3.43.76 \(\int \frac {e^{\frac {-5 x-x^2}{3-4 x+e^{e x+x^2} x-x^2}} (-15-6 x-x^2+e^{e x+x^2} (-x^2+10 x^3+2 x^4+e (5 x^2+x^3)))}{9-24 x+10 x^2+e^{2 e x+2 x^2} x^2+8 x^3+x^4+e^{e x+x^2} (6 x-8 x^2-2 x^3)} \, dx\)

Optimal. Leaf size=30 \[ e^{\frac {5+x}{3-e^{x (e+x)}-\frac {3-x}{x}+x}} \]

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Rubi [F]  time = 25.71, 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 {e^{\frac {-5 x-x^2}{3-4 x+e^{e x+x^2} x-x^2}} \left (-15-6 x-x^2+e^{e x+x^2} \left (-x^2+10 x^3+2 x^4+e \left (5 x^2+x^3\right )\right )\right )}{9-24 x+10 x^2+e^{2 e x+2 x^2} x^2+8 x^3+x^4+e^{e x+x^2} \left (6 x-8 x^2-2 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-5*x - x^2)/(3 - 4*x + E^(E*x + x^2)*x - x^2))*(-15 - 6*x - x^2 + E^(E*x + x^2)*(-x^2 + 10*x^3 + 2*x^
4 + E*(5*x^2 + x^3))))/(9 - 24*x + 10*x^2 + E^(2*E*x + 2*x^2)*x^2 + 8*x^3 + x^4 + E^(E*x + x^2)*(6*x - 8*x^2 -
2*x^3)),x]

[Out]

