3.65.19 \(\int \frac {e^{\frac {1}{4} (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 (-4-4 x-x^2))} (e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}} (2+2 e^x)+e^{2+2 x} (-50-25 x)+e^{2+x} (20-90 x-50 x^2)+e^2 (-2+19 x-40 x^2-25 x^3))}{2+50 e^{2 x}-20 x+50 x^2+e^x (-20+100 x)} \, dx\)

Optimal. Leaf size=32 \[ e^{e^{3+\frac {1}{-5+\frac {1}{e^x+x}}}-e^2 \left (1+\frac {x}{2}\right )^2} \]

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Rubi [F]  time = 51.66, 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 {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right ) \left (e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}} \left (2+2 e^x\right )+e^{2+2 x} (-50-25 x)+e^{2+x} \left (20-90 x-50 x^2\right )+e^2 \left (-2+19 x-40 x^2-25 x^3\right )\right )}{2+50 e^{2 x}-20 x+50 x^2+e^x (-20+100 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((4*E^((-3 + 14*E^x + 14*x)/(-1 + 5*E^x + 5*x)) + E^2*(-4 - 4*x - x^2))/4)*(E^((-3 + 14*E^x + 14*x)/(-1
+ 5*E^x + 5*x))*(2 + 2*E^x) + E^(2 + 2*x)*(-50 - 25*x) + E^(2 + x)*(20 - 90*x - 50*x^2) + E^2*(-2 + 19*x - 40
*x^2 - 25*x^3)))/(2 + 50*E^(2*x) - 20*x + 50*x^2 + E^x*(-20 + 100*x)),x]

[Out]

