Integrand size = 102, antiderivative size = 36 \[ \int \frac {e^{\frac {1}{18} \left (225-30 x+x^2\right )} \left (e^{3 x} (60-4 x)+15 x^3-x^4+e^{2 x} \left (36+84 x-8 x^2\right )+e^x \left (18 x+57 x^2-5 x^3\right )\right )}{225 e^{3 x}+675 e^{2 x} x+675 e^x x^2+225 x^3} \, dx=5-\frac {1}{25} e^{\frac {1}{2} \left (-5+\frac {x}{3}\right )^2} \left (1+\frac {e^x}{e^x+x}\right )^2 \]
Time = 8.92 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\frac {1}{18} \left (225-30 x+x^2\right )} \left (e^{3 x} (60-4 x)+15 x^3-x^4+e^{2 x} \left (36+84 x-8 x^2\right )+e^x \left (18 x+57 x^2-5 x^3\right )\right )}{225 e^{3 x}+675 e^{2 x} x+675 e^x x^2+225 x^3} \, dx=-\frac {e^{\frac {1}{18} (-15+x)^2} \left (2 e^x+x\right )^2}{25 \left (e^x+x\right )^2} \]
Integrate[(E^((225 - 30*x + x^2)/18)*(E^(3*x)*(60 - 4*x) + 15*x^3 - x^4 + E^(2*x)*(36 + 84*x - 8*x^2) + E^x*(18*x + 57*x^2 - 5*x^3)))/(225*E^(3*x) + 675*E^(2*x)*x + 675*E^x*x^2 + 225*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.
\(\displaystyle \int \frac {e^{\frac {1}{18} \left (x^2-30 x+225\right )} \left (-x^4+15 x^3+e^{2 x} \left (-8 x^2+84 x+36\right )+e^x \left (-5 x^3+57 x^2+18 x\right )+e^{3 x} (60-4 x)\right )}{225 x^3+675 e^x x^2+675 e^{2 x} x+225 e^{3 x}} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{\frac {1}{18} \left (x^2-30 x+225\right )} \left (-x^4+15 x^3+e^{2 x} \left (-8 x^2+84 x+36\right )+e^x \left (-5 x^3+57 x^2+18 x\right )+e^{3 x} (60-4 x)\right )}{225 \left (x+e^x\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{225} \int \frac {e^{\frac {1}{18} \left (x^2-30 x+225\right )} \left (-x^4+15 x^3+4 e^{3 x} (15-x)+4 e^{2 x} \left (-2 x^2+21 x+9\right )+e^x \left (-5 x^3+57 x^2+18 x\right )\right )}{\left (x+e^x\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{225} \int \left (-\frac {18 e^{\frac {1}{18} \left (x^2-30 x+225\right )} (x-1) x^2}{\left (x+e^x\right )^3}-\frac {e^{\frac {1}{18} \left (x^2-30 x+225\right )} \left (x^2-69 x+54\right ) x}{\left (x+e^x\right )^2}-4 e^{\frac {1}{18} \left (x^2-30 x+225\right )} (x-15)+\frac {4 e^{\frac {1}{18} \left (x^2-30 x+225\right )} \left (x^2-24 x+9\right )}{x+e^x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{225} \left (18 \int \frac {e^{\frac {1}{18} \left (x^2-30 x+225\right )} x^2}{\left (x+e^x\right )^3}dx-54 \int \frac {e^{\frac {1}{18} \left (x^2-30 x+225\right )} x}{\left (x+e^x\right )^2}dx+69 \int \frac {e^{\frac {1}{18} \left (x^2-30 x+225\right )} x^2}{\left (x+e^x\right )^2}dx+36 \int \frac {e^{\frac {1}{18} \left (x^2-30 x+225\right )}}{x+e^x}dx-96 \int \frac {e^{\frac {1}{18} \left (x^2-30 x+225\right )} x}{x+e^x}dx+4 \int \frac {e^{\frac {1}{18} \left (x^2-30 x+225\right )} x^2}{x+e^x}dx-18 \int \frac {e^{\frac {1}{18} \left (x^2-30 x+225\right )} x^3}{\left (x+e^x\right )^3}dx-\int \frac {e^{\frac {1}{18} \left (x^2-30 x+225\right )} x^3}{\left (x+e^x\right )^2}dx-36 e^{\frac {x^2}{18}-\frac {5 x}{3}+\frac {25}{2}}\right )\) |
