3.80.70 \(\int \frac {e^{\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}} (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2)}{25 x^2-10 e^{-3+x} x^2+e^{-6+2 x} x^2} \, dx\)

Optimal. Leaf size=31 \[ 7+e^3+e^{-\frac {e^x}{5-e^{-3+x}}+\frac {3+x}{x}} \]

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Rubi [F]  time = 11.08, 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 {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}\right ) \left (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2\right )}{25 x^2-10 e^{-3+x} x^2+e^{-6+2 x} x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-15 - 5*x + E^x*x + E^(-3 + x)*(3 + x))/(-5*x + E^(-3 + x)*x))*(-75 + 30*E^(-3 + x) - 3*E^(-6 + 2*x)
- 5*E^x*x^2))/(25*x^2 - 10*E^(-3 + x)*x^2 + E^(-6 + 2*x)*x^2),x]

[Out]

-25*Defer[Int][E^(9 + (-15 - 5*x + E^x*x + E^(-3 + x)*(3 + x))/(-5*x + E^(-3 + x)*x))/(5*E^3 - E^x)^2, x] - 5*
Defer[Int][E^(6 + (-15 - 5*x + E^x*x + E^(-3 + x)*(3 + x))/(-5*x + E^(-3 + x)*x))/(-5*E^3 + E^x), x] - 3*Defer
[Int][E^((-15 - 5*x + E^x*x + E^(-3 + x)*(3 + x))/(-5*x + E^(-3 + x)*x))/x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (6+\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}\right ) \left (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2\right )}{\left (5 e^3-e^x\right )^2 x^2} \, dx\\ &=\int \left (-\frac {25 \exp \left (9+\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}\right )}{\left (5 e^3-e^x\right )^2}-\frac {5 \exp \left (6+\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}\right )}{-5 e^3+e^x}-\frac {3 \exp \left (\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}\right )}{x^2}\right ) \, dx\\ &=-\left (3 \int \frac {\exp \left (\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}\right )}{x^2} \, dx\right )-5 \int \frac {\exp \left (6+\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}\right )}{-5 e^3+e^x} \, dx-25 \int \frac {\exp \left (9+\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}\right )}{\left (5 e^3-e^x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 28, normalized size = 0.90 \begin {gather*} e^{1+e^3+\frac {5 e^6}{-5 e^3+e^x}+\frac {3}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-15 - 5*x + E^x*x + E^(-3 + x)*(3 + x))/(-5*x + E^(-3 + x)*x))*(-75 + 30*E^(-3 + x) - 3*E^(-6 +
2*x) - 5*E^x*x^2))/(25*x^2 - 10*E^(-3 + x)*x^2 + E^(-6 + 2*x)*x^2),x]

[Out]

E^(1 + E^3 + (5*E^6)/(-5*E^3 + E^x) + 3/x)

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fricas [A]  time = 0.90, size = 34, normalized size = 1.10 \begin {gather*} e^{\left (\frac {5 \, {\left (x + 3\right )} e^{3} - {\left (x e^{3} + x + 3\right )} e^{x}}{5 \, x e^{3} - x e^{x}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*exp(x)*x^2-3*exp(x-3)^2+30*exp(x-3)-75)*exp((exp(x)*x+(3+x)*exp(x-3)-5*x-15)/(x*exp(x-3)-5*x))/(
x^2*exp(x-3)^2-10*x^2*exp(x-3)+25*x^2),x, algorithm="fricas")

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*exp(x)*x^2-3*exp(x-3)^2+30*exp(x-3)-75)*exp((exp(x)*x+(3+x)*exp(x-3)-5*x-15)/(x*exp(x-3)-5*x))/(
x^2*exp(x-3)^2-10*x^2*exp(x-3)+25*x^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Evaluation time: 0.45Unable to divide, perhaps due to rounding error%%%{-375,[0,10,12,2]%%%}+%%%{13125,[0,9
,12,3]%%%}+

