3.13.12 \(\int \frac {390625+e^{10/x}-625000 x+437500 x^2-175000 x^3+43750 x^4-7000 x^5+700 x^6-40 x^7+x^8+e^{5/x} (1250-1000 x+300 x^2-40 x^3+2 x^4)+e^{\frac {1250+2 e^{5/x}-1000 x+301 x^2-40 x^3+2 x^4}{625+e^{5/x}-500 x+150 x^2-20 x^3+x^4}} (e^{5/x} (-80-32 x)-20000 x+8000 x^2-320 x^4+32 x^5)+e^{\frac {2 (1250+2 e^{5/x}-1000 x+301 x^2-40 x^3+2 x^4)}{625+e^{5/x}-500 x+150 x^2-20 x^3+x^4}} (10000 x-4000 x^2+160 x^4-16 x^5+e^{5/x} (40+16 x))}{390625+e^{10/x}-625000 x+437500 x^2-175000 x^3+43750 x^4-7000 x^5+700 x^6-40 x^7+x^8+e^{5/x} (1250-1000 x+300 x^2-40 x^3+2 x^4)} \, dx\)

Optimal. Leaf size=36 \[ -1+4 \left (2-e^{2+\frac {x^2}{e^{5/x}+(5-x)^4}}\right )^2+x \]

________________________________________________________________________________________

Rubi [F]  time = 92.88, 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 {390625+e^{10/x}-625000 x+437500 x^2-175000 x^3+43750 x^4-7000 x^5+700 x^6-40 x^7+x^8+e^{5/x} \left (1250-1000 x+300 x^2-40 x^3+2 x^4\right )+\exp \left (\frac {1250+2 e^{5/x}-1000 x+301 x^2-40 x^3+2 x^4}{625+e^{5/x}-500 x+150 x^2-20 x^3+x^4}\right ) \left (e^{5/x} (-80-32 x)-20000 x+8000 x^2-320 x^4+32 x^5\right )+\exp \left (\frac {2 \left (1250+2 e^{5/x}-1000 x+301 x^2-40 x^3+2 x^4\right )}{625+e^{5/x}-500 x+150 x^2-20 x^3+x^4}\right ) \left (10000 x-4000 x^2+160 x^4-16 x^5+e^{5/x} (40+16 x)\right )}{390625+e^{10/x}-625000 x+437500 x^2-175000 x^3+43750 x^4-7000 x^5+700 x^6-40 x^7+x^8+e^{5/x} \left (1250-1000 x+300 x^2-40 x^3+2 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(390625 + E^(10/x) - 625000*x + 437500*x^2 - 175000*x^3 + 43750*x^4 - 7000*x^5 + 700*x^6 - 40*x^7 + x^8 +
E^(5/x)*(1250 - 1000*x + 300*x^2 - 40*x^3 + 2*x^4) + E^((1250 + 2*E^(5/x) - 1000*x + 301*x^2 - 40*x^3 + 2*x^4)
/(625 + E^(5/x) - 500*x + 150*x^2 - 20*x^3 + x^4))*(E^(5/x)*(-80 - 32*x) - 20000*x + 8000*x^2 - 320*x^4 + 32*x
^5) + E^((2*(1250 + 2*E^(5/x) - 1000*x + 301*x^2 - 40*x^3 + 2*x^4))/(625 + E^(5/x) - 500*x + 150*x^2 - 20*x^3
+ x^4))*(10000*x - 4000*x^2 + 160*x^4 - 16*x^5 + E^(5/x)*(40 + 16*x)))/(390625 + E^(10/x) - 625000*x + 437500*
x^2 - 175000*x^3 + 43750*x^4 - 7000*x^5 + 700*x^6 - 40*x^7 + x^8 + E^(5/x)*(1250 - 1000*x + 300*x^2 - 40*x^3 +
2*x^4)),x]

[Out]

