3.30.66 \(\int \frac {e^{-x+\frac {-20+25 x+e^{2 e^{-x} x} (-4 x+4 x^2)}{-4+5 x}} (e^x (16-40 x+25 x^2)+e^{2 e^{-x} x} (32 x^2-104 x^3+112 x^4-40 x^5+e^x (16 x-32 x^2+20 x^3)))}{16-40 x+25 x^2} \, dx\)

Optimal. Leaf size=34 \[ e^{5+\frac {4 e^{2 e^{-x} x} x^2}{5 x+\frac {x}{-1+x}}} x \]

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Rubi [F]  time = 39.09, 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 (-x+\frac {-20+25 x+e^{2 e^{-x} x} \left (-4 x+4 x^2\right )}{-4+5 x}\right ) \left (e^x \left (16-40 x+25 x^2\right )+e^{2 e^{-x} x} \left (32 x^2-104 x^3+112 x^4-40 x^5+e^x \left (16 x-32 x^2+20 x^3\right )\right )\right )}{16-40 x+25 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-x + (-20 + 25*x + E^((2*x)/E^x)*(-4*x + 4*x^2))/(-4 + 5*x))*(E^x*(16 - 40*x + 25*x^2) + E^((2*x)/E^x)
*(32*x^2 - 104*x^3 + 112*x^4 - 40*x^5 + E^x*(16*x - 32*x^2 + 20*x^3))))/(16 - 40*x + 25*x^2),x]

[Out]

Defer[Int][E^((-20 + 25*x + E^((2*x)/E^x)*(-4*x + 4*x^2))/(-4 + 5*x)), x] - (32*Defer[Int][E^(-x + (2*x)/E^x +
(-20 + 25*x + E^((2*x)/E^x)*(-4*x + 4*x^2))/(-4 + 5*x)), x])/625 + (4*Defer[Int][E^((2*x)/E^x + (-20 + 25*x +
E^((2*x)/E^x)*(-4*x + 4*x^2))/(-4 + 5*x))*x, x])/5 - (8*Defer[Int][E^(-x + (2*x)/E^x + (-20 + 25*x + E^((2*x)
/E^x)*(-4*x + 4*x^2))/(-4 + 5*x))*x, x])/125 + (48*Defer[Int][E^(-x + (2*x)/E^x + (-20 + 25*x + E^((2*x)/E^x)*
(-4*x + 4*x^2))/(-4 + 5*x))*x^2, x])/25 - (8*Defer[Int][E^(-x + (2*x)/E^x + (-20 + 25*x + E^((2*x)/E^x)*(-4*x
+ 4*x^2))/(-4 + 5*x))*x^3, x])/5 + (64*Defer[Int][E^((2*x)/E^x + (-20 + 25*x + E^((2*x)/E^x)*(-4*x + 4*x^2))/(
-4 + 5*x))/(-4 + 5*x)^2, x])/25 + (16*Defer[Int][E^((2*x)/E^x + (-20 + 25*x + E^((2*x)/E^x)*(-4*x + 4*x^2))/(-
4 + 5*x))/(-4 + 5*x), x])/25 - (128*Defer[Int][E^(-x + (2*x)/E^x + (-20 + 25*x + E^((2*x)/E^x)*(-4*x + 4*x^2))
/(-4 + 5*x))/(-4 + 5*x), x])/625

