x^2 + 1875000*x^3 - 500000*x^4 + 50000*x^5) - exp(5/x)*(7812500*
x - 7812500*x^2 + 2812500*x^3 - 437500*x^4 + 25000*x^5) - exp(15/x)*(7812500*x - 10937500*x^2 + 5625000*x^3 -
1250000*x^4 + 100000*x^5) + exp(10/x)*(11718750*x - 14062500*x^2 + 6093750*x^3 - 1125000*x^4 + 75000*x^5) - 15
62500*x^2 + 468750*x^3 - 62500*x^4 + 3125*x^5) - 9765625*x^2 + 3906250*x^3 - 781250*x^4 + 78125*x^5 - 3125*x^6
- exp(2*x)*(exp(15/x)*(156250*x - 187500*x^2 + 75000*x^3 - 10000*x^4) - 156250*x + exp(5/x)*(468750*x - 37500
0*x^2 + 93750*x^3 - 7500*x^4) - exp(10/x)*(468750*x - 468750*x^2 + 150000*x^3 - 15000*x^4) + 93750*x^2 - 18750
*x^3 + 1250*x^4)), x)
________________________________________________________________________________________
sympy [B] time = 1.70, size = 427, normalized size = 15.25 \begin {gather*} \frac {x}{10000 x^{4} e^{\frac {20}{x}} - 20000 x^{4} e^{\frac {15}{x}} + 15000 x^{4} e^{\frac {10}{x}} - 5000 x^{4} e^{\frac {5}{x}} + 625 x^{4} - 100000 x^{3} e^{\frac {20}{x}} + 250000 x^{3} e^{\frac {15}{x}} - 225000 x^{3} e^{\frac {10}{x}} + 87500 x^{3} e^{\frac {5}{x}} - 12500 x^{3} + 375000 x^{2} e^{\frac {20}{x}} - 1125000 x^{2} e^{\frac {15}{x}} + 1218750 x^{2} e^{\frac {10}{x}} - 562500 x^{2} e^{\frac {5}{x}} + 93750 x^{2} - 625000 x e^{\frac {20}{x}} + 2187500 x e^{\frac {15}{x}} - 2812500 x e^{\frac {10}{x}} + 1562500 x e^{\frac {5}{x}} - 312500 x + \left (40 x e^{\frac {5}{x}} - 20 x - 100 e^{\frac {5}{x}} + 100\right ) e^{3 x} + \left (600 x^{2} e^{\frac {10}{x}} - 600 x^{2} e^{\frac {5}{x}} + 150 x^{2} - 3000 x e^{\frac {10}{x}} + 4500 x e^{\frac {5}{x}} - 1500 x + 3750 e^{\frac {10}{x}} - 7500 e^{\frac {5}{x}} + 3750\right ) e^{2 x} + \left (4000 x^{3} e^{\frac {15}{x}} - 6000 x^{3} e^{\frac {10}{x}} + 3000 x^{3} e^{\frac {5}{x}} - 500 x^{3} - 30000 x^{2} e^{\frac {15}{x}} + 60000 x^{2} e^{\frac {10}{x}} - 37500 x^{2} e^{\frac {5}{x}} + 7500 x^{2} + 75000 x e^{\frac {15}{x}} - 187500 x e^{\frac {10}{x}} + 150000 x e^{\frac {5}{x}} - 37500 x - 62500 e^{\frac {15}{x}} + 187500 e^{\frac {10}{x}} - 187500 e^{\frac {5}{x}} + 62500\right ) e^{x} + 390625 e^{\frac {20}{x}} - 1562500 e^{\frac {15}{x}} + 2343750 e^{\frac {10}{x}} - 1562500 e^{\frac {5}{x}} + e^{4 x} + 390625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-4*x**2+x)*exp(x)+(-30*x**2+175*x-500)*exp(5/x)+15*x**2+25*x)/(x*exp(x)**5+((50*x**2-125*x)*exp(5/
