3.1.1 \(\int \frac {e^{-\frac {-10+25 x+e^x (-2+5 x)}{60-50 x+50 x^2+e^x (-10 x+10 x^2)}} (720-1400 x+1500 x^2-750 x^3+500 x^4+e^{2 x} (4 x+12 x^2-30 x^3+20 x^4)+e^x (-236 x+300 x^2-300 x^3+200 x^4))}{180-300 x+425 x^2-250 x^3+125 x^4+e^{2 x} (5 x^2-10 x^3+5 x^4)+e^x (-60 x+110 x^2-100 x^3+50 x^4)} \, dx\)

Optimal. Leaf size=33 \[ 4+4 e^{-\frac {-\frac {2}{5}+x}{\frac {12}{5+e^x}+x (-2+2 x)}} x \]

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Rubi [F]  time = 28.21, 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 {\exp \left (-\frac {-10+25 x+e^x (-2+5 x)}{60-50 x+50 x^2+e^x \left (-10 x+10 x^2\right )}\right ) \left (720-1400 x+1500 x^2-750 x^3+500 x^4+e^{2 x} \left (4 x+12 x^2-30 x^3+20 x^4\right )+e^x \left (-236 x+300 x^2-300 x^3+200 x^4\right )\right )}{180-300 x+425 x^2-250 x^3+125 x^4+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )+e^x \left (-60 x+110 x^2-100 x^3+50 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(720 - 1400*x + 1500*x^2 - 750*x^3 + 500*x^4 + E^(2*x)*(4*x + 12*x^2 - 30*x^3 + 20*x^4) + E^x*(-236*x + 30
0*x^2 - 300*x^3 + 200*x^4))/(E^((-10 + 25*x + E^x*(-2 + 5*x))/(60 - 50*x + 50*x^2 + E^x*(-10*x + 10*x^2)))*(18
0 - 300*x + 425*x^2 - 250*x^3 + 125*x^4 + E^(2*x)*(5*x^2 - 10*x^3 + 5*x^4) + E^x*(-60*x + 110*x^2 - 100*x^3 +
50*x^4))),x]

[Out]