390625*Defer[Int][(E^(5/x) + (-5 + x)^4)^(-2), x] + 1250*Defer[Int][E^(5/x)/(E^(5/x) + (-5 + x)^4)^2, x] + Def
er[Int][E^(10/x)/(E^(5/x) + (-5 + x)^4)^2, x] + 50000*Defer[Int][E^((1250 + 2*E^(5/x) - 1000*x + 301*x^2 - 40*
x^3 + 2*x^4)/(E^(5/x) + (-5 + x)^4))/(E^(5/x) + (-5 + x)^4)^2, x] - 25000*Defer[Int][E^((2*(1250 + 2*E^(5/x) -
1000*x + 301*x^2 - 40*x^3 + 2*x^4))/(E^(5/x) + (-5 + x)^4))/(E^(5/x) + (-5 + x)^4)^2, x] - 80*Defer[Int][E^((
1250 + 2*E^(5/x) - 1000*x + 301*x^2 - 40*x^3 + 2*x^4)/(E^(5/x) + (-5 + x)^4))/(E^(5/x) + (-5 + x)^4), x] + 40*
Defer[Int][E^((2*(1250 + 2*E^(5/x) - 1000*x + 301*x^2 - 40*x^3 + 2*x^4))/(E^(5/x) + (-5 + x)^4))/(E^(5/x) + (-
5 + x)^4), x] - 625000*Defer[Int][x/(E^(5/x) + (-5 + x)^4)^2, x] - 1000*Defer[Int][(E^(5/x)*x)/(E^(5/x) + (-5
+ x)^4)^2, x] - 40000*Defer[Int][(E^((1250 + 2*E^(5/x) - 1000*x + 301*x^2 - 40*x^3 + 2*x^4)/(E^(5/x) + (-5 + x
)^4))*x)/(E^(5/x) + (-5 + x)^4)^2, x] + 20000*Defer[Int][(E^((2*(1250 + 2*E^(5/x) - 1000*x + 301*x^2 - 40*x^3
+ 2*x^4))/(E^(5/x) + (-5 + x)^4))*x)/(E^(5/x) + (-5 + x)^4)^2, x] - 32*Defer[Int][(E^((1250 + 2*E^(5/x) - 1000
*x + 301*x^2 - 40*x^3 + 2*x^4)/(E^(5/x) + (-5 + x)^4))*x)/(E^(5/x) + (-5 + x)^4), x] + 16*Defer[Int][(E^((2*(1
250 + 2*E^(5/x) - 1000*x + 301*x^2 - 40*x^3 + 2*x^4))/(E^(5/x) + (-5 + x)^4))*x)/(E^(5/x) + (-5 + x)^4), x] +
437500*Defer[Int][x^2/(E^(5/x) + (-5 + x)^4)^2, x] + 300*Defer[Int][(E^(5/x)*x^2)/(E^(5/x) + (-5 + x)^4)^2, x]
+ 4000*Defer[Int][(E^((1250 + 2*E^(5/x) - 1000*x + 301*x^2 - 40*x^3 + 2*x^4)/(E^(5/x) + (-5 + x)^4))*x^2)/(E^
(5/x) + (-5 + x)^4)^2, x] - 2000*Defer[Int][(E^((2*(1250 + 2*E^(5/x) - 1000*x + 301*x^2 - 40*x^3 + 2*x^4))/(E^
(5/x) + (-5 + x)^4))*x^2)/(E^(5/x) + (-5 + x)^4)^2, x] - 175000*Defer[Int][x^3/(E^(5/x) + (-5 + x)^4)^2, x] -
40*Defer[Int][(E^(5/x)*x^3)/(E^(5/x) + (-5 + x)^4)^2, x] + 3200*Defer[Int][(E^((1250 + 2*E^(5/x) - 1000*x + 30
1*x^2 - 40*x^3 + 2*x^4)/(E^(5/x) + (-5 + x)^4))*x^3)/(E^(5/x) + (-5 + x)^4)^2, x] - 1600*Defer[Int][(E^((2*(12
50 + 2*E^(5/x) - 1000*x + 301*x^2 - 40*x^3 + 2*x^4))/(E^(5/x) + (-5 + x)^4))*x^3)/(E^(5/x) + (-5 + x)^4)^2, x]
+ 43750*Defer[Int][x^4/(E^(5/x) + (-5 + x)^4)^2, x] + 2*Defer[Int][(E^(5/x)*x^4)/(E^(5/x) + (-5 + x)^4)^2, x]
- 880*Defer[Int][(E^((1250 + 2*E^(5/x) - 1000*x + 301*x^2 - 40*x^3 + 2*x^4)/(E^(5/x) + (-5 + x)^4))*x^4)/(E^(
5/x) + (-5 + x)^4)^2, x] + 440*Defer[Int][(E^((2*(1250 + 2*E^(5/x) - 1000*x + 301*x^2 - 40*x^3 + 2*x^4))/(E^(5
/x) + (-5 + x)^4))*x^4)/(E^(5/x) + (-5 + x)^4)^2, x] - 7000*Defer[Int][x^5/(E^(5/x) + (-5 + x)^4)^2, x] + 64*D
efer[Int][(E^((1250 + 2*E^(5/x) - 1000*x + 301*x^2 - 40*x^3 + 2*x^4)/(E^(5/x) + (-5 + x)^4))*x^5)/(E^(5/x) + (
-5 + x)^4)^2, x] - 32*Defer[Int][(E^((2*(1250 + 2*E^(5/x) - 1000*x + 301*x^2 - 40*x^3 + 2*x^4))/(E^(5/x) + (-5
+ x)^4))*x^5)/(E^(5/x) + (-5 + x)^4)^2, x] + 700*Defer[Int][x^6/(E^(5/x) + (-5 + x)^4)^2, x] - 40*Defer[Int][
x^7/(E^(5/x) + (-5 + x)^4)^2, x] + Defer[Int][x^8/(E^(5/x) + (-5 + x)^4)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {390625+e^{10/x}+2 e^{5/x} (-5+x)^4-625000 x+437500 x^2-175000 x^3+43750 x^4-7000 x^5+700 x^6-40 x^7+x^8+\exp \left (\frac {1250+2 e^{5/x}-1000 x+301 x^2-40 x^3+2 x^4}{e^{5/x}+(-5+x)^4}\right ) \left (32 (-5+x)^3 x (5+x)-16 e^{5/x} (5+2 x)\right )+8 \exp \left (\frac {2500+4 e^{5/x}-2000 x+602 x^2-80 x^3+4 x^4}{e^{5/x}+(-5+x)^4}\right ) \left (-2 (-5+x)^3 x (5+x)+e^{5/x} (5+2 x)\right )}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx\\ &=\int \left (\frac {390625}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2}+\frac {e^{10/x}}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2}+\frac {2 e^{5/x} (-5+x)^4}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2}-\frac {625000 x}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2}+\frac {437500 x^2}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2}-\frac {175000 x^3}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2}+\frac {43750 x^4}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2}-\frac {7000 x^5}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2}+\frac {700 