x)-25*x**2+125*x)*exp(x)**4+((1000*x**3-5000*x**2+6250*x)*exp(5/x)**2+(-1000*x**3+7500*x**2-12500*x)*exp(5/x)+
250*x**3-2500*x**2+6250*x)*exp(x)**3+((10000*x**4-75000*x**3+187500*x**2-156250*x)*exp(5/x)**3+(-15000*x**4+15
0000*x**3-468750*x**2+468750*x)*exp(5/x)**2+(7500*x**4-93750*x**3+375000*x**2-468750*x)*exp(5/x)-1250*x**4+187
50*x**3-93750*x**2+156250*x)*exp(x)**2+((50000*x**5-500000*x**4+1875000*x**3-3125000*x**2+1953125*x)*exp(5/x)*
*4+(-100000*x**5+1250000*x**4-5625000*x**3+10937500*x**2-7812500*x)*exp(5/x)**3+(75000*x**5-1125000*x**4+60937
50*x**3-14062500*x**2+11718750*x)*exp(5/x)**2+(-25000*x**5+437500*x**4-2812500*x**3+7812500*x**2-7812500*x)*ex
p(5/x)+3125*x**5-62500*x**4+468750*x**3-1562500*x**2+1953125*x)*exp(x)+(100000*x**6-1250000*x**5+6250000*x**4-
15625000*x**3+19531250*x**2-9765625*x)*exp(5/x)**5+(-250000*x**6+3750000*x**5-21875000*x**4+62500000*x**3-8789
0625*x**2+48828125*x)*exp(5/x)**4+(250000*x**6-4375000*x**5+29687500*x**4-97656250*x**3+156250000*x**2-9765625
0*x)*exp(5/x)**3+(-125000*x**6+2500000*x**5-19531250*x**4+74218750*x**3-136718750*x**2+97656250*x)*exp(5/x)**2
+(31250*x**6-703125*x**5+6250000*x**4-27343750*x**3+58593750*x**2-48828125*x)*exp(5/x)-3125*x**6+78125*x**5-78
1250*x**4+3906250*x**3-9765625*x**2+9765625*x),x)
[Out]
x/(10000*x**4*exp(20/x) - 20000*x**4*exp(15/x) + 15000*x**4*exp(10/x) - 5000*x**4*exp(5/x) + 625*x**4 - 100000
*x**3*exp(20/x) + 250000*x**3*exp(15/x) - 225000*x**3*exp(10/x) + 87500*x**3*exp(5/x) - 12500*x**3 + 375000*x*
*2*exp(20/x) - 1125000*x**2*exp(15/x) + 1218750*x**2*exp(10/x) - 562500*x**2*exp(5/x) + 93750*x**2 - 625000*x*
exp(20/x) + 2187500*x*exp(15/x) - 2812500*x*exp(10/x) + 1562500*x*exp(5/x) - 312500*x + (40*x*exp(5/x) - 20*x
- 100*exp(5/x) + 100)*exp(3*x) + (600*x**2*exp(10/x) - 600*x**2*exp(5/x) + 150*x**2 - 3000*x*exp(10/x) + 4500*
x*exp(5/x) - 1500*x + 3750*exp(10/x) - 7500*exp(5/x) + 3750)*exp(2*x) + (4000*x**3*exp(15/x) - 6000*x**3*exp(1
0/x) + 3000*x**3*exp(5/x) - 500*x**3 - 30000*x**2*exp(15/x) + 60000*x**2*exp(10/x) - 37500*x**2*exp(5/x) + 750
0*x**2 + 75000*x*exp(15/x) - 187500*x*exp(10/x) + 150000*x*exp(5/x) - 37500*x - 62500*exp(15/x) + 187500*exp(1
0/x) - 187500*exp(5/x) + 62500)*exp(x) + 390625*exp(20/x) - 1562500*exp(15/x) + 2343750*exp(10/x) - 1562500*ex
p(5/x) + exp(4*x) + 390625)
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