3.85.21 $\int \frac{25x+15x^2+e^{5/x}(-500+175x-30x^2)+e^x(x-4x^2)}{9765625x+e^{5x}x-9765625x^2+3906250x^3-781250x^4+78125x^5-3125x^6+e^{20/x}(48828125x-87890625x^2+62500000x^3-21875000x^4+3750000x^5-250000x^6)+e^{10/x}(97656250x-136718750x^2+74218750x^3-19531250x^4+2500000x^5-125000x^6)+e^{5/x}(-48828125x+58593750x^2-27343750x^3+6250000x^4-703125x^5+31250x^6)+e^{25/x}(-9765625x+19531250x^2-15625000x^3+6250000x^4-1250000x^5+100000x^6)+e^{15/x}(-97656250x+156250000x^2-97656250x^3+29687500x^4-4375000x^5+250000x^6)+e^{4x}(125x-25x^2+e^{5/x}(-125x+50x^2))+e^{3x}(6250x-2500x^2+250x^3+e^{5/x}(-12500x+7500x^2-1000x^3)+e^{10/x}(6250x-5000x^2+1000x^3))+e^{2x}(156250x-93750x^2+18750x^3-1250x^4+e^{10/x}(468750x-468750x^2+150000x^3-15000x^4)+e^{5/x}(-468750x+375000x^2-93750x^3+7500x^4)+e^{15/x}(-156250x+187500x^2-75000x^3+10000x^4))+e^x(1953125x-1562500x^2+468750x^3-62500x^4+3125x^5+e^{15/x}(-7812500x+10937500x^2-5625000x^3+1250000x^4-100000x^5)+e^{5/x}(-7812500x+7812500x^2-2812500x^3+437500x^4-25000x^5)+e^{20/x}(1953125x-3125000x^2+1875000x^3-500000x^4+50000x^5)+e^{10/x}(11718750x-14062500x^2+6093750x^3-1125000x^4+75000x^5))}dx$

Optimal. Leaf size=28 $$\frac{x}{{(e^x+5(5-x+e^{5/x}(-5+2x)))}^4}$$

________________________________________________________________________________________

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 = {} $$\begin{array}{c}\int \frac{25x+15x^2+e^{5/x}(-500+175x-30x^2)+e^x(x-4x^2)}{9765625x+e^{5x}x-9765625x^2+3906250x^3-781250x^4+78125x^5-3125x^6+e^{20/x}(48828125x-87890625x^2+62500000x^3-21875000x^4+3750000x^5-250000x^6)+e^{10/x}(97656250x-136718750x^2+74218750x^3-19531250x^4+2500000x^5-125000x^6)+e^{5/x}(-48828125x+58593750x^2-27343750x^3+6250000x^4-703125x^5+31250x^6)+e^{25/x}(-9765625x+19531250x^2-15625000x^3+6250000x^4-1250000x^5+100000x^6)+e^{15/x}(-97656250x+156250000x^2-97656250x^3+29687500x^4-4375000x^5+250000x^6)+e^{4x}(125x-25x^2+e^{5/x}(-125x+50x^2))+e^{3x}(6250x-2500x^2+250x^3+e^{5/x}(-12500x+7500x^2-1000x^3)+e^{10/x}(6250x-5000x^2+1000x^3))+e^{2x}(156250x-93750x^2+18750x^3-1250x^4+e^{10/x}(468750x-468750x^2+150000x^3-15000x^4)+e^{5/x}(-468750x+375000x^2-93750x^3+7500x^4)+e^{15/x}(-156250x+187500x^2-75000x^3+10000x^4))+e^x(1953125x-1562500x^2+468750x^3-62500x^4+3125x^5+e^{15/x}(-7812500x+10937500x^2-5625000x^3+1250000x^4-100000x^5)+e^{5/x}(-7812500x+7812500x^2-2812500x^3+437500x^4-25000x^5)+e^{20/x}(1953125x-3125000x^2+1875000x^3-500000x^4+50000x^5)+e^{10/x}(11718750x-14062500x^2+6093750x^3-1125000x^4+75000x^5))}dx\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{5x(5+3x)+e^x(x-4x^2)-5e^{5/x}(100-35x+6x^2)}{x{(e^x-5(-5+x)+5e^{5/x}(-5+2x))}^5}dx\\ & =\int (-\frac{-1+4x}{{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^4}+\frac{20(-25e^{5/x}+10e^{5/x}x+6x^2-7e^{5/x}{x}^2-x^3+2e^{5/x}{x}^3)}{x{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^5})dx\\ & =20\int \frac{-25e^{5/x}+10e^{5/x}x+6x^2-7e^{5/x}{x}^2-x^3+2e^{5/x}{x}^3}{x{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^5}dx-\int \frac{-1+4x}{{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^4}dx\\ & =20\int \frac{-((-6+x)x^2)+e^{5/x}(-25+10x-7x^2+2x^3)}{x{(e^x-5(-5+x)+5e^{5/x}(-5+2x))}^5}dx-\int (-\frac{1}{{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^4}+\frac{4x}{{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^4})dx\\ & =-(4\int \frac{x}{{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^4}dx)+20\int (\frac{10e^{5/x}}{{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^5}-\frac{25e^{5/x}}{x{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^5}+\frac{6x}{{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^5}-\frac{7e^{5/x}x}{{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^5}-\frac{x^2}{{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^5}+\frac{2e^{5/x}{x}^2}{{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^5})dx+\int \frac{1}{{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^4}dx\\ & =-(4\int \frac{x}{{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^4}dx)-20\int \frac{x^2}{{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^5}dx+40\int \frac{e^{5/x}{x}^2}{{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^5}dx+120\int \frac{x}{{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^5}dx-140\int \frac{e^{5/x}x}{{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^5}dx+200\int \frac{e^{5/x}}{{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^5}dx-500\int \frac{e^{5/x}}{x{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^5}dx+\int \frac{1}{{(25-25e^{5/x}+e^x-5x+10e^{5/x}x)}^4}dx\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 1.03, size = 27, normalized size = 0.96 $$\begin{array}{c}\frac{x}{{(e^x-5(-5+x)+5e^{5/x}(-5+2x))}^4}\end{array}$$