x^6}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2}-\frac {40 x^7}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2}+\frac {x^8}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2}+\frac {8 \exp \left (\frac {2 \left (1250+2 e^{5/x}-1000 x+301 x^2-40 x^3+2 x^4\right )}{625+e^{5/x}-500 x+150 x^2-20 x^3+x^4}\right ) \left (5 e^{5/x}+1250 x+2 e^{5/x} x-500 x^2+20 x^4-2 x^5\right )}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2}+\frac {16 \exp \left (\frac {1250+2 e^{5/x}-1000 x+301 x^2-40 x^3+2 x^4}{625+e^{5/x}-500 x+150 x^2-20 x^3+x^4}\right ) \left (-5 e^{5/x}-1250 x-2 e^{5/x} x+500 x^2-20 x^4+2 x^5\right )}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2}\right ) \, dx\\ &=2 \int \frac {e^{5/x} (-5+x)^4}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2} \, dx+8 \int \frac {\exp \left (\frac {2 \left (1250+2 e^{5/x}-1000 x+301 x^2-40 x^3+2 x^4\right )}{625+e^{5/x}-500 x+150 x^2-20 x^3+x^4}\right ) \left (5 e^{5/x}+1250 x+2 e^{5/x} x-500 x^2+20 x^4-2 x^5\right )}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2} \, dx+16 \int \frac {\exp \left (\frac {1250+2 e^{5/x}-1000 x+301 x^2-40 x^3+2 x^4}{625+e^{5/x}-500 x+150 x^2-20 x^3+x^4}\right ) \left (-5 e^{5/x}-1250 x-2 e^{5/x} x+500 x^2-20 x^4+2 x^5\right )}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2} \, dx-40 \int \frac {x^7}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2} \, dx+700 \int \frac {x^6}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2} \, dx-7000 \int \frac {x^5}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2} \, dx+43750 \int \frac {x^4}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2} \, dx-175000 \int \frac {x^3}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2} \, dx+390625 \int \frac {1}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2} \, dx+437500 \int \frac {x^2}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2} \, dx-625000 \int \frac {x}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2} \, dx+\int \frac {e^{10/x}}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2} \, dx+\int \frac {x^8}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2} \, dx\\ &=2 \int \frac {e^{5/x} (5-x)^4}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx+8 \int \frac {\exp \left (\frac {2 \left (1250+2 e^{5/x}-1000 x+301 x^2-40 x^3+2 x^4\right )}{e^{5/x}+(-5+x)^4}\right ) \left (-2 (-5+x)^3 x (5+x)+e^{5/x} (5+2 x)\right )}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx+16 \int \frac {\exp \left (\frac {1250+2 e^{5/x}-1000 x+301 x^2-40 x^3+2 x^4}{e^{5/x}+(-5+x)^4}\right ) \left (2 (-5+x)^3 x (5+x)-e^{5/x} (5+2 x)\right )}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx-40 \int \frac {x^7}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx+700 \int \frac {x^6}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx-7000 \int \frac {x^5}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx+43750 \int \frac {x^4}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx-175000 \int \frac {x^3}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx+390625 \int \frac {1}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx+437500 \int \frac {x^2}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx-625000 \int \frac {x}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx+\int \frac {e^{10/x}}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx+\int \frac {x^8}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx\\ &=2 \int \left (\frac {625 e^{5/x}}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2}-\frac {500 e^{5/x} x}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2}+\frac {150 e^{5/x} x^2}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2}-\frac {20 e^{5/x} x^3}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2}+\frac {e^{5/x} x^4}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2}\right ) \, dx+8 \int \left (-\frac {\exp \left (\frac {2 \left (1250+2 e^{5/x}-1000 x+301 x^2-40 x^3+2 x^4\right )}{e^{5/x}+(-5+x)^4}\right ) (-5+x)^3 \left (-25+5 x+4 x^2\right )}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2}+\frac {\exp \left (\frac {2 \left (1250+2 e^{5/x}-1000 x+301 x^2-40 x^3+2 x^4\right )}{e^{5/x}+(-5+x)^4}\right ) (5+2 x)}{625+e^{5/x}-500 x+150 x^2-20 x^3+x^4}\right ) \, dx+16 \int \left (\frac {\exp \left (\frac {1250+2 e^{5/x}-1000 x+301 x^2-40 x^3+2 x^4}{e^{5/x}+(-5+x)^4}\right ) (-5+x)^3 \left (-25+5 x+4 x^2\right )}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2}-\frac {\exp \left (\frac {1250+2 e^{5/x}-1000 x+301 x^2-40 x^3+2 x^4}{e^{5/x}+(-5+x)^4}\right ) (5+2 x)}{625+e^{5/x}-500 x+150 x^2-20 x^3+x^4}\right ) \, dx-40 \int \frac {x^7}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx+700 \int \frac {x^6}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx-7000 \int \frac {x^5}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx+43750 \int \frac {x^4}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx-175000 \int \frac {x^3}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx+390625 \int \frac {1}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx+437500 \int \frac {x^2}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx-625000 \int \frac {x}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx+\int \frac {e^{10/x}}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx+\int \frac {x^8}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx\\ &=2 \int \frac {e^{5/x} x^4}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2} \, dx-8 \int \frac {e^{\frac {2 \left (1250+2 e^{5/x}-1000 x+301 x^2-40 x^3+2 x^4\right )}{e^{5/x}+(-5+x)^4}} (-5+x)^3 \left (-25+5 x+4 x^2\right )}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2} \, dx+8 \int \frac {e^{\frac {2 \left (1250+2 e^{5/x}-1000 x+301 x^2-40 x^3+2 x^4\right )}{e^{5/x}+(-5+x)^4}} (5+2 x)}{625+e^{5/x}-500 x+150 x^2-20 x^3+x^4} \, dx+16 \int \frac {e^{\frac {1250+2 e^{5/x}-1000 x+301 x^2-40 x^3+2 x^4}{e^{5/x}+(-5+x)^4}} (-5+x)^3 \left (-25+5 x+4 x^2\right )}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2} \, dx-16 \int \frac {e^{\frac {1250+2 e^{5/x}-1000 x+301 x^2-40 x^3+2 x^4}{e^{5/x}+(-5+x)^4}} (5+2 x)}{625+e^{5/x}-500 x+150 x^2-20 x^3+x^4} \, dx-40 \int \frac {x^7}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx-40 \int \frac {e^{5/x} x^3}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2} \, dx+300 \int \frac {e^{5/x} x^2}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2} \, dx+700 \int \frac {x^6}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx-1000 \int \frac {e^{5/x} x}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2} \, dx+1250 \int \frac {e^{5/x}}{\left (625+e^{5/x}-500 x+150 x^2-20 x^3+x^4\right )^2} \, dx-7000 \int \frac {x^5}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx+43750 \int \frac {x^4}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx-175000 \int \frac {x^3}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx+390625 \int \frac {1}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx+437500 \int \frac {x^2}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx-625000 \int \frac {x}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx+\int \frac {e^{10/x}}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx+\int \frac {x^8}{\left (e^{5/x}+(-5+x)^4\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.89, size = 53, normalized size = 1.47 \begin {gather*} -16 e^{2+\frac {x^2}{e^{5/x}+(-5+x)^4}}+4 e^{4+\frac {2 x^2}{e^{5/x}+(-5+x)^4}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(390625 + E^(10/x) - 625000*x + 437500*x^2 - 175000*x^3 + 43750*x^4 - 7000*x^5 + 700*x^6 - 40*x^7 +
x^8 + E^(5/x)*(1250 - 1000*x + 300*x^2 - 40*x^3 + 2*x^4) + E^((1250 + 2*E^(5/x) - 1000*x + 301*x^2 - 40*x^3 +
2*x^4)/(625 + E^(5/x) - 500*x + 150*x^2 - 20*x^3 + x^4))*(E^(5/x)*(-80 - 32*x) - 20000*x + 8000*x^2 - 320*x^4
+ 32*x^5) + E^((2*(1250 + 2*E^(5/x) - 1000*x + 301*x^2 - 40*x^3 + 2*x^4))/(625 + E^(5/x) - 500*x + 150*x^2 - 2
0*x^3 + x^4))*(10000*x - 4000*x^2 + 160*x^4 - 16*x^5 + E^(5/x)*(40 + 16*x)))/(390625 + E^(10/x) - 625000*x + 4
37500*x^2 - 175000*x^3 + 43750*x^4 - 7000*x^5 + 700*x^6 - 40*x^7 + x^8 + E^(5/x)*(1250 - 1000*x + 300*x^2 - 40
*x^3 + 2*x^4)),x]

