\(\int \frac {e^{\frac {160016-128625 x+38900 x^2-5270 x^3+276 x^4-x^5}{625-500 x+150 x^2-20 x^3+x^4}} (-3061+3125 x-1250 x^2+250 x^3-25 x^4+x^5)}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx\) [4238]

Optimal result

Integrand size = 97, antiderivative size = 19 \[ \int \frac {e^{\frac {160016-128625 x+38900 x^2-5270 x^3+276 x^4-x^5}{625-500 x+150 x^2-20 x^3+x^4}} \left (-3061+3125 x-1250 x^2+250 x^3-25 x^4+x^5\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=3+e-e^{256+\frac {16}{(-5+x)^4}-x} \]

[Out]

-exp(256-x+16/(-5+x)^4)+3+exp(1)

Rubi [F]

\[ \int \frac {e^{\frac {160016-128625 x+38900 x^2-5270 x^3+276 x^4-x^5}{625-500 x+150 x^2-20 x^3+x^4}} \left (-3061+3125 x-1250 x^2+250 x^3-25 x^4+x^5\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=\int \frac {\exp \left (\frac {160016-128625 x+38900 x^2-5270 x^3+276 x^4-x^5}{625-500 x+150 x^2-20 x^3+x^4}\right ) \left (-3061+3125 x-1250 x^2+250 x^3-25 x^4+x^5\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx \]

[In]

Int[(E^((160016 - 128625*x + 38900*x^2 - 5270*x^3 + 276*x^4 - x^5)/(625 - 500*x + 150*x^2 - 20*x^3 + x^4))*(-3
061 + 3125*x - 1250*x^2 + 250*x^3 - 25*x^4 + x^5))/(-3125 + 3125*x - 1250*x^2 + 250*x^3 - 25*x^4 + x^5),x]

[Out]

Defer[Int][E^((160016 - 128625*x + 38900*x^2 - 5270*x^3 + 276*x^4 - x^5)/(-5 + x)^4), x] + 64*Defer[Int][E^((1
60016 - 128625*x + 38900*x^2 - 5270*x^3 + 276*x^4 - x^5)/(-5 + x)^4)/(-5 + x)^5, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {160016-128625 x+38900 x^2-5270 x^3+276 x^4-x^5}{(-5+x)^4}\right ) \left (3061-3125 x+1250 x^2-250 x^3+25 x^4-x^5\right )}{(5-x)^5} \, dx \\ & = \int \left (\exp \left (\frac {160016-128625 x+38900 x^2-5270 x^3+276 x^4-x^5}{(-5+x)^4}\right )+\frac {64 \exp \left (\frac {160016-128625 x+38900 x^2-5270 x^3+276 x^4-x^5}{(-5+x)^4}\right )}{(-5+x)^5}\right ) \, dx \\ & = 64 \int \frac {\exp \left (\frac {160016-128625 x+38900 x^2-5270 x^3+276 x^4-x^5}{(-5+x)^4}\right )}{(-5+x)^5} \, dx+\int \exp \left (\frac {160016-128625 x+38900 x^2-5270 x^3+276 x^4-x^5}{(-5+x)^4}\right ) \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {e^{\frac {160016-128625 x+38900 x^2-5270 x^3+276 x^4-x^5}{625-500 x+150 x^2-20 x^3+x^4}} \left (-3061+3125 x-1250 x^2+250 x^3-25 x^4+x^5\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=-e^{256+\frac {16}{(-5+x)^4}-x} \]

[In]

Integrate[(E^((160016 - 128625*x + 38900*x^2 - 5270*x^3 + 276*x^4 - x^5)/(625 - 500*x + 150*x^2 - 20*x^3 + x^4
))*(-3061 + 3125*x - 1250*x^2 + 250*x^3 - 25*x^4 + x^5))/(-3125 + 3125*x - 1250*x^2 + 250*x^3 - 25*x^4 + x^5),
x]

[Out]

-E^(256 + 16/(-5 + x)^4 - x)

Maple [A] (verified)