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (-x+\frac {-20+25 x+e^{2 e^{-x} x} \left (-4 x+4 x^2\right )}{-4+5 x}\right ) \left (e^x \left (16-40 x+25 x^2\right )+e^{2 e^{-x} x} \left (32 x^2-104 x^3+112 x^4-40 x^5+e^x \left (16 x-32 x^2+20 x^3\right )\right )\right )}{(-4+5 x)^2} \, dx\\ &=\int \left (\exp \left (\frac {-20+25 x+e^{2 e^{-x} x} \left (-4 x+4 x^2\right )}{-4+5 x}\right )-\frac {4 \exp \left (-x+2 e^{-x} x+\frac {-20+25 x+e^{2 e^{-x} x} \left (-4 x+4 x^2\right )}{-4+5 x}\right ) x \left (-4 e^x-8 x+8 e^x x+26 x^2-5 e^x x^2-28 x^3+10 x^4\right )}{(-4+5 x)^2}\right ) \, dx\\ &=-\left (4 \int \frac {\exp \left (-x+2 e^{-x} x+\frac {-20+25 x+e^{2 e^{-x} x} \left (-4 x+4 x^2\right )}{-4+5 x}\right ) x \left (-4 e^x-8 x+8 e^x x+26 x^2-5 e^x x^2-28 x^3+10 x^4\right )}{(-4+5 x)^2} \, dx\right )+\int \exp \left (\frac {-20+25 x+e^{2 e^{-x} x} \left (-4 x+4 x^2\right )}{-4+5 x}\right ) \, dx\\ &=-\left (4 \int \left (-\frac {8 \exp \left (-x+2 e^{-x} x+\frac {-20+25 x+e^{2 e^{-x} x} \left (-4 x+4 x^2\right )}{-4+5 x}\right ) x^2}{(-4+5 x)^2}+\frac {26 \exp \left (-x+2 e^{-x} x+\frac {-20+25 x+e^{2 e^{-x} x} \left (-4 x+4 x^2\right )}{-4+5 x}\right ) x^3}{(-4+5 x)^2}-\frac {28 \exp \left (-x+2 e^{-x} x+\frac {-20+25 x+e^{2 e^{-x} x} \left (-4 x+4 x^2\right )}{-4+5 x}\right ) x^4}{(-4+5 x)^2}+\frac {10 \exp \left (-x+2 e^{-x} x+\frac {-20+25 x+e^{2 e^{-x} x} \left (-4 x+4 x^2\right )}{-4+5 x}\right ) x^5}{(-4+5 x)^2}-\frac {\exp \left (2 e^{-x} x+\frac {-20+25 x+e^{2 e^{-x} x} \left (-4 x+4 x^2\right )}{-4+5 x}\right ) x \left (4-8 x+5 x^2\right )}{(-4+5 x)^2}\right ) \, dx\right )+\int \exp \left (\frac {-20+25 x+e^{2 e^{-x} x} \left (-4 x+4 x^2\right )}{-4+5 x}\right ) \, dx\\ &=4 \int \frac {\exp \left (2 e^{-x} x+\frac {-20+25 x+e^{2 e^{-x} x} \left (-4 x+4 x^2\right )}{-4+5 x}\right ) x \left (4-8 x+5 x^2\right )}{(-4+5 x)^2} \, dx+32 \int \frac {\exp \left (-x+2 e^{-x} x+\frac {-20+25 x+e^{2 e^{-x} x} \left (-4 x+4 x^2\right )}{-4+5 x}\right ) x^2}{(-4+5 x)^2} \, dx-40 \int \frac {\exp \left (-x+2 e^{-x} x+\frac {-20+25 x+e^{2 e^{-x} x} \left (-4 x+4 x^2\right )}{-4+5 x}\right ) x^5}{(-4+5 x)^2} \, dx-104 \int \frac {\exp \left (-x+2 e^{-x} x+\frac {-20+25 x+e^{2 e^{-x} x} \left (-4 x+4 x^2\right )}{-4+5 x}\right ) x^3}{(-4+5 x)^2} \, dx+112 \int \frac {\exp \left (-x+2 e^{-x} x+\frac {-20+25 x+e^{2 e^{-x} x} \left (-4 x+4 x^2\right )}{-4+5 x}\right ) x^4}{(-4+5 x)^2} \, dx+\int \exp \left (\frac {-20+25 x+e^{2 e^{-x} x} \left (-4 x+4 x^2\right )}{-4+5 x}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.26, size = 29, normalized size = 0.85 \begin {gather*} e^{5+\frac {4 e^{2 e^{-x} x} (-1+x) x}{-4+5 x}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-x + (-20 + 25*x + E^((2*x)/E^x)*(-4*x + 4*x^2))/(-4 + 5*x))*(E^x*(16 - 40*x + 25*x^2) + E^((2*x
)/E^x)*(32*x^2 - 104*x^3 + 112*x^4 - 40*x^5 + E^x*(16*x - 32*x^2 + 20*x^3))))/(16 - 40*x + 25*x^2),x]

[Out]

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

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fricas [A]  time = 1.09, size = 41, normalized size = 1.21 \begin {gather*} x e^{\left (x - \frac {5 \, x^{2} - 4 \, {\left (x^{2} - x\right )} e^{\left (2 \, x e^{\left (-x\right )}\right )} - 29 \, x + 20}{5 \, x - 4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((20*x^3-32*x^2+16*x)*exp(x)-40*x^5+112*x^4-104*x^3+32*x^2)*exp(x/exp(x))^2+(25*x^2-40*x+16)*exp(x)
)*exp(((4*x^2-4*x)*exp(x/exp(x))^2+25*x-20)/(5*x-4))/(25*x^2-40*x+16)/exp(x),x, algorithm="fricas")

[Out]

x*e^(x - (5*x^2 - 4*(x^2 - x)*e^(2*x*e^(-x)) - 29*x + 20)/(5*x - 4))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (4 \, {\left (10 \, x^{5} - 28 \, x^{4} + 26 \, x^{3} - 8 \, x^{2} - {\left (5 \, x^{3} - 8 \, x^{2} + 4 \, x\right )} e^{x}\right )} e^{\left (2 \, x e^{\left (-x\right )}\right )} - {\left (25 \, x^{2} - 40 \, x + 16\right )} e^{x}\right )} e^{\left (-x + \frac {4 \, {\left (x^{2} - x\right )} e^{\left (2 \, x e^{\left (-x\right )}\right )} + 25 \, x - 20}{5 \, x - 4}\right )}}{25 \, x^{2} - 40 \, x + 16}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((20*x^3-32*x^2+16*x)*exp(x)-40*x^5+112*x^4-104*x^3+32*x^2)*exp(x/exp(x))^2+(25*x^2-40*x+16)*exp(x)
)*exp(((4*x^2-4*x)*exp(x/exp(x))^2+25*x-20)/(5*x-4))/(25*x^2-40*x+16)/exp(x),x, algorithm="giac")

