\(\int \frac {e^{\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}} (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2)}{25 x^2-10 e^{-3+x} x^2+e^{-6+2 x} x^2} \, dx\) [7970]

Optimal result

Integrand size = 91, antiderivative size = 31 \[ \int \frac {e^{\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}} \left (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2\right )}{25 x^2-10 e^{-3+x} x^2+e^{-6+2 x} x^2} \, dx=7+e^3+e^{-\frac {e^x}{5-e^{-3+x}}+\frac {3+x}{x}} \]

[Out]

exp(3)+7+exp((3+x)/x-exp(x)/(5-exp(-3+x)))

Rubi [F]

\[ \int \frac {e^{\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}} \left (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2\right )}{25 x^2-10 e^{-3+x} x^2+e^{-6+2 x} x^2} \, dx=\int \frac {\exp \left (\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}\right ) \left (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2\right )}{25 x^2-10 e^{-3+x} x^2+e^{-6+2 x} x^2} \, dx \]

[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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}} \left (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2\right )}{25 x^2-10 e^{-3+x} x^2+e^{-6+2 x} x^2} \, dx=e^{1+e^3+\frac {5 e^6}{-5 e^3+e^x}+\frac {3}{x}} \]

[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)

Maple [A] (verified)

Time = 1.96 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\({\mathrm e}^{\frac {{\mathrm e}^{x} x +\left (3+x \right ) {\mathrm e}^{-3+x}-5 x -15}{x \left ({\mathrm e}^{-3+x}-5\right )}}\)	\(31\)
risch	\({\mathrm e}^{\frac {{\mathrm e}^{x} x +x \,{\mathrm e}^{-3+x}+3 \,{\mathrm e}^{-3+x}-5 x -15}{x \left ({\mathrm e}^{-3+x}-5\right )}}\)	\(35\)
norman	\(\frac {\left (5 x \,{\mathrm e}^{6} {\mathrm e}^{\frac {{\mathrm e}^{x} x +\left (3+x \right ) {\mathrm e}^{-3} {\mathrm e}^{x}-5 x -15}{{\mathrm e}^{-3} {\mathrm e}^{x} x -5 x}}-x \,{\mathrm e}^{3} {\mathrm e}^{x} {\mathrm e}^{\frac {{\mathrm e}^{x} x +\left (3+x \right ) {\mathrm e}^{-3} {\mathrm e}^{x}-5 x -15}{{\mathrm e}^{-3} {\mathrm e}^{x} x -5 x}}\right ) {\mathrm e}^{-3}}{x \left (5 \,{\mathrm e}^{3}-{\mathrm e}^{x}\right )}\)	\(97\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {e^{\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}} \left (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2\right )}{25 x^2-10 e^{-3+x} x^2+e^{-6+2 x} x^2} \, dx=e^{\left (\frac {5 \, {\left (x + 3\right )} e^{3} - {\left (x e^{3} + x + 3\right )} e^{x}}{5 \, x e^{3} - x e^{x}}\right )} \]

[In]

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

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {e^{\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}} \left (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2\right )}{25 x^2-10 e^{-3+x} x^2+e^{-6+2 x} x^2} \, dx=e^{\frac {x e^{x} - 5 x + \frac {\left (x + 3\right ) e^{x}}{e^{3}} - 15}{\frac {x e^{x}}{e^{3}} - 5 x}} \]

[In]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (24) = 48\).

Time = 0.35 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.68 \[ \int \frac {e^{\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}} \left (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2\right )}{25 x^2-10 e^{-3+x} x^2+e^{-6+2 x} x^2} \, dx=e^{\left (\frac {15 \, e^{3}}{5 \, x e^{3} - x e^{x}} + \frac {5 \, e^{3}}{5 \, e^{3} - e^{x}} - \frac {e^{\left (x + 3\right )}}{5 \, e^{3} - e^{x}} - \frac {3 \, e^{x}}{5 \, x e^{3} - x e^{x}} - \frac {e^{x}}{5 \, e^{3} - e^{x}}\right )} \]

[In]

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

Giac [F]

\[ \int \frac {e^{\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}} \left (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2\right )}{25 x^2-10 e^{-3+x} x^2+e^{-6+2 x} x^2} \, dx=\int { -\frac {{\left (5 \, x^{2} e^{x} + 3 \, e^{\left (2 \, x - 6\right )} - 30 \, e^{\left (x - 3\right )} + 75\right )} e^{\left (\frac {{\left (x + 3\right )} e^{\left (x - 3\right )} + x e^{x} - 5 \, x - 15}{x e^{\left (x - 3\right )} - 5 \, x}\right )}}{x^{2} e^{\left (2 \, x - 6\right )} - 10 \, x^{2} e^{\left (x - 3\right )} + 25 \, x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.37 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.81 \[ \int \frac {e^{\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}} \left (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2\right )}{25 x^2-10 e^{-3+x} x^2+e^{-6+2 x} x^2} \, dx={\mathrm {e}}^{-\frac {3\,{\mathrm {e}}^x}{5\,x\,{\mathrm {e}}^3-x\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^x}{5\,{\mathrm {e}}^3-{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^3\,{\mathrm {e}}^x}{5\,{\mathrm {e}}^3-{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {15\,{\mathrm {e}}^3}{5\,x\,{\mathrm {e}}^3-x\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {5\,{\mathrm {e}}^3}{5\,{\mathrm {e}}^3-{\mathrm {e}}^x}} \]

[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)))