[Out]

-16*E^(2 + x^2/(E^(5/x) + (-5 + x)^4)) + 4*E^(4 + (2*x^2)/(E^(5/x) + (-5 + x)^4)) + x

________________________________________________________________________________________

fricas [B]  time = 0.86, size = 119, normalized size = 3.31 \begin {gather*} x + 4 \, e^{\left (\frac {2 \, {\left (2 \, x^{4} - 40 \, x^{3} + 301 \, x^{2} - 1000 \, x + 2 \, e^{\frac {5}{x}} + 1250\right )}}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + e^{\frac {5}{x}} + 625}\right )} - 16 \, e^{\left (\frac {2 \, x^{4} - 40 \, x^{3} + 301 \, x^{2} - 1000 \, x + 2 \, e^{\frac {5}{x}} + 1250}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + e^{\frac {5}{x}} + 625}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x+40)*exp(5/x)-16*x^5+160*x^4-4000*x^2+10000*x)*exp((2*exp(5/x)+2*x^4-40*x^3+301*x^2-1000*x+12
50)/(exp(5/x)+x^4-20*x^3+150*x^2-500*x+625))^2+((-32*x-80)*exp(5/x)+32*x^5-320*x^4+8000*x^2-20000*x)*exp((2*ex
p(5/x)+2*x^4-40*x^3+301*x^2-1000*x+1250)/(exp(5/x)+x^4-20*x^3+150*x^2-500*x+625))+exp(5/x)^2+(2*x^4-40*x^3+300
*x^2-1000*x+1250)*exp(5/x)+x^8-40*x^7+700*x^6-7000*x^5+43750*x^4-175000*x^3+437500*x^2-625000*x+390625)/(exp(5
/x)^2+(2*x^4-40*x^3+300*x^2-1000*x+1250)*exp(5/x)+x^8-40*x^7+700*x^6-7000*x^5+43750*x^4-175000*x^3+437500*x^2-
625000*x+390625),x, algorithm="fricas")

[Out]

x + 4*e^(2*(2*x^4 - 40*x^3 + 301*x^2 - 1000*x + 2*e^(5/x) + 1250)/(x^4 - 20*x^3 + 150*x^2 - 500*x + e^(5/x) +
625)) - 16*e^((2*x^4 - 40*x^3 + 301*x^2 - 1000*x + 2*e^(5/x) + 1250)/(x^4 - 20*x^3 + 150*x^2 - 500*x + e^(5/x)
+ 625))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8} - 40 \, x^{7} + 700 \, x^{6} - 7000 \, x^{5} + 43750 \, x^{4} - 175000 \, x^{3} + 437500 \, x^{2} - 8 \, {\left (2 \, x^{5} - 20 \, x^{4} + 500 \, x^{2} - {\left (2 \, x + 5\right )} e^{\frac {5}{x}} - 1250 \, x\right )} e^{\left (\frac {2 \, {\left (2 \, x^{4} - 40 \, x^{3} + 301 \, x^{2} - 1000 \, x + 2 \, e^{\frac {5}{x}} + 1250\right )}}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + e^{\frac {5}{x}} + 625}\right )} + 16 \, {\left (2 \, x^{5} - 20 \, x^{4} + 500 \, x^{2} - {\left (2 \, x + 5\right )} e^{\frac {5}{x}} - 1250 \, x\right )} e^{\left (\frac {2 \, x^{4} - 40 \, x^{3} + 301 \, x^{2} - 1000 \, x + 2 \, e^{\frac {5}{x}} + 1250}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + e^{\frac {5}{x}} + 625}\right )} + 2 \, {\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )} e^{\frac {5}{x}} - 625000 \, x + e^{\frac {10}{x}} + 390625}{x^{8} - 40 \, x^{7} + 700 \, x^{6} - 7000 \, x^{5} + 43750 \, x^{4} - 175000 \, x^{3} + 437500 \, x^{2} + 2 \, {\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )} e^{\frac {5}{x}} - 625000 \, x + e^{\frac {10}{x}} + 390625}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x+40)*exp(5/x)-16*x^5+160*x^4-4000*x^2+10000*x)*exp((2*exp(5/x)+2*x^4-40*x^3+301*x^2-1000*x+12
50)/(exp(5/x)+x^4-20*x^3+150*x^2-500*x+625))^2+((-32*x-80)*exp(5/x)+32*x^5-320*x^4+8000*x^2-20000*x)*exp((2*ex
p(5/x)+2*x^4-40*x^3+301*x^2-1000*x+1250)/(exp(5/x)+x^4-20*x^3+150*x^2-500*x+625))+exp(5/x)^2+(2*x^4-40*x^3+300
*x^2-1000*x+1250)*exp(5/x)+x^8-40*x^7+700*x^6-7000*x^5+43750*x^4-175000*x^3+437500*x^2-625000*x+390625)/(exp(5
/x)^2+(2*x^4-40*x^3+300*x^2-1000*x+1250)*exp(5/x)+x^8-40*x^7+700*x^6-7000*x^5+43750*x^4-175000*x^3+437500*x^2-
625000*x+390625),x, algorithm="giac")