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maple [A]  time = 0.38, size = 35, normalized size = 1.13

method	result	size
risch	\({\mathrm e}^{\frac {{\mathrm e}^{x} x +x \,{\mathrm e}^{x -3}+3 \,{\mathrm e}^{x -3}-5 x -15}{x \left ({\mathrm e}^{x -3}-5\right )}}\)	\(35\)
norman	\(\frac {\left (5 x \,{\mathrm e}^{6} {\mathrm e}^{\frac {{\mathrm e}^{x} x +\left (3+x \right ) {\mathrm e}^{-3} {\mathrm e}^{x}-5 x -15}{{\mathrm e}^{-3} {\mathrm e}^{x} x -5 x}}-x \,{\mathrm e}^{3} {\mathrm e}^{x} {\mathrm e}^{\frac {{\mathrm e}^{x} x +\left (3+x \right ) {\mathrm e}^{-3} {\mathrm e}^{x}-5 x -15}{{\mathrm e}^{-3} {\mathrm e}^{x} x -5 x}}\right ) {\mathrm e}^{-3}}{x \left (5 \,{\mathrm e}^{3}-{\mathrm e}^{x}\right )}\)	\(97\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-5*exp(x)*x^2-3*exp(x-3)^2+30*exp(x-3)-75)*exp((exp(x)*x+(3+x)*exp(x-3)-5*x-15)/(x*exp(x-3)-5*x))/(x^2*ex
p(x-3)^2-10*x^2*exp(x-3)+25*x^2),x,method=_RETURNVERBOSE)

[Out]

exp((exp(x)*x+x*exp(x-3)+3*exp(x-3)-5*x-15)/x/(exp(x-3)-5))

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maxima [B]  time = 0.53, size = 83, normalized size = 2.68 \begin {gather*} e^{\left (\frac {15 \, e^{3}}{5 \, x e^{3} - x e^{x}} + \frac {5 \, e^{3}}{5 \, e^{3} - e^{x}} - \frac {e^{\left (x + 3\right )}}{5 \, e^{3} - e^{x}} - \frac {3 \, e^{x}}{5 \, x e^{3} - x e^{x}} - \frac {e^{x}}{5 \, e^{3} - e^{x}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*exp(x)*x^2-3*exp(x-3)^2+30*exp(x-3)-75)*exp((exp(x)*x+(3+x)*exp(x-3)-5*x-15)/(x*exp(x-3)-5*x))/(
x^2*exp(x-3)^2-10*x^2*exp(x-3)+25*x^2),x, algorithm="maxima")

[Out]

e^(15*e^3/(5*x*e^3 - x*e^x) + 5*e^3/(5*e^3 - e^x) - e^(x + 3)/(5*e^3 - e^x) - 3*e^x/(5*x*e^3 - x*e^x) - e^x/(5
*e^3 - e^x))

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mupad [B]  time = 7.43, size = 87, normalized size = 2.81 \begin {gather*} {\mathrm {e}}^{-\frac {3\,{\mathrm {e}}^x}{5\,x\,{\mathrm {e}}^3-x\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^x}{5\,{\mathrm {e}}^3-{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^3\,{\mathrm {e}}^x}{5\,{\mathrm {e}}^3-{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {15\,{\mathrm {e}}^3}{5\,x\,{\mathrm {e}}^3-x\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {5\,{\mathrm {e}}^3}{5\,{\mathrm {e}}^3-{\mathrm {e}}^x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((5*x - exp(x - 3)*(x + 3) - x*exp(x) + 15)/(5*x - x*exp(x - 3)))*(3*exp(2*x - 6) - 30*exp(x - 3) + 5
*x^2*exp(x) + 75))/(x^2*exp(2*x - 6) - 10*x^2*exp(x - 3) + 25*x^2),x)

[Out]

exp(-(3*exp(x))/(5*x*exp(3) - x*exp(x)))*exp(-exp(x)/(5*exp(3) - exp(x)))*exp(-(exp(3)*exp(x))/(5*exp(3) - exp
(x)))*exp((15*exp(3))/(5*x*exp(3) - x*exp(x)))*exp((5*exp(3))/(5*exp(3) - exp(x)))

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sympy [A]  time = 0.53, size = 32, normalized size = 1.03 \begin {gather*} e^{\frac {x e^{x} - 5 x + \frac {\left (x + 3\right ) e^{x}}{e^{3}} - 15}{\frac {x e^{x}}{e^{3}} - 5 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*exp(x)*x**2-3*exp(x-3)**2+30*exp(x-3)-75)*exp((exp(x)*x+(3+x)*exp(x-3)-5*x-15)/(x*exp(x-3)-5*x))
/(x**2*exp(x-3)**2-10*x**2*exp(x-3)+25*x**2),x)

[Out]

exp((x*exp(x) - 5*x + (x + 3)*exp(-3)*exp(x) - 15)/(x*exp(-3)*exp(x) - 5*x))

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