-Defer[Int][E^(2 + (4*E^((-3 + 14*E^x + 14*x)/(-1 + 5*E^x + 5*x)) + E^2*(-4 - 4*x - x^2))/4), x] - Defer[Int][
E^(2 + (4*E^((-3 + 14*E^x + 14*x)/(-1 + 5*E^x + 5*x)) + E^2*(-4 - 4*x - x^2))/4)*x, x]/2 + (6*Defer[Int][E^((-
3 + 14*E^x + 14*x)/(-1 + 5*E^x + 5*x) + (4*E^((-3 + 14*E^x + 14*x)/(-1 + 5*E^x + 5*x)) + E^2*(-4 - 4*x - x^2))
/4)/(-1 + 5*E^x + 5*x)^2, x])/5 - Defer[Int][(E^((-3 + 14*E^x + 14*x)/(-1 + 5*E^x + 5*x) + (4*E^((-3 + 14*E^x
+ 14*x)/(-1 + 5*E^x + 5*x)) + E^2*(-4 - 4*x - x^2))/4)*x)/(-1 + 5*E^x + 5*x)^2, x] + Defer[Int][E^((-3 + 14*E^
x + 14*x)/(-1 + 5*E^x + 5*x) + (4*E^((-3 + 14*E^x + 14*x)/(-1 + 5*E^x + 5*x)) + E^2*(-4 - 4*x - x^2))/4)/(-1 +
5*E^x + 5*x), x]/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right ) \left (e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}} \left (2+2 e^x\right )+e^{2+2 x} (-50-25 x)+e^{2+x} \left (20-90 x-50 x^2\right )+e^2 \left (-2+19 x-40 x^2-25 x^3\right )\right )}{2 \left (1-5 e^x-5 x\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {\exp \left (\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right ) \left (e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}} \left (2+2 e^x\right )+e^{2+2 x} (-50-25 x)+e^{2+x} \left (20-90 x-50 x^2\right )+e^2 \left (-2+19 x-40 x^2-25 x^3\right )\right )}{\left (1-5 e^x-5 x\right )^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {2 \exp \left (\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right ) \left (1+e^x\right )}{\left (1-5 e^x-5 x\right )^2}-\exp \left (2+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right ) (2+x)\right ) \, dx\\ &=-\left (\frac {1}{2} \int \exp \left (2+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right ) (2+x) \, dx\right )+\int \frac {\exp \left (\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right ) \left (1+e^x\right )}{\left (1-5 e^x-5 x\right )^2} \, dx\\ &=-\left (\frac {1}{2} \int \left (2 \exp \left (2+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right )+\exp \left (2+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right ) x\right ) \, dx\right )+\int \left (-\frac {\exp \left (\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right ) (-6+5 x)}{5 \left (-1+5 e^x+5 x\right )^2}+\frac {\exp \left (\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right )}{5 \left (-1+5 e^x+5 x\right )}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {\exp \left (\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right ) (-6+5 x)}{\left (-1+5 e^x+5 x\right )^2} \, dx\right )+\frac {1}{5} \int \frac {\exp \left (\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right )}{-1+5 e^x+5 x} \, dx-\frac {1}{2} \int \exp \left (2+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right ) x \, dx-\int \exp \left (2+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right ) \, dx\\ &=\frac {1}{5} \int \frac {\exp \left (\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right )}{-1+5 e^x+5 x} \, dx-\frac {1}{5} \int \left (-\frac {6 \exp \left (\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right )}{\left (-1+5 e^x+5 x\right )^2}+\frac {5 \exp \left (\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right ) x}{\left (-1+5 e^x+5 x\right )^2}\right ) \, dx-\frac {1}{2} \int \exp \left (2+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right ) x \, dx-\int \exp \left (2+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right ) \, dx\\ &=\frac {1}{5} \int \frac {\exp \left (\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right )}{-1+5 e^x+5 x} \, dx-\frac {1}{2} \int \exp \left (2+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right ) x \, dx+\frac {6}{5} \int \frac {\exp \left (\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right )}{\left (-1+5 e^x+5 x\right )^2} \, dx-\int \exp \left (2+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right ) \, dx-\int \frac {\exp \left (\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}+\frac {1}{4} \left (4 e^{\frac {-3+14 e^x+14 x}{-1+5 e^x+5 x}}+e^2 \left (-4-4 x-x^2\right )\right )\right ) x}{\left (-1+5 e^x+5 x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.49, size = 37, normalized size = 1.16 \begin {gather*} e^{\frac {1}{4} e^2 \left (4 e^{\frac {4}{5}+\frac {1}{5-25 e^x-25 x}}-(2+x)^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((4*E^((-3 + 14*E^x + 14*x)/(-1 + 5*E^x + 5*x)) + E^2*(-4 - 4*x - x^2))/4)*(E^((-3 + 14*E^x + 14*
x)/(-1 + 5*E^x + 5*x))*(2 + 2*E^x) + E^(2 + 2*x)*(-50 - 25*x) + E^(2 + x)*(20 - 90*x - 50*x^2) + E^2*(-2 + 19*
x - 40*x^2 - 25*x^3)))/(2 + 50*E^(2*x) - 20*x + 50*x^2 + E^x*(-20 + 100*x)),x]

[Out]

E^((E^2*(4*E^(4/5 + (5 - 25*E^x - 25*x)^(-1)) - (2 + x)^2))/4)

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fricas [A]  time = 0.75, size = 48, normalized size = 1.50 \begin {gather*} e^{\left (-\frac {1}{4} \, {\left (x^{2} + 4 \, x + 4\right )} e^{2} + e^{\left (\frac {{\left (14 \, x - 3\right )} e^{2} + 14 \, e^{\left (x + 2\right )}}{{\left (5 \, x - 1\right )} e^{2} + 5 \, e^{\left (x + 2\right )}}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(x)+2)*exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+(-25*x-50)*exp(1)^2*exp(x)^2+(-50*x^2-90*x+20
)*exp(1)^2*exp(x)+(-25*x^3-40*x^2+19*x-2)*exp(1)^2)*exp(exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+1/4*(-x^2-4*x
-4)*exp(1)^2)/(50*exp(x)^2+(100*x-20)*exp(x)+50*x^2-20*x+2),x, algorithm="fricas")