-15*Defer[Int][E^((x*(5 + x))/(-3 - (-4 + E^(x*(E + x)))*x + x^2))/(-3 + 4*x - E^(x*(E + x))*x + x^2)^2, x] -
3*Defer[Int][(E^((x*(5 + x))/(-3 - (-4 + E^(x*(E + x)))*x + x^2))*x)/(-3 + 4*x - E^(x*(E + x))*x + x^2)^2, x]
- (35 + 3*E)*Defer[Int][(E^((x*(5 + x))/(-3 - (-4 + E^(x*(E + x)))*x + x^2))*x^2)/(-3 + 4*x - E^(x*(E + x))*x
+ x^2)^2, x] + 5*Defer[Int][(E^(1 + E*x + x^2 + (x*(5 + x))/(-3 - (-4 + E^(x*(E + x)))*x + x^2))*x^2)/(-3 + 4*
x - E^(x*(E + x))*x + x^2)^2, x] + (33 + 4*E)*Defer[Int][(E^((x*(5 + x))/(-3 - (-4 + E^(x*(E + x)))*x + x^2))*
x^3)/(-3 + 4*x - E^(x*(E + x))*x + x^2)^2, x] + (18 + E)*Defer[Int][(E^((x*(5 + x))/(-3 - (-4 + E^(x*(E + x)))
*x + x^2))*x^4)/(-3 + 4*x - E^(x*(E + x))*x + x^2)^2, x] + 2*Defer[Int][(E^((x*(5 + x))/(-3 - (-4 + E^(x*(E +
x)))*x + x^2))*x^5)/(-3 + 4*x - E^(x*(E + x))*x + x^2)^2, x] + Defer[Int][(E^((x*(5 + x))/(-3 - (-4 + E^(x*(E
+ x)))*x + x^2))*x)/(-3 + 4*x - E^(x*(E + x))*x + x^2), x] - (10 + E)*Defer[Int][(E^((x*(5 + x))/(-3 - (-4 + E
^(x*(E + x)))*x + x^2))*x^2)/(-3 + 4*x - E^(x*(E + x))*x + x^2), x] - 2*Defer[Int][(E^((x*(5 + x))/(-3 - (-4 +
E^(x*(E + x)))*x + x^2))*x^3)/(-3 + 4*x - E^(x*(E + x))*x + x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) \left (-15-6 x-\left (1+e^{x (e+x)}-5 e^{1+e x+x^2}\right ) x^2+e^{x (e+x)} (10+e) x^3+2 e^{x (e+x)} x^4\right )}{\left (3+\left (-4+e^{x (e+x)}\right ) x-x^2\right )^2} \, dx\\ &=\int \left (\frac {\exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) x \left (-1+(10+e) x+2 x^2\right )}{3-4 x+e^{x (e+x)} x-x^2}+\frac {\exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) \left (-15-3 x-35 \left (1+\frac {3 e}{35}\right ) x^2+5 e^{1+e x+x^2} x^2+33 \left (1+\frac {4 e}{33}\right ) x^3+18 \left (1+\frac {e}{18}\right ) x^4+2 x^5\right )}{\left (3-4 x+e^{x (e+x)} x-x^2\right )^2}\right ) \, dx\\ &=\int \frac {\exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) x \left (-1+(10+e) x+2 x^2\right )}{3-4 x+e^{x (e+x)} x-x^2} \, dx+\int \frac {\exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) \left (-15-3 x-35 \left (1+\frac {3 e}{35}\right ) x^2+5 e^{1+e x+x^2} x^2+33 \left (1+\frac {4 e}{33}\right ) x^3+18 \left (1+\frac {e}{18}\right ) x^4+2 x^5\right )}{\left (3-4 x+e^{x (e+x)} x-x^2\right )^2} \, dx\\ &=\int \left (-\frac {15 \exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right )}{\left (-3+4 x-e^{x (e+x)} x+x^2\right )^2}-\frac {3 \exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) x}{\left (-3+4 x-e^{x (e+x)} x+x^2\right )^2}+\frac {5 \exp \left (1+e x+x^2+\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) x^2}{\left (-3+4 x-e^{x (e+x)} x+x^2\right )^2}-\frac {\exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) (35+3 e) x^2}{\left (-3+4 x-e^{x (e+x)} x+x^2\right )^2}+\frac {\exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) (33+4 e) x^3}{\left (-3+4 x-e^{x (e+x)} x+x^2\right )^2}+\frac {\exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) (18+e) x^4}{\left (-3+4 x-e^{x (e+x)} x+x^2\right )^2}+\frac {2 \exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) x^5}{\left (-3+4 x-e^{x (e+x)} x+x^2\right )^2}\right ) \, dx+\int \left (\frac {\exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) x}{-3+4 x-e^{x (e+x)} x+x^2}-\frac {\exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) (10+e) x^2}{-3+4 x-e^{x (e+x)} x+x^2}-\frac {2 \exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) x^3}{-3+4 x-e^{x (e+x)} x+x^2}\right ) \, dx\\ &=2 \int \frac {\exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) x^5}{\left (-3+4 x-e^{x (e+x)} x+x^2\right )^2} \, dx-2 \int \frac {\exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) x^3}{-3+4 x-e^{x (e+x)} x+x^2} \, dx-3 \int \frac {\exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) x}{\left (-3+4 x-e^{x (e+x)} x+x^2\right )^2} \, dx+5 \int \frac {\exp \left (1+e x+x^2+\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) x^2}{\left (-3+4 x-e^{x (e+x)} x+x^2\right )^2} \, dx-15 \int \frac {\exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right )}{\left (-3+4 x-e^{x (e+x)} x+x^2\right )^2} \, dx+(-35-3 e) \int \frac {\exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) x^2}{\left (-3+4 x-e^{x (e+x)} x+x^2\right )^2} \, dx+(-10-e) \int \frac {\exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) x^2}{-3+4 x-e^{x (e+x)} x+x^2} \, dx+(18+e) \int \frac {\exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) x^4}{\left (-3+4 x-e^{x (e+x)} x+x^2\right )^2} \, dx+(33+4 e) \int \frac {\exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) x^3}{\left (-3+4 x-e^{x (e+x)} x+x^2\right )^2} \, dx+\int \frac {\exp \left (\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}\right ) x}{-3+4 x-e^{x (e+x)} x+x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 26, normalized size = 0.87 \begin {gather*} e^{\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-5*x - x^2)/(3 - 4*x + E^(E*x + x^2)*x - x^2))*(-15 - 6*x - x^2 + E^(E*x + x^2)*(-x^2 + 10*x^3
+ 2*x^4 + E*(5*x^2 + x^3))))/(9 - 24*x + 10*x^2 + E^(2*E*x + 2*x^2)*x^2 + 8*x^3 + x^4 + E^(E*x + x^2)*(6*x - 8
*x^2 - 2*x^3)),x]

[Out]

E^((x*(5 + x))/(-3 - (-4 + E^(x*(E + x)))*x + x^2))

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fricas [A]  time = 0.61, size = 31, normalized size = 1.03 \begin {gather*} e^{\left (\frac {x^{2} + 5 \, x}{x^{2} - x e^{\left (x^{2} + x e\right )} + 4 \, x - 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3+5*x^2)*exp(1)+2*x^4+10*x^3-x^2)*exp(x*exp(1)+x^2)-x^2-6*x-15)*exp((-x^2-5*x)/(x*exp(x*exp(1)+
x^2)-x^2-4*x+3))/(x^2*exp(x*exp(1)+x^2)^2+(-2*x^3-8*x^2+6*x)*exp(x*exp(1)+x^2)+x^4+8*x^3+10*x^2-24*x+9),x, alg
orithm="fricas")

[Out]