4*Defer[Int][E^(-1/10*((5 + E^x)*(-2 + 5*x))/(6 - (5 + E^x)*x + (5 + E^x)*x^2)), x] + (6*Defer[Int][1/(E^(((5
+ E^x)*(-2 + 5*x))/(10*(6 - (5 + E^x)*x + (5 + E^x)*x^2)))*(-1 + x)^2), x])/5 + (6*Defer[Int][1/(E^(((5 + E^x)
*(-2 + 5*x))/(10*(6 - (5 + E^x)*x + (5 + E^x)*x^2)))*(-1 + x)), x])/5 + (4*Defer[Int][1/(E^(((5 + E^x)*(-2 + 5
*x))/(10*(6 - (5 + E^x)*x + (5 + E^x)*x^2)))*x), x])/5 + 72*Defer[Int][1/(E^(((5 + E^x)*(-2 + 5*x))/(10*(6 - (
5 + E^x)*x + (5 + E^x)*x^2)))*(6 - 5*x - E^x*x + 5*x^2 + E^x*x^2)^2), x] + (216*Defer[Int][1/(E^(((5 + E^x)*(-
2 + 5*x))/(10*(6 - (5 + E^x)*x + (5 + E^x)*x^2)))*(-1 + x)^2*(6 - 5*x - E^x*x + 5*x^2 + E^x*x^2)^2), x])/5 + (
792*Defer[Int][1/(E^(((5 + E^x)*(-2 + 5*x))/(10*(6 - (5 + E^x)*x + (5 + E^x)*x^2)))*(-1 + x)*(6 - 5*x - E^x*x
+ 5*x^2 + E^x*x^2)^2), x])/5 + (144*Defer[Int][1/(E^(((5 + E^x)*(-2 + 5*x))/(10*(6 - (5 + E^x)*x + (5 + E^x)*x
^2)))*x*(6 - 5*x - E^x*x + 5*x^2 + E^x*x^2)^2), x])/5 - 24*Defer[Int][x/(E^(((5 + E^x)*(-2 + 5*x))/(10*(6 - (5
+ E^x)*x + (5 + E^x)*x^2)))*(6 - 5*x - E^x*x + 5*x^2 + E^x*x^2)^2), x] + 60*Defer[Int][x^2/(E^(((5 + E^x)*(-2
+ 5*x))/(10*(6 - (5 + E^x)*x + (5 + E^x)*x^2)))*(6 - 5*x - E^x*x + 5*x^2 + E^x*x^2)^2), x] - 12*Defer[Int][1/
(E^(((5 + E^x)*(-2 + 5*x))/(10*(6 - (5 + E^x)*x + (5 + E^x)*x^2)))*(6 - 5*x - E^x*x + 5*x^2 + E^x*x^2)), x] -
(72*Defer[Int][1/(E^(((5 + E^x)*(-2 + 5*x))/(10*(6 - (5 + E^x)*x + (5 + E^x)*x^2)))*(-1 + x)^2*(6 - 5*x - E^x*
x + 5*x^2 + E^x*x^2)), x])/5 - (168*Defer[Int][1/(E^(((5 + E^x)*(-2 + 5*x))/(10*(6 - (5 + E^x)*x + (5 + E^x)*x
^2)))*(-1 + x)*(6 - 5*x - E^x*x + 5*x^2 + E^x*x^2)), x])/5 - (48*Defer[Int][1/(E^(((5 + E^x)*(-2 + 5*x))/(10*(
6 - (5 + E^x)*x + (5 + E^x)*x^2)))*x*(6 - 5*x - E^x*x + 5*x^2 + E^x*x^2)), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \exp \left (-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}\right ) \left (360+2 \left (-350-59 e^x+e^{2 x}\right ) x+6 \left (125+25 e^x+e^{2 x}\right ) x^2-15 \left (5+e^x\right )^2 x^3+10 \left (5+e^x\right )^2 x^4\right )}{5 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )^2} \, dx\\ &=\frac {2}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}\right ) \left (360+2 \left (-350-59 e^x+e^{2 x}\right ) x+6 \left (125+25 e^x+e^{2 x}\right ) x^2-15 \left (5+e^x\right )^2 x^3+10 \left (5+e^x\right )^2 x^4\right )}{\left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )^2} \, dx\\ &=\frac {2}{5} \int \left (-\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}\right ) \left (4-11 x+8 x^2+5 x^3\right )}{(-1+x)^2 x \left (6-5 x-e^x x+5 x^2+e^x x^2\right )}+\frac {\exp \left (-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}\right ) \left (2+6 x-15 x^2+10 x^3\right )}{(-1+x)^2 x}+\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}\right ) \left (12-42 x+8 x^2+75 x^3-60 x^4+25 x^5\right )}{(-1+x)^2 x \left (6-5 x-e^x x+5 x^2+e^x x^2\right )^2}\right ) \, dx\\ &=\frac {2}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}\right ) \left (2+6 x-15 x^2+10 x^3\right )}{(-1+x)^2 x} \, dx-\frac {12}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}\right ) \left (4-11 x+8 x^2+5 x^3\right )}{(-1+x)^2 x \left (6-5 x-e^x x+5 x^2+e^x x^2\right )} \, dx+\frac {12}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}\right ) \left (12-42 x+8 x^2+75 x^3-60 x^4+25 x^5\right )}{(-1+x)^2 x \left (6-5 x-e^x x+5 x^2+e^x x^2\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 40, normalized size = 1.21 \begin {gather*} 4 e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(720 - 1400*x + 1500*x^2 - 750*x^3 + 500*x^4 + E^(2*x)*(4*x + 12*x^2 - 30*x^3 + 20*x^4) + E^x*(-236*
x + 300*x^2 - 300*x^3 + 200*x^4))/(E^((-10 + 25*x + E^x*(-2 + 5*x))/(60 - 50*x + 50*x^2 + E^x*(-10*x + 10*x^2)
))*(180 - 300*x + 425*x^2 - 250*x^3 + 125*x^4 + E^(2*x)*(5*x^2 - 10*x^3 + 5*x^4) + E^x*(-60*x + 110*x^2 - 100*
x^3 + 50*x^4))),x]

[Out]

(4*x)/E^(((5 + E^x)*(-2 + 5*x))/(10*(6 - (5 + E^x)*x + (5 + E^x)*x^2)))