[Out]

e^(-1/4*(x^2 + 4*x + 4)*e^2 + e^(((14*x - 3)*e^2 + 14*e^(x + 2))/((5*x - 1)*e^2 + 5*e^(x + 2))))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left ({\left (25 \, x^{3} + 40 \, x^{2} - 19 \, x + 2\right )} e^{2} + 25 \, {\left (x + 2\right )} e^{\left (2 \, x + 2\right )} + 10 \, {\left (5 \, x^{2} + 9 \, x - 2\right )} e^{\left (x + 2\right )} - 2 \, {\left (e^{x} + 1\right )} e^{\left (\frac {14 \, x + 14 \, e^{x} - 3}{5 \, x + 5 \, e^{x} - 1}\right )}\right )} e^{\left (-\frac {1}{4} \, {\left (x^{2} + 4 \, x + 4\right )} e^{2} + e^{\left (\frac {14 \, x + 14 \, e^{x} - 3}{5 \, x + 5 \, e^{x} - 1}\right )}\right )}}{2 \, {\left (25 \, x^{2} + 10 \, {\left (5 \, x - 1\right )} e^{x} - 10 \, x + 25 \, e^{\left (2 \, x\right )} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(x)+2)*exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+(-25*x-50)*exp(1)^2*exp(x)^2+(-50*x^2-90*x+20
)*exp(1)^2*exp(x)+(-25*x^3-40*x^2+19*x-2)*exp(1)^2)*exp(exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+1/4*(-x^2-4*x
-4)*exp(1)^2)/(50*exp(x)^2+(100*x-20)*exp(x)+50*x^2-20*x+2),x, algorithm="giac")

[Out]

integrate(-1/2*((25*x^3 + 40*x^2 - 19*x + 2)*e^2 + 25*(x + 2)*e^(2*x + 2) + 10*(5*x^2 + 9*x - 2)*e^(x + 2) - 2
*(e^x + 1)*e^((14*x + 14*e^x - 3)/(5*x + 5*e^x - 1)))*e^(-1/4*(x^2 + 4*x + 4)*e^2 + e^((14*x + 14*e^x - 3)/(5*
x + 5*e^x - 1)))/(25*x^2 + 10*(5*x - 1)*e^x - 10*x + 25*e^(2*x) + 1), x)

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maple [A]  time = 0.25, size = 41, normalized size = 1.28

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{\frac {14 \,{\mathrm e}^{x}+14 x -3}{5 \,{\mathrm e}^{x}+5 x -1}}-\frac {x^{2} {\mathrm e}^{2}}{4}-{\mathrm e}^{2} x -{\mathrm e}^{2}}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*exp(x)+2)*exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+(-25*x-50)*exp(1)^2*exp(x)^2+(-50*x^2-90*x+20)*exp(
1)^2*exp(x)+(-25*x^3-40*x^2+19*x-2)*exp(1)^2)*exp(exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+1/4*(-x^2-4*x-4)*ex
p(1)^2)/(50*exp(x)^2+(100*x-20)*exp(x)+50*x^2-20*x+2),x,method=_RETURNVERBOSE)

[Out]

exp(exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))-1/4*x^2*exp(2)-exp(2)*x-exp(2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {1}{2} \, \int \frac {{\left ({\left (25 \, x^{3} + 40 \, x^{2} - 19 \, x + 2\right )} e^{2} + 25 \, {\left (x + 2\right )} e^{\left (2 \, x + 2\right )} + 10 \, {\left (5 \, x^{2} + 9 \, x - 2\right )} e^{\left (x + 2\right )} - 2 \, {\left (e^{x} + 1\right )} e^{\left (\frac {14 \, x + 14 \, e^{x} - 3}{5 \, x + 5 \, e^{x} - 1}\right )}\right )} e^{\left (-\frac {1}{4} \, {\left (x^{2} + 4 \, x + 4\right )} e^{2} + e^{\left (\frac {14 \, x + 14 \, e^{x} - 3}{5 \, x + 5 \, e^{x} - 1}\right )}\right )}}{25 \, x^{2} + 10 \, {\left (5 \, x - 1\right )} e^{x} - 10 \, x + 25 \, e^{\left (2 \, x\right )} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(x)+2)*exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+(-25*x-50)*exp(1)^2*exp(x)^2+(-50*x^2-90*x+20
)*exp(1)^2*exp(x)+(-25*x^3-40*x^2+19*x-2)*exp(1)^2)*exp(exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+1/4*(-x^2-4*x
-4)*exp(1)^2)/(50*exp(x)^2+(100*x-20)*exp(x)+50*x^2-20*x+2),x, algorithm="maxima")