Int[(E^((225 - 30*x + x^2)/18)*(E^(3*x)*(60 - 4*x) + 15*x^3 - x^4 + E^(2*x )*(36 + 84*x - 8*x^2) + E^x*(18*x + 57*x^2 - 5*x^3)))/(225*E^(3*x) + 675*E ^(2*x)*x + 675*E^x*x^2 + 225*x^3),x]
3.26.67.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89
method | result | size |
risch | \(-\frac {\left (x^{2}+4 \,{\mathrm e}^{x} x +4 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{\frac {\left (x -15\right )^{2}}{18}}}{25 \left ({\mathrm e}^{x}+x \right )^{2}}\) | \(32\) |
parallelrisch | \(-\frac {18 \,{\mathrm e}^{\frac {1}{18} x^{2}-\frac {5}{3} x +\frac {25}{2}} x^{2}+72 \,{\mathrm e}^{\frac {1}{18} x^{2}-\frac {5}{3} x +\frac {25}{2}} {\mathrm e}^{x} x +72 \,{\mathrm e}^{\frac {1}{18} x^{2}-\frac {5}{3} x +\frac {25}{2}} {\mathrm e}^{2 x}}{450 \left ({\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} x +x^{2}\right )}\) | \(74\) |
int(((-4*x+60)*exp(x)^3+(-8*x^2+84*x+36)*exp(x)^2+(-5*x^3+57*x^2+18*x)*exp (x)-x^4+15*x^3)*exp(1/36*x^2-5/6*x+25/4)^2/(225*exp(x)^3+675*x*exp(x)^2+67 5*exp(x)*x^2+225*x^3),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\frac {1}{18} \left (225-30 x+x^2\right )} \left (e^{3 x} (60-4 x)+15 x^3-x^4+e^{2 x} \left (36+84 x-8 x^2\right )+e^x \left (18 x+57 x^2-5 x^3\right )\right )}{225 e^{3 x}+675 e^{2 x} x+675 e^x x^2+225 x^3} \, dx=-\frac {{\left (x^{2} + 4 \, x e^{x} + 4 \, e^{\left (2 \, x\right )}\right )} e^{\left (\frac {1}{18} \, x^{2} - \frac {5}{3} \, x + \frac {25}{2}\right )}}{25 \, {\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )}} \]
integrate(((-4*x+60)*exp(x)^3+(-8*x^2+84*x+36)*exp(x)^2+(-5*x^3+57*x^2+18* x)*exp(x)-x^4+15*x^3)*exp(1/36*x^2-5/6*x+25/4)^2/(225*exp(x)^3+675*x*exp(x )^2+675*exp(x)*x^2+225*x^3),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
Time = 0.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.42 \[ \int \frac {e^{\frac {1}{18} \left (225-30 x+x^2\right )} \left (e^{3 x} (60-4 x)+15 x^3-x^4+e^{2 x} \left (36+84 x-8 x^2\right )+e^x \left (18 x+57 x^2-5 x^3\right )\right )}{225 e^{3 x}+675 e^{2 x} x+675 e^x x^2+225 x^3} \, dx=\frac {\left (- x^{2} - 4 x e^{x} - 4 e^{2 x}\right ) e^{\frac {x^{2}}{18} - \frac {5 x}{3} + \frac {25}{2}}}{25 x^{2} + 50 x e^{x} + 25 e^{2 x}} \]
integrate(((-4*x+60)*exp(x)**3+(-8*x**2+84*x+36)*exp(x)**2+(-5*x**3+57*x** 2+18*x)*exp(x)-x**4+15*x**3)*exp(1/36*x**2-5/6*x+25/4)**2/(225*exp(x)**3+6 75*x*exp(x)**2+675*exp(x)*x**2+225*x**3),x)
(-x**2 - 4*x*exp(x) - 4*exp(2*x))*exp(x**2/18 - 5*x/3 + 25/2)/(25*x**2 + 5 0*x*exp(x) + 25*exp(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (25) = 50\).