[Out]

integrate((x^8 - 40*x^7 + 700*x^6 - 7000*x^5 + 43750*x^4 - 175000*x^3 + 437500*x^2 - 8*(2*x^5 - 20*x^4 + 500*x
^2 - (2*x + 5)*e^(5/x) - 1250*x)*e^(2*(2*x^4 - 40*x^3 + 301*x^2 - 1000*x + 2*e^(5/x) + 1250)/(x^4 - 20*x^3 + 1
50*x^2 - 500*x + e^(5/x) + 625)) + 16*(2*x^5 - 20*x^4 + 500*x^2 - (2*x + 5)*e^(5/x) - 1250*x)*e^((2*x^4 - 40*x
^3 + 301*x^2 - 1000*x + 2*e^(5/x) + 1250)/(x^4 - 20*x^3 + 150*x^2 - 500*x + e^(5/x) + 625)) + 2*(x^4 - 20*x^3
+ 150*x^2 - 500*x + 625)*e^(5/x) - 625000*x + e^(10/x) + 390625)/(x^8 - 40*x^7 + 700*x^6 - 7000*x^5 + 43750*x^
4 - 175000*x^3 + 437500*x^2 + 2*(x^4 - 20*x^3 + 150*x^2 - 500*x + 625)*e^(5/x) - 625000*x + e^(10/x) + 390625)
, x)

________________________________________________________________________________________

maple [B]  time = 0.37, size = 120, normalized size = 3.33

method	result	size
risch	\(4 \,{\mathrm e}^{\frac {4 \,{\mathrm e}^{\frac {5}{x}}+4 x^{4}-80 x^{3}+602 x^{2}-2000 x +2500}{{\mathrm e}^{\frac {5}{x}}+x^{4}-20 x^{3}+150 x^{2}-500 x +625}}+x -16 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{\frac {5}{x}}+2 x^{4}-40 x^{3}+301 x^{2}-1000 x +1250}{{\mathrm e}^{\frac {5}{x}}+x^{4}-20 x^{3}+150 x^{2}-500 x +625}}\)	\(120\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((16*x+40)*exp(5/x)-16*x^5+160*x^4-4000*x^2+10000*x)*exp((2*exp(5/x)+2*x^4-40*x^3+301*x^2-1000*x+1250)/(e
xp(5/x)+x^4-20*x^3+150*x^2-500*x+625))^2+((-32*x-80)*exp(5/x)+32*x^5-320*x^4+8000*x^2-20000*x)*exp((2*exp(5/x)
+2*x^4-40*x^3+301*x^2-1000*x+1250)/(exp(5/x)+x^4-20*x^3+150*x^2-500*x+625))+exp(5/x)^2+(2*x^4-40*x^3+300*x^2-1
000*x+1250)*exp(5/x)+x^8-40*x^7+700*x^6-7000*x^5+43750*x^4-175000*x^3+437500*x^2-625000*x+390625)/(exp(5/x)^2+
(2*x^4-40*x^3+300*x^2-1000*x+1250)*exp(5/x)+x^8-40*x^7+700*x^6-7000*x^5+43750*x^4-175000*x^3+437500*x^2-625000
*x+390625),x,method=_RETURNVERBOSE)

[Out]