[Out]

integrate(-(4*(10*x^5 - 28*x^4 + 26*x^3 - 8*x^2 - (5*x^3 - 8*x^2 + 4*x)*e^x)*e^(2*x*e^(-x)) - (25*x^2 - 40*x +
16)*e^x)*e^(-x + (4*(x^2 - x)*e^(2*x*e^(-x)) + 25*x - 20)/(5*x - 4))/(25*x^2 - 40*x + 16), x)

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maple [A]  time = 0.52, size = 41, normalized size = 1.21

method	result	size
risch	\(x \,{\mathrm e}^{\frac {4 \,{\mathrm e}^{2 x \,{\mathrm e}^{-x}} x^{2}-4 \,{\mathrm e}^{2 x \,{\mathrm e}^{-x}} x +25 x -20}{5 x -4}}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((20*x^3-32*x^2+16*x)*exp(x)-40*x^5+112*x^4-104*x^3+32*x^2)*exp(x/exp(x))^2+(25*x^2-40*x+16)*exp(x))*exp(
((4*x^2-4*x)*exp(x/exp(x))^2+25*x-20)/(5*x-4))/(25*x^2-40*x+16)/exp(x),x,method=_RETURNVERBOSE)

[Out]

x*exp((4*exp(2*x*exp(-x))*x^2-4*exp(2*x*exp(-x))*x+25*x-20)/(5*x-4))

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maxima [A]  time = 0.97, size = 43, normalized size = 1.26 \begin {gather*} x e^{\left (\frac {4}{5} \, x e^{\left (2 \, x e^{\left (-x\right )}\right )} - \frac {16 \, e^{\left (2 \, x e^{\left (-x\right )}\right )}}{25 \, {\left (5 \, x - 4\right )}} - \frac {4}{25} \, e^{\left (2 \, x e^{\left (-x\right )}\right )} + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((20*x^3-32*x^2+16*x)*exp(x)-40*x^5+112*x^4-104*x^3+32*x^2)*exp(x/exp(x))^2+(25*x^2-40*x+16)*exp(x)
)*exp(((4*x^2-4*x)*exp(x/exp(x))^2+25*x-20)/(5*x-4))/(25*x^2-40*x+16)/exp(x),x, algorithm="maxima")

[Out]

x*e^(4/5*x*e^(2*x*e^(-x)) - 16/25*e^(2*x*e^(-x))/(5*x - 4) - 4/25*e^(2*x*e^(-x)) + 5)

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mupad [B]  time = 2.11, size = 63, normalized size = 1.85 \begin {gather*} x\,{\mathrm {e}}^{-\frac {4\,x\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{-x}}}{5\,x-4}}\,{\mathrm {e}}^{-\frac {20}{5\,x-4}}\,{\mathrm {e}}^{\frac {4\,x^2\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{-x}}}{5\,x-4}}\,{\mathrm {e}}^{\frac {25\,x}{5\,x-4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x)*exp(-(exp(2*x*exp(-x))*(4*x - 4*x^2) - 25*x + 20)/(5*x - 4))*(exp(x)*(25*x^2 - 40*x + 16) + exp(2
*x*exp(-x))*(32*x^2 - 104*x^3 + 112*x^4 - 40*x^5 + exp(x)*(16*x - 32*x^2 + 20*x^3))))/(25*x^2 - 40*x + 16),x)

[Out]

x*exp(-(4*x*exp(2*x*exp(-x)))/(5*x - 4))*exp(-20/(5*x - 4))*exp((4*x^2*exp(2*x*exp(-x)))/(5*x - 4))*exp((25*x)
/(5*x - 4))

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sympy [A]  time = 109.77, size = 29, normalized size = 0.85 \begin {gather*} x e^{\frac {25 x + \left (4 x^{2} - 4 x\right ) e^{2 x e^{- x}} - 20}{5 x - 4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((20*x**3-32*x**2+16*x)*exp(x)-40*x**5+112*x**4-104*x**3+32*x**2)*exp(x/exp(x))**2+(25*x**2-40*x+16
)*exp(x))*exp(((4*x**2-4*x)*exp(x/exp(x))**2+25*x-20)/(5*x-4))/(25*x**2-40*x+16)/exp(x),x)

[Out]

x*exp((25*x + (4*x**2 - 4*x)*exp(2*x*exp(-x)) - 20)/(5*x - 4))

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