Time = 3.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.79

method	result	size
risch	\(-{\mathrm e}^{-\frac {x^{5}-276 x^{4}+5270 x^{3}-38900 x^{2}+128625 x -160016}{\left (-5+x \right )^{4}}}\)	\(34\)
gosper	\(-{\mathrm e}^{-\frac {x^{5}-276 x^{4}+5270 x^{3}-38900 x^{2}+128625 x -160016}{x^{4}-20 x^{3}+150 x^{2}-500 x +625}}\)	\(49\)
parallelrisch	\(-{\mathrm e}^{-\frac {x^{5}-276 x^{4}+5270 x^{3}-38900 x^{2}+128625 x -160016}{x^{4}-20 x^{3}+150 x^{2}-500 x +625}}\)	\(49\)
norman	\(\frac {500 x \,{\mathrm e}^{\frac {-x^{5}+276 x^{4}-5270 x^{3}+38900 x^{2}-128625 x +160016}{x^{4}-20 x^{3}+150 x^{2}-500 x +625}}-150 x^{2} {\mathrm e}^{\frac {-x^{5}+276 x^{4}-5270 x^{3}+38900 x^{2}-128625 x +160016}{x^{4}-20 x^{3}+150 x^{2}-500 x +625}}+20 x^{3} {\mathrm e}^{\frac {-x^{5}+276 x^{4}-5270 x^{3}+38900 x^{2}-128625 x +160016}{x^{4}-20 x^{3}+150 x^{2}-500 x +625}}-x^{4} {\mathrm e}^{\frac {-x^{5}+276 x^{4}-5270 x^{3}+38900 x^{2}-128625 x +160016}{x^{4}-20 x^{3}+150 x^{2}-500 x +625}}-625 \,{\mathrm e}^{\frac {-x^{5}+276 x^{4}-5270 x^{3}+38900 x^{2}-128625 x +160016}{x^{4}-20 x^{3}+150 x^{2}-500 x +625}}}{\left (-5+x \right )^{4}}\)	\(263\)

[In]

int((x^5-25*x^4+250*x^3-1250*x^2+3125*x-3061)*exp((-x^5+276*x^4-5270*x^3+38900*x^2-128625*x+160016)/(x^4-20*x^
3+150*x^2-500*x+625))/(x^5-25*x^4+250*x^3-1250*x^2+3125*x-3125),x,method=_RETURNVERBOSE)

[Out]

-exp(-(x^5-276*x^4+5270*x^3-38900*x^2+128625*x-160016)/(-5+x)^4)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (19) = 38\).

Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.53 \[ \int \frac {e^{\frac {160016-128625 x+38900 x^2-5270 x^3+276 x^4-x^5}{625-500 x+150 x^2-20 x^3+x^4}} \left (-3061+3125 x-1250 x^2+250 x^3-25 x^4+x^5\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=-e^{\left (-\frac {x^{5} - 276 \, x^{4} + 5270 \, x^{3} - 38900 \, x^{2} + 128625 \, x - 160016}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625}\right )} \]

[In]

integrate((x^5-25*x^4+250*x^3-1250*x^2+3125*x-3061)*exp((-x^5+276*x^4-5270*x^3+38900*x^2-128625*x+160016)/(x^4
-20*x^3+150*x^2-500*x+625))/(x^5-25*x^4+250*x^3-1250*x^2+3125*x-3125),x, algorithm="fricas")

[Out]

-e^(-(x^5 - 276*x^4 + 5270*x^3 - 38900*x^2 + 128625*x - 160016)/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (15) = 30\).

Time = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.32 \[ \int \frac {e^{\frac {160016-128625 x+38900 x^2-5270 x^3+276 x^4-x^5}{625-500 x+150 x^2-20 x^3+x^4}} \left (-3061+3125 x-1250 x^2+250 x^3-25 x^4+x^5\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=- e^{\frac {- x^{5} + 276 x^{4} - 5270 x^{3} + 38900 x^{2} - 128625 x + 160016}{x^{4} - 20 x^{3} + 150 x^{2} - 500 x + 625}} \]

[In]

integrate((x**5-25*x**4+250*x**3-1250*x**2+3125*x-3061)*exp((-x**5+276*x**4-5270*x**3+38900*x**2-128625*x+1600
16)/(x**4-20*x**3+150*x**2-500*x+625))/(x**5-25*x**4+250*x**3-1250*x**2+3125*x-3125),x)

[Out]

-exp((-x**5 + 276*x**4 - 5270*x**3 + 38900*x**2 - 128625*x + 160016)/(x**4 - 20*x**3 + 150*x**2 - 500*x + 625)
)