4*exp(2*(2*exp(5/x)+2*x^4-40*x^3+301*x^2-1000*x+1250)/(exp(5/x)+x^4-20*x^3+150*x^2-500*x+625))+x-16*exp((2*exp
(5/x)+2*x^4-40*x^3+301*x^2-1000*x+1250)/(exp(5/x)+x^4-20*x^3+150*x^2-500*x+625))

________________________________________________________________________________________

maxima [B]  time = 0.54, size = 73, normalized size = 2.03 \begin {gather*} x + 4 \, e^{\left (\frac {2 \, x^{2}}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + e^{\frac {5}{x}} + 625} + 4\right )} - 16 \, e^{\left (\frac {x^{2}}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + e^{\frac {5}{x}} + 625} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x+40)*exp(5/x)-16*x^5+160*x^4-4000*x^2+10000*x)*exp((2*exp(5/x)+2*x^4-40*x^3+301*x^2-1000*x+12
50)/(exp(5/x)+x^4-20*x^3+150*x^2-500*x+625))^2+((-32*x-80)*exp(5/x)+32*x^5-320*x^4+8000*x^2-20000*x)*exp((2*ex
p(5/x)+2*x^4-40*x^3+301*x^2-1000*x+1250)/(exp(5/x)+x^4-20*x^3+150*x^2-500*x+625))+exp(5/x)^2+(2*x^4-40*x^3+300
*x^2-1000*x+1250)*exp(5/x)+x^8-40*x^7+700*x^6-7000*x^5+43750*x^4-175000*x^3+437500*x^2-625000*x+390625)/(exp(5
/x)^2+(2*x^4-40*x^3+300*x^2-1000*x+1250)*exp(5/x)+x^8-40*x^7+700*x^6-7000*x^5+43750*x^4-175000*x^3+437500*x^2-
625000*x+390625),x, algorithm="maxima")

[Out]

x + 4*e^(2*x^2/(x^4 - 20*x^3 + 150*x^2 - 500*x + e^(5/x) + 625) + 4) - 16*e^(x^2/(x^4 - 20*x^3 + 150*x^2 - 500
*x + e^(5/x) + 625) + 2)

________________________________________________________________________________________

mupad [B]  time = 1.33, size = 386, normalized size = 10.72 \begin {gather*} x-16\,{\mathrm {e}}^{\frac {2\,x^4}{{\mathrm {e}}^{5/x}-500\,x+150\,x^2-20\,x^3+x^4+625}}\,{\mathrm {e}}^{-\frac {40\,x^3}{{\mathrm {e}}^{5/x}-500\,x+150\,x^2-20\,x^3+x^4+625}}\,{\mathrm {e}}^{\frac {301\,x^2}{{\mathrm {e}}^{5/x}-500\,x+150\,x^2-20\,x^3+x^4+625}}\,{\mathrm {e}}^{\frac {1250}{{\mathrm {e}}^{5/x}-500\,x+150\,x^2-20\,x^3+x^4+625}}\,{\mathrm {e}}^{-\frac {1000\,x}{{\mathrm {e}}^{5/x}-500\,x+150\,x^2-20\,x^3+x^4+625}}\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{5/x}}{{\mathrm {e}}^{5/x}-500\,x+150\,x^2-20\,x^3+x^4+625}}+4\,{\mathrm {e}}^{\frac {4\,x^4}{{\mathrm {e}}^{5/x}-500\,x+150\,x^2-20\,x^3+x^4+625}}\,{\mathrm {e}}^{-\frac {80\,x^3}{{\mathrm {e}}^{5/x}-500\,x+150\,x^2-20\,x^3+x^4+625}}\,{\mathrm {e}}^{\frac {602\,x^2}{{\mathrm {e}}^{5/x}-500\,x+150\,x^2-20\,x^3+x^4+625}}\,{\mathrm {e}}^{\frac {2500}{{\mathrm {e}}^{5/x}-500\,x+150\,x^2-20\,x^3+x^4+625}}\,{\mathrm {e}}^{-\frac {2000\,x}{{\mathrm {e}}^{5/x}-500\,x+150\,x^2-20\,x^3+x^4+625}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{5/x}}{{\mathrm {e}}^{5/x}-500\,x+150\,x^2-20\,x^3+x^4+625}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(10/x) - 625000*x + exp((2*(2*exp(5/x) - 1000*x + 301*x^2 - 40*x^3 + 2*x^4 + 1250))/(exp(5/x) - 500*x
+ 150*x^2 - 20*x^3 + x^4 + 625))*(10000*x + exp(5/x)*(16*x + 40) - 4000*x^2 + 160*x^4 - 16*x^5) - exp((2*exp(5
/x) - 1000*x + 301*x^2 - 40*x^3 + 2*x^4 + 1250)/(exp(5/x) - 500*x + 150*x^2 - 20*x^3 + x^4 + 625))*(20000*x +
exp(5/x)*(32*x + 80) - 8000*x^2 + 320*x^4 - 32*x^5) + exp(5/x)*(300*x^2 - 1000*x - 40*x^3 + 2*x^4 + 1250) + 43
7500*x^2 - 175000*x^3 + 43750*x^4 - 7000*x^5 + 700*x^6 - 40*x^7 + x^8 + 390625)/(exp(10/x) - 625000*x + exp(5/
x)*(300*x^2 - 1000*x - 40*x^3 + 2*x^4 + 1250) + 437500*x^2 - 175000*x^3 + 43750*x^4 - 7000*x^5 + 700*x^6 - 40*
x^7 + x^8 + 390625),x)