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

________________________________________________________________________________________

fricas [B]  time = 0.71, size = 298, normalized size = 10.64 $$\begin{array}{c}\frac{x}{625x^4-12500x^3+93750x^2+20((2x-5)e^{\frac{5}{x}}-x+5)e^{\left(3x\right)}+150(x^2+(4x^2-20x+25)e^{\frac{10}{x}}-2(2x^2-15x+25)e^{\frac{5}{x}}-10x+25)e^{\left(2x\right)}-500(x^3-15x^2-(8x^3-60x^2+150x-125)e^{\frac{15}{x}}+3(4x^3-40x^2+125x-125)e^{\frac{10}{x}}-3(2x^3-25x^2+100x-125)e^{\frac{5}{x}}+75x-125)e^x+625(16x^4-160x^3+600x^2-1000x+625)e^{\frac{20}{x}}-2500(8x^4-100x^3+450x^2-875x+625)e^{\frac{15}{x}}+3750(4x^4-60x^3+325x^2-750x+625)e^{\frac{10}{x}}-2500(2x^4-35x^3+225x^2-625x+625)e^{\frac{5}{x}}-312500x+e^{\left(4x\right)}+390625}\end{array}$$

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))

________________________________________________________________________________________

maple [A]  time = 0.71, size = 29, normalized size = 1.04

method	result	size
risch	$\frac{x}{{(10x{\mathrm{e}}^{\frac{5}{x}}+{\mathrm{e}}^x-25{\mathrm{e}}^{\frac{5}{x}}-5x+25)}^4}$	$29$

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

________________________________________________________________________________________

maxima [B]  time = 2.04, size = 279, normalized size = 9.96 $$\begin{array}{c}\frac{x}{625x^4-12500x^3+93750x^2-20(x-5)e^{\left(3x\right)}+150(x^2-10x+25)e^{\left(2x\right)}-500(x^3-15x^2+75x-125)e^x+625(16x^4-160x^3+600x^2-1000x+625)e^{\frac{20}{x}}-500(40x^4-500x^3+2250x^2-(8x^3-60x^2+150x-125)e^x-4375x+3125)e^{\frac{15}{x}}+150(100x^4-1500x^3+8125x^2+(4x^2-20x+25)e^{\left(2x\right)}-10(4x^3-40x^2+125x-125)e^x-18750x+15625)e^{\frac{10}{x}}-20(250x^4-4375x^3+28125x^2-(2x-5)e^{\left(3x\right)}+15(2x^2-15x+25)e^{\left(2x\right)}-75(2x^3-25x^2+100x-125)e^x-78125x+78125)e^{\frac{5}{x}}-312500x+e^{\left(4x\right)}+390625}\end{array}$$