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

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giac [B]  time = 1.58, size = 53, normalized size = 1.77 \begin {gather*} e^{\left (\frac {x^{2}}{x^{2} - x e^{\left (x^{2} + x e\right )} + 4 \, x - 3} + \frac {5 \, x}{x^{2} - x e^{\left (x^{2} + x e\right )} + 4 \, x - 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3+5*x^2)*exp(1)+2*x^4+10*x^3-x^2)*exp(x*exp(1)+x^2)-x^2-6*x-15)*exp((-x^2-5*x)/(x*exp(x*exp(1)+
x^2)-x^2-4*x+3))/(x^2*exp(x*exp(1)+x^2)^2+(-2*x^3-8*x^2+6*x)*exp(x*exp(1)+x^2)+x^4+8*x^3+10*x^2-24*x+9),x, alg
orithm="giac")

[Out]

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

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maple [A]  time = 0.47, size = 27, normalized size = 0.90

method	result	size
risch	\({\mathrm e}^{\frac {\left (5+x \right ) x}{-x \,{\mathrm e}^{x \left (x +{\mathrm e}\right )}+x^{2}+4 x -3}}\)	\(27\)
norman	\(\frac {x^{2} {\mathrm e}^{\frac {-x^{2}-5 x}{x \,{\mathrm e}^{x \,{\mathrm e}+x^{2}}-x^{2}-4 x +3}}+4 x \,{\mathrm e}^{\frac {-x^{2}-5 x}{x \,{\mathrm e}^{x \,{\mathrm e}+x^{2}}-x^{2}-4 x +3}}-x \,{\mathrm e}^{x \,{\mathrm e}+x^{2}} {\mathrm e}^{\frac {-x^{2}-5 x}{x \,{\mathrm e}^{x \,{\mathrm e}+x^{2}}-x^{2}-4 x +3}}-3 \,{\mathrm e}^{\frac {-x^{2}-5 x}{x \,{\mathrm e}^{x \,{\mathrm e}+x^{2}}-x^{2}-4 x +3}}}{x^{2}-x \,{\mathrm e}^{x \,{\mathrm e}+x^{2}}+4 x -3}\)	\(182\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^3+5*x^2)*exp(1)+2*x^4+10*x^3-x^2)*exp(x*exp(1)+x^2)-x^2-6*x-15)*exp((-x^2-5*x)/(x*exp(x*exp(1)+x^2)-x
^2-4*x+3))/(x^2*exp(x*exp(1)+x^2)^2+(-2*x^3-8*x^2+6*x)*exp(x*exp(1)+x^2)+x^4+8*x^3+10*x^2-24*x+9),x,method=_RE
TURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.49, size = 84, normalized size = 2.80 \begin {gather*} e^{\left (\frac {x e^{\left (x^{2} + x e\right )}}{x^{2} - x e^{\left (x^{2} + x e\right )} + 4 \, x - 3} + \frac {x}{x^{2} - x e^{\left (x^{2} + x e\right )} + 4 \, x - 3} + \frac {3}{x^{2} - x e^{\left (x^{2} + x e\right )} + 4 \, x - 3} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3+5*x^2)*exp(1)+2*x^4+10*x^3-x^2)*exp(x*exp(1)+x^2)-x^2-6*x-15)*exp((-x^2-5*x)/(x*exp(x*exp(1)+
x^2)-x^2-4*x+3))/(x^2*exp(x*exp(1)+x^2)^2+(-2*x^3-8*x^2+6*x)*exp(x*exp(1)+x^2)+x^4+8*x^3+10*x^2-24*x+9),x, alg
orithm="maxima")

[Out]

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

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mupad [B]  time = 3.49, size = 31, normalized size = 1.03 \begin {gather*} {\mathrm {e}}^{\frac {x^2+5\,x}{4\,x+x^2-x\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{x\,\mathrm {e}}-3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((5*x + x^2)/(4*x - x*exp(x*exp(1) + x^2) + x^2 - 3))*(6*x - exp(x*exp(1) + x^2)*(exp(1)*(5*x^2 + x^3
) - x^2 + 10*x^3 + 2*x^4) + x^2 + 15))/(x^2*exp(2*x*exp(1) + 2*x^2) - 24*x + 10*x^2 + 8*x^3 + x^4 - exp(x*exp(
1) + x^2)*(8*x^2 - 6*x + 2*x^3) + 9),x)

[Out]

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

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sympy [A]  time = 0.99, size = 29, normalized size = 0.97 \begin {gather*} e^{\frac {- x^{2} - 5 x}{- x^{2} + x e^{x^{2} + e x} - 4 x + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**3+5*x**2)*exp(1)+2*x**4+10*x**3-x**2)*exp(x*exp(1)+x**2)-x**2-6*x-15)*exp((-x**2-5*x)/(x*exp(x
*exp(1)+x**2)-x**2-4*x+3))/(x**2*exp(x*exp(1)+x**2)**2+(-2*x**3-8*x**2+6*x)*exp(x*exp(1)+x**2)+x**4+8*x**3+10*
x**2-24*x+9),x)

[Out]

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

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