[Out]

-1/2*integrate(((25*x^3 + 40*x^2 - 19*x + 2)*e^2 + 25*(x + 2)*e^(2*x + 2) + 10*(5*x^2 + 9*x - 2)*e^(x + 2) - 2
*(e^x + 1)*e^((14*x + 14*e^x - 3)/(5*x + 5*e^x - 1)))*e^(-1/4*(x^2 + 4*x + 4)*e^2 + e^((14*x + 14*e^x - 3)/(5*
x + 5*e^x - 1)))/(25*x^2 + 10*(5*x - 1)*e^x - 10*x + 25*e^(2*x) + 1), x)

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mupad [B]  time = 4.60, size = 67, normalized size = 2.09 \begin {gather*} {\mathrm {e}}^{-\frac {x^2\,{\mathrm {e}}^2}{4}}\,{\mathrm {e}}^{-{\mathrm {e}}^2}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^2}\,{\mathrm {e}}^{{\mathrm {e}}^{\frac {14\,x}{5\,x+5\,{\mathrm {e}}^x-1}}\,{\mathrm {e}}^{\frac {14\,{\mathrm {e}}^x}{5\,x+5\,{\mathrm {e}}^x-1}}\,{\mathrm {e}}^{-\frac {3}{5\,x+5\,{\mathrm {e}}^x-1}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp((14*x + 14*exp(x) - 3)/(5*x + 5*exp(x) - 1)) - (exp(2)*(4*x + x^2 + 4))/4)*(exp(2)*(40*x^2 - 19*
x + 25*x^3 + 2) - exp((14*x + 14*exp(x) - 3)/(5*x + 5*exp(x) - 1))*(2*exp(x) + 2) + exp(2)*exp(x)*(90*x + 50*x
^2 - 20) + exp(2*x)*exp(2)*(25*x + 50)))/(50*exp(2*x) - 20*x + exp(x)*(100*x - 20) + 50*x^2 + 2),x)

[Out]

exp(-(x^2*exp(2))/4)*exp(-exp(2))*exp(-x*exp(2))*exp(exp((14*x)/(5*x + 5*exp(x) - 1))*exp((14*exp(x))/(5*x + 5
*exp(x) - 1))*exp(-3/(5*x + 5*exp(x) - 1)))

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sympy [A]  time = 2.01, size = 36, normalized size = 1.12 \begin {gather*} e^{\left (- \frac {x^{2}}{4} - x - 1\right ) e^{2} + e^{\frac {14 x + 14 e^{x} - 3}{5 x + 5 e^{x} - 1}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*exp(x)+2)*exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+(-25*x-50)*exp(1)**2*exp(x)**2+(-50*x**2-90*x
+20)*exp(1)**2*exp(x)+(-25*x**3-40*x**2+19*x-2)*exp(1)**2)*exp(exp((14*exp(x)+14*x-3)/(5*exp(x)+5*x-1))+1/4*(-
x**2-4*x-4)*exp(1)**2)/(50*exp(x)**2+(100*x-20)*exp(x)+50*x**2-20*x+2),x)

[Out]

exp((-x**2/4 - x - 1)*exp(2) + exp((14*x + 14*exp(x) - 3)/(5*x + 5*exp(x) - 1)))

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