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fricas [A]  time = 0.55, size = 41, normalized size = 1.24 \begin {gather*} 4 \, x e^{\left (-\frac {{\left (5 \, x - 2\right )} e^{x} + 25 \, x - 10}{10 \, {\left (5 \, x^{2} + {\left (x^{2} - x\right )} e^{x} - 5 \, x + 6\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^4-30*x^3+12*x^2+4*x)*exp(x)^2+(200*x^4-300*x^3+300*x^2-236*x)*exp(x)+500*x^4-750*x^3+1500*x^2
-1400*x+720)/((5*x^4-10*x^3+5*x^2)*exp(x)^2+(50*x^4-100*x^3+110*x^2-60*x)*exp(x)+125*x^4-250*x^3+425*x^2-300*x
+180)/exp(((5*x-2)*exp(x)+25*x-10)/((10*x^2-10*x)*exp(x)+50*x^2-50*x+60)),x, algorithm="fricas")

[Out]

4*x*e^(-1/10*((5*x - 2)*e^x + 25*x - 10)/(5*x^2 + (x^2 - x)*e^x - 5*x + 6))

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giac [A]  time = 2.66, size = 56, normalized size = 1.70 \begin {gather*} 4 \, x e^{\left (-\frac {5 \, x^{2} e^{x} + 25 \, x^{2} + 10 \, x e^{x} + 50 \, x - 6 \, e^{x}}{30 \, {\left (x^{2} e^{x} + 5 \, x^{2} - x e^{x} - 5 \, x + 6\right )}} + \frac {1}{6}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^4-30*x^3+12*x^2+4*x)*exp(x)^2+(200*x^4-300*x^3+300*x^2-236*x)*exp(x)+500*x^4-750*x^3+1500*x^2
-1400*x+720)/((5*x^4-10*x^3+5*x^2)*exp(x)^2+(50*x^4-100*x^3+110*x^2-60*x)*exp(x)+125*x^4-250*x^3+425*x^2-300*x
+180)/exp(((5*x-2)*exp(x)+25*x-10)/((10*x^2-10*x)*exp(x)+50*x^2-50*x+60)),x, algorithm="giac")

[Out]

4*x*e^(-1/30*(5*x^2*e^x + 25*x^2 + 10*x*e^x + 50*x - 6*e^x)/(x^2*e^x + 5*x^2 - x*e^x - 5*x + 6) + 1/6)

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maple [A]  time = 0.33, size = 39, normalized size = 1.18

method	result	size
risch	\(4 x \,{\mathrm e}^{-\frac {\left (5 x -2\right ) \left ({\mathrm e}^{x}+5\right )}{10 \left ({\mathrm e}^{x} x^{2}-{\mathrm e}^{x} x +5 x^{2}-5 x +6\right )}}\)	\(39\)
norman	\(\frac {\left (24 x -20 x^{2}+20 x^{3}-4 \,{\mathrm e}^{x} x^{2}+4 \,{\mathrm e}^{x} x^{3}\right ) {\mathrm e}^{-\frac {\left (5 x -2\right ) {\mathrm e}^{x}+25 x -10}{\left (10 x^{2}-10 x \right ) {\mathrm e}^{x}+50 x^{2}-50 x +60}}}{{\mathrm e}^{x} x^{2}-{\mathrm e}^{x} x +5 x^{2}-5 x +6}\)	\(94\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((20*x^4-30*x^3+12*x^2+4*x)*exp(x)^2+(200*x^4-300*x^3+300*x^2-236*x)*exp(x)+500*x^4-750*x^3+1500*x^2-1400*
x+720)/((5*x^4-10*x^3+5*x^2)*exp(x)^2+(50*x^4-100*x^3+110*x^2-60*x)*exp(x)+125*x^4-250*x^3+425*x^2-300*x+180)/
exp(((5*x-2)*exp(x)+25*x-10)/((10*x^2-10*x)*exp(x)+50*x^2-50*x+60)),x,method=_RETURNVERBOSE)

[Out]

4*x*exp(-1/10*(5*x-2)*(exp(x)+5)/(exp(x)*x^2-exp(x)*x+5*x^2-5*x+6))

