Integrand size = 91, antiderivative size = 31 \[ \int \frac {e^{\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}} \left (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2\right )}{25 x^2-10 e^{-3+x} x^2+e^{-6+2 x} x^2} \, dx=7+e^3+e^{-\frac {e^x}{5-e^{-3+x}}+\frac {3+x}{x}} \] Output:
exp(3)+7+exp((3+x)/x-exp(x)/(5-exp(-3+x)))
Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}} \left (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2\right )}{25 x^2-10 e^{-3+x} x^2+e^{-6+2 x} x^2} \, dx=e^{1+e^3+\frac {5 e^6}{-5 e^3+e^x}+\frac {3}{x}} \] Input:
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]
Output:
E^(1 + E^3 + (5*E^6)/(-5*E^3 + E^x) + 3/x)
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (-5 e^x x^2+30 e^{x-3}-3 e^{2 x-6}-75\right ) \exp \left (\frac {e^x x-5 x+e^{x-3} (x+3)-15}{e^{x-3} x-5 x}\right )}{-10 e^{x-3} x^2+e^{2 x-6} x^2+25 x^2} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (-5 e^x x^2+30 e^{x-3}-3 e^{2 x-6}-75\right ) \exp \left (\frac {e^x x-5 x+e^{x-3} (x+3)-15}{e^{x-3} x-5 x}+6\right )}{\left (5 e^3-e^x\right )^2 x^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {3 \exp \left (\frac {e^x x-5 x+e^{x-3} (x+3)-15}{e^{x-3} x-5 x}\right )}{x^2}-\frac {25 \exp \left (\frac {e^x x-5 x+e^{x-3} (x+3)-15}{e^{x-3} x-5 x}+9\right )}{\left (5 e^3-e^x\right )^2}-\frac {5 \exp \left (\frac {e^x x-5 x+e^{x-3} (x+3)-15}{e^{x-3} x-5 x}+6\right )}{e^x-5 e^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -3 \int \frac {\exp \left (\frac {e^x x-5 x+e^{x-3} (x+3)-15}{e^{x-3} x-5 x}\right )}{x^2}dx-25 \int \frac {\exp \left (\frac {e^x x-5 x+e^{x-3} (x+3)-15}{e^{x-3} x-5 x}+9\right )}{\left (5 e^3-e^x\right )^2}dx-5 \int \frac {\exp \left (\frac {e^x x-5 x+e^{x-3} (x+3)-15}{e^{x-3} x-5 x}+6\right )}{-5 e^3+e^x}dx\) |
Input:
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]
Output:
$Aborted
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 | \(\frac {\left (5 x \,{\mathrm e}^{6} {\mathrm e}^{\frac {{\mathrm e}^{x} x +\left (3+x \right ) {\mathrm e}^{-3} {\mathrm e}^{x}-5 x -15}{{\mathrm e}^{-3} {\mathrm e}^{x} x -5 x}}-x \,{\mathrm e}^{3} {\mathrm e}^{x} {\mathrm e}^{\frac {{\mathrm e}^{x} x +\left (3+x \right ) {\mathrm e}^{-3} {\mathrm e}^{x}-5 x -15}{{\mathrm e}^{-3} {\mathrm e}^{x} x -5 x}}\right ) {\mathrm e}^{-3}}{x \left (5 \,{\mathrm e}^{3}-{\mathrm e}^{x}\right )}\) | \(97\) |
Input:
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)
Output:
exp((exp(x)*x+(3+x)*exp(-3+x)-5*x-15)/x/(exp(-3+x)-5))
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {e^{\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}} \left (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2\right )}{25 x^2-10 e^{-3+x} x^2+e^{-6+2 x} x^2} \, dx=e^{\left (\frac {5 \, {\left (x + 3\right )} e^{3} - {\left (x e^{3} + x + 3\right )} e^{x}}{5 \, x e^{3} - x e^{x}}\right )} \] Input:
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)+2 5*x^2),x, algorithm="fricas")
Output:
e^((5*(x + 3)*e^3 - (x*e^3 + x + 3)*e^x)/(5*x*e^3 - x*e^x))
Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {e^{\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}} \left (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2\right )}{25 x^2-10 e^{-3+x} x^2+e^{-6+2 x} x^2} \, dx=e^{\frac {x e^{x} - 5 x + \frac {\left (x + 3\right ) e^{x}}{e^{3}} - 15}{\frac {x e^{x}}{e^{3}} - 5 x}} \] Input:
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)
Output:
exp((x*exp(x) - 5*x + (x + 3)*exp(-3)*exp(x) - 15)/(x*exp(-3)*exp(x) - 5*x ))
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (24) = 48\).