[Out]

x - 16*exp((2*x^4)/(exp(5/x) - 500*x + 150*x^2 - 20*x^3 + x^4 + 625))*exp(-(40*x^3)/(exp(5/x) - 500*x + 150*x^
2 - 20*x^3 + x^4 + 625))*exp((301*x^2)/(exp(5/x) - 500*x + 150*x^2 - 20*x^3 + x^4 + 625))*exp(1250/(exp(5/x) -
500*x + 150*x^2 - 20*x^3 + x^4 + 625))*exp(-(1000*x)/(exp(5/x) - 500*x + 150*x^2 - 20*x^3 + x^4 + 625))*exp((
2*exp(5/x))/(exp(5/x) - 500*x + 150*x^2 - 20*x^3 + x^4 + 625)) + 4*exp((4*x^4)/(exp(5/x) - 500*x + 150*x^2 - 2
0*x^3 + x^4 + 625))*exp(-(80*x^3)/(exp(5/x) - 500*x + 150*x^2 - 20*x^3 + x^4 + 625))*exp((602*x^2)/(exp(5/x) -
500*x + 150*x^2 - 20*x^3 + x^4 + 625))*exp(2500/(exp(5/x) - 500*x + 150*x^2 - 20*x^3 + x^4 + 625))*exp(-(2000
*x)/(exp(5/x) - 500*x + 150*x^2 - 20*x^3 + x^4 + 625))*exp((4*exp(5/x))/(exp(5/x) - 500*x + 150*x^2 - 20*x^3 +
x^4 + 625))

________________________________________________________________________________________

sympy [B]  time = 1.22, size = 109, normalized size = 3.03 \begin {gather*} x + 4 e^{\frac {2 \left (2 x^{4} - 40 x^{3} + 301 x^{2} - 1000 x + 2 e^{\frac {5}{x}} + 1250\right )}{x^{4} - 20 x^{3} + 150 x^{2} - 500 x + e^{\frac {5}{x}} + 625}} - 16 e^{\frac {2 x^{4} - 40 x^{3} + 301 x^{2} - 1000 x + 2 e^{\frac {5}{x}} + 1250}{x^{4} - 20 x^{3} + 150 x^{2} - 500 x + e^{\frac {5}{x}} + 625}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x+40)*exp(5/x)-16*x**5+160*x**4-4000*x**2+10000*x)*exp((2*exp(5/x)+2*x**4-40*x**3+301*x**2-100
0*x+1250)/(exp(5/x)+x**4-20*x**3+150*x**2-500*x+625))**2+((-32*x-80)*exp(5/x)+32*x**5-320*x**4+8000*x**2-20000
*x)*exp((2*exp(5/x)+2*x**4-40*x**3+301*x**2-1000*x+1250)/(exp(5/x)+x**4-20*x**3+150*x**2-500*x+625))+exp(5/x)*
*2+(2*x**4-40*x**3+300*x**2-1000*x+1250)*exp(5/x)+x**8-40*x**7+700*x**6-7000*x**5+43750*x**4-175000*x**3+43750
0*x**2-625000*x+390625)/(exp(5/x)**2+(2*x**4-40*x**3+300*x**2-1000*x+1250)*exp(5/x)+x**8-40*x**7+700*x**6-7000
*x**5+43750*x**4-175000*x**3+437500*x**2-625000*x+390625),x)

[Out]

x + 4*exp(2*(2*x**4 - 40*x**3 + 301*x**2 - 1000*x + 2*exp(5/x) + 1250)/(x**4 - 20*x**3 + 150*x**2 - 500*x + ex
p(5/x) + 625)) - 16*exp((2*x**4 - 40*x**3 + 301*x**2 - 1000*x + 2*exp(5/x) + 1250)/(x**4 - 20*x**3 + 150*x**2
- 500*x + exp(5/x) + 625))

________________________________________________________________________________________