Maxima [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.58 \[ \int \frac {e^{\frac {160016-128625 x+38900 x^2-5270 x^3+276 x^4-x^5}{625-500 x+150 x^2-20 x^3+x^4}} \left (-3061+3125 x-1250 x^2+250 x^3-25 x^4+x^5\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=-e^{\left (-x + \frac {16}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625} + 256\right )} \]

[In]

integrate((x^5-25*x^4+250*x^3-1250*x^2+3125*x-3061)*exp((-x^5+276*x^4-5270*x^3+38900*x^2-128625*x+160016)/(x^4
-20*x^3+150*x^2-500*x+625))/(x^5-25*x^4+250*x^3-1250*x^2+3125*x-3125),x, algorithm="maxima")

[Out]

-e^(-x + 16/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625) + 256)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (19) = 38\).

Time = 0.26 (sec) , antiderivative size = 149, normalized size of antiderivative = 7.84 \[ \int \frac {e^{\frac {160016-128625 x+38900 x^2-5270 x^3+276 x^4-x^5}{625-500 x+150 x^2-20 x^3+x^4}} \left (-3061+3125 x-1250 x^2+250 x^3-25 x^4+x^5\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=-e^{\left (-\frac {x^{5}}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625} + \frac {276 \, x^{4}}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625} - \frac {5270 \, x^{3}}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625} + \frac {38900 \, x^{2}}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625} - \frac {128625 \, x}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625} + \frac {160016}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625}\right )} \]

[In]

integrate((x^5-25*x^4+250*x^3-1250*x^2+3125*x-3061)*exp((-x^5+276*x^4-5270*x^3+38900*x^2-128625*x+160016)/(x^4
-20*x^3+150*x^2-500*x+625))/(x^5-25*x^4+250*x^3-1250*x^2+3125*x-3125),x, algorithm="giac")

[Out]

-e^(-x^5/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625) + 276*x^4/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625) - 5270*x^3/(x
^4 - 20*x^3 + 150*x^2 - 500*x + 625) + 38900*x^2/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625) - 128625*x/(x^4 - 20*x
^3 + 150*x^2 - 500*x + 625) + 160016/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625))

Mupad [B] (verification not implemented)

Time = 10.64 (sec) , antiderivative size = 153, normalized size of antiderivative = 8.05 \[ \int \frac {e^{\frac {160016-128625 x+38900 x^2-5270 x^3+276 x^4-x^5}{625-500 x+150 x^2-20 x^3+x^4}} \left (-3061+3125 x-1250 x^2+250 x^3-25 x^4+x^5\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=-{\mathrm {e}}^{-\frac {128625\,x}{x^4-20\,x^3+150\,x^2-500\,x+625}}\,{\mathrm {e}}^{-\frac {x^5}{x^4-20\,x^3+150\,x^2-500\,x+625}}\,{\mathrm {e}}^{\frac {276\,x^4}{x^4-20\,x^3+150\,x^2-500\,x+625}}\,{\mathrm {e}}^{-\frac {5270\,x^3}{x^4-20\,x^3+150\,x^2-500\,x+625}}\,{\mathrm {e}}^{\frac {38900\,x^2}{x^4-20\,x^3+150\,x^2-500\,x+625}}\,{\mathrm {e}}^{\frac {160016}{x^4-20\,x^3+150\,x^2-500\,x+625}} \]

[In]

int((exp(-(128625*x - 38900*x^2 + 5270*x^3 - 276*x^4 + x^5 - 160016)/(150*x^2 - 500*x - 20*x^3 + x^4 + 625))*(
3125*x - 1250*x^2 + 250*x^3 - 25*x^4 + x^5 - 3061))/(3125*x - 1250*x^2 + 250*x^3 - 25*x^4 + x^5 - 3125),x)

[Out]

-exp(-(128625*x)/(150*x^2 - 500*x - 20*x^3 + x^4 + 625))*exp(-x^5/(150*x^2 - 500*x - 20*x^3 + x^4 + 625))*exp(
(276*x^4)/(150*x^2 - 500*x - 20*x^3 + x^4 + 625))*exp(-(5270*x^3)/(150*x^2 - 500*x - 20*x^3 + x^4 + 625))*exp(
(38900*x^2)/(150*x^2 - 500*x - 20*x^3 + x^4 + 625))*exp(160016/(150*x^2 - 500*x - 20*x^3 + x^4 + 625))