Time = 0.23 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.68 \[ \int \frac {e^{\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}} \left (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2\right )}{25 x^2-10 e^{-3+x} x^2+e^{-6+2 x} x^2} \, dx=e^{\left (\frac {15 \, e^{3}}{5 \, x e^{3} - x e^{x}} + \frac {5 \, e^{3}}{5 \, e^{3} - e^{x}} - \frac {e^{\left (x + 3\right )}}{5 \, e^{3} - e^{x}} - \frac {3 \, e^{x}}{5 \, x e^{3} - x e^{x}} - \frac {e^{x}}{5 \, e^{3} - e^{x}}\right )} \] Input:
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)+2 5*x^2),x, algorithm="maxima")
Output:
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))
\[ \int \frac {e^{\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}} \left (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2\right )}{25 x^2-10 e^{-3+x} x^2+e^{-6+2 x} x^2} \, dx=\int { -\frac {{\left (5 \, x^{2} e^{x} + 3 \, e^{\left (2 \, x - 6\right )} - 30 \, e^{\left (x - 3\right )} + 75\right )} e^{\left (\frac {{\left (x + 3\right )} e^{\left (x - 3\right )} + x e^{x} - 5 \, x - 15}{x e^{\left (x - 3\right )} - 5 \, x}\right )}}{x^{2} e^{\left (2 \, x - 6\right )} - 10 \, x^{2} e^{\left (x - 3\right )} + 25 \, x^{2}} \,d x } \] Input:
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)+2 5*x^2),x, algorithm="giac")
Output:
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)
Time = 3.24 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.81 \[ \int \frac {e^{\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}} \left (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2\right )}{25 x^2-10 e^{-3+x} x^2+e^{-6+2 x} x^2} \, dx={\mathrm {e}}^{-\frac {3\,{\mathrm {e}}^x}{5\,x\,{\mathrm {e}}^3-x\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^x}{5\,{\mathrm {e}}^3-{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^3\,{\mathrm {e}}^x}{5\,{\mathrm {e}}^3-{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {15\,{\mathrm {e}}^3}{5\,x\,{\mathrm {e}}^3-x\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {5\,{\mathrm {e}}^3}{5\,{\mathrm {e}}^3-{\mathrm {e}}^x}} \] Input:
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)
Output:
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)))
Time = 0.16 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81 \[ \int \frac {e^{\frac {-15-5 x+e^x x+e^{-3+x} (3+x)}{-5 x+e^{-3+x} x}} \left (-75+30 e^{-3+x}-3 e^{-6+2 x}-5 e^x x^2\right )}{25 x^2-10 e^{-3+x} x^2+e^{-6+2 x} x^2} \, dx=\frac {e^{\frac {e^{x} e^{3} x +3 e^{x}}{e^{x} x -5 e^{3} x}} e}{e^{\frac {15 e^{3}}{e^{x} x -5 e^{3} x}}} \] Input:
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)
Output:
(e**((e**x*e**3*x + 3*e**x)/(e**x*x - 5*e**3*x))*e)/e**((15*e**3)/(e**x*x - 5*e**3*x))