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 $$\begin{array}{c}\int \frac{25x+{\mathrm{e}}^x(x-4x^2)-{\mathrm{e}}^{5/x}(30x^2-175x+500)+15x^2}{9765625x+x{\mathrm{e}}^{5x}-{\mathrm{e}}^{25/x}(-100000x^6+1250000x^5-6250000x^4+15625000x^3-19531250x^2+9765625x)-{\mathrm{e}}^{5/x}(-31250x^6+703125x^5-6250000x^4+27343750x^3-58593750x^2+48828125x)+{\mathrm{e}}^{20/x}(-250000x^6+3750000x^5-21875000x^4+62500000x^3-87890625x^2+48828125x)+{\mathrm{e}}^{10/x}(-125000x^6+2500000x^5-19531250x^4+74218750x^3-136718750x^2+97656250x)-{\mathrm{e}}^{15/x}(-250000x^6+4375000x^5-29687500x^4+97656250x^3-156250000x^2+97656250x)-{\mathrm{e}}^{4x}({\mathrm{e}}^{5/x}(125x-50x^2)-125x+25x^2)+{\mathrm{e}}^{3x}(6250x+{\mathrm{e}}^{10/x}(1000x^3-5000x^2+6250x)-{\mathrm{e}}^{5/x}(1000x^3-7500x^2+12500x)-2500x^2+250x^3)+{\mathrm{e}}^x(1953125x+{\mathrm{e}}^{20/x}(50000x^5-500000x^4+1875000x^3-3125000x^2+1953125x)-{\mathrm{e}}^{5/x}(25000x^5-437500x^4+2812500x^3-7812500x^2+7812500x)-{\mathrm{e}}^{15/x}(100000x^5-1250000x^4+5625000x^3-10937500x^2+7812500x)+{\mathrm{e}}^{10/x}(75000x^5-1125000x^4+6093750x^3-14062500x^2+11718750x)-1562500x^2+468750x^3-62500x^4+3125x^5)-9765625x^2+3906250x^3-781250x^4+78125x^5-3125x^6-{\mathrm{e}}^{2x}({\mathrm{e}}^{15/x}(-10000x^4+75000x^3-187500x^2+156250x)-156250x+{\mathrm{e}}^{5/x}(-7500x^4+93750x^3-375000x^2+468750x)-{\mathrm{e}}^{10/x}(-15000x^4+150000x^3-468750x^2+468750x)+93750x^2-18750x^3+1250x^4)}dx\end{array}$$

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 $$\begin{array}{c}\frac{x}{10000x^4{e}^{\frac{20}{x}}-20000x^4{e}^{\frac{15}{x}}+15000x^4{e}^{\frac{10}{x}}-5000x^4{e}^{\frac{5}{x}}+625x^4-100000x^3{e}^{\frac{20}{x}}+250000x^3{e}^{\frac{15}{x}}-225000x^3{e}^{\frac{10}{x}}+87500x^3{e}^{\frac{5}{x}}-12500x^3+375000x^2{e}^{\frac{20}{x}}-1125000x^2{e}^{\frac{15}{x}}+1218750x^2{e}^{\frac{10}{x}}-562500x^2{e}^{\frac{5}{x}}+93750x^2-625000x{e}^{\frac{20}{x}}+2187500x{e}^{\frac{15}{x}}-2812500x{e}^{\frac{10}{x}}+1562500x{e}^{\frac{5}{x}}-312500x+(40x{e}^{\frac{5}{x}}-20x-100e^{\frac{5}{x}}+100)e^{3x}+(600x^2{e}^{\frac{10}{x}}-600x^2{e}^{\frac{5}{x}}+150x^2-3000x{e}^{\frac{10}{x}}+4500x{e}^{\frac{5}{x}}-1500x+3750e^{\frac{10}{x}}-7500e^{\frac{5}{x}}+3750)e^{2x}+(4000x^3{e}^{\frac{15}{x}}-6000x^3{e}^{\frac{10}{x}}+3000x^3{e}^{\frac{5}{x}}-500x^3-30000x^2{e}^{\frac{15}{x}}+60000x^2{e}^{\frac{10}{x}}-37500x^2{e}^{\frac{5}{x}}+7500x^2+75000x{e}^{\frac{15}{x}}-187500x{e}^{\frac{10}{x}}+150000x{e}^{\frac{5}{x}}-37500x-62500e^{\frac{15}{x}}+187500e^{\frac{10}{x}}-187500e^{\frac{5}{x}}+62500)e^x+390625e^{\frac{20}{x}}-1562500e^{\frac{15}{x}}+2343750e^{\frac{10}{x}}-1562500e^{\frac{5}{x}}+e^{4x}+390625}\end{array}$$

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)

________________________________________________________________________________________