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maxima [B]  time = 1.01, size = 105, normalized size = 3.18 \begin {gather*} 4 \, x e^{\left (-\frac {x e^{x}}{2 \, {\left (5 \, x^{2} + {\left (x^{2} - x\right )} e^{x} - 5 \, x + 6\right )}} - \frac {5 \, x}{2 \, {\left (5 \, x^{2} + {\left (x^{2} - x\right )} e^{x} - 5 \, x + 6\right )}} + \frac {e^{x}}{5 \, {\left (5 \, x^{2} + {\left (x^{2} - x\right )} e^{x} - 5 \, x + 6\right )}} + \frac {1}{5 \, x^{2} + {\left (x^{2} - x\right )} e^{x} - 5 \, x + 6}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^4-30*x^3+12*x^2+4*x)*exp(x)^2+(200*x^4-300*x^3+300*x^2-236*x)*exp(x)+500*x^4-750*x^3+1500*x^2
-1400*x+720)/((5*x^4-10*x^3+5*x^2)*exp(x)^2+(50*x^4-100*x^3+110*x^2-60*x)*exp(x)+125*x^4-250*x^3+425*x^2-300*x
+180)/exp(((5*x-2)*exp(x)+25*x-10)/((10*x^2-10*x)*exp(x)+50*x^2-50*x+60)),x, algorithm="maxima")

[Out]

4*x*e^(-1/2*x*e^x/(5*x^2 + (x^2 - x)*e^x - 5*x + 6) - 5/2*x/(5*x^2 + (x^2 - x)*e^x - 5*x + 6) + 1/5*e^x/(5*x^2
+ (x^2 - x)*e^x - 5*x + 6) + 1/(5*x^2 + (x^2 - x)*e^x - 5*x + 6))

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mupad [B]  time = 1.54, size = 113, normalized size = 3.42 \begin {gather*} 4\,x\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{5\,x^2\,{\mathrm {e}}^x-25\,x-5\,x\,{\mathrm {e}}^x+25\,x^2+30}}\,{\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^x}{2\,x^2\,{\mathrm {e}}^x-10\,x-2\,x\,{\mathrm {e}}^x+10\,x^2+12}}\,{\mathrm {e}}^{-\frac {5\,x}{2\,x^2\,{\mathrm {e}}^x-10\,x-2\,x\,{\mathrm {e}}^x+10\,x^2+12}}\,{\mathrm {e}}^{\frac {1}{x^2\,{\mathrm {e}}^x-5\,x-x\,{\mathrm {e}}^x+5\,x^2+6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((25*x + exp(x)*(5*x - 2) - 10)/(50*x + exp(x)*(10*x - 10*x^2) - 50*x^2 - 60))*(exp(2*x)*(4*x + 12*x^2
- 30*x^3 + 20*x^4) - exp(x)*(236*x - 300*x^2 + 300*x^3 - 200*x^4) - 1400*x + 1500*x^2 - 750*x^3 + 500*x^4 + 7
20))/(exp(2*x)*(5*x^2 - 10*x^3 + 5*x^4) - exp(x)*(60*x - 110*x^2 + 100*x^3 - 50*x^4) - 300*x + 425*x^2 - 250*x
^3 + 125*x^4 + 180),x)

[Out]

4*x*exp(exp(x)/(5*x^2*exp(x) - 25*x - 5*x*exp(x) + 25*x^2 + 30))*exp(-(x*exp(x))/(2*x^2*exp(x) - 10*x - 2*x*ex
p(x) + 10*x^2 + 12))*exp(-(5*x)/(2*x^2*exp(x) - 10*x - 2*x*exp(x) + 10*x^2 + 12))*exp(1/(x^2*exp(x) - 5*x - x*
exp(x) + 5*x^2 + 6))

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sympy [A]  time = 20.74, size = 39, normalized size = 1.18 \begin {gather*} 4 x e^{- \frac {25 x + \left (5 x - 2\right ) e^{x} - 10}{50 x^{2} - 50 x + \left (10 x^{2} - 10 x\right ) e^{x} + 60}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x**4-30*x**3+12*x**2+4*x)*exp(x)**2+(200*x**4-300*x**3+300*x**2-236*x)*exp(x)+500*x**4-750*x**3
+1500*x**2-1400*x+720)/((5*x**4-10*x**3+5*x**2)*exp(x)**2+(50*x**4-100*x**3+110*x**2-60*x)*exp(x)+125*x**4-250
*x**3+425*x**2-300*x+180)/exp(((5*x-2)*exp(x)+25*x-10)/((10*x**2-10*x)*exp(x)+50*x**2-50*x+60)),x)

[Out]

4*x*exp(-(25*x + (5*x - 2)*exp(x) - 10)/(50*x**2 - 50*x + (10*x**2 - 10*x)*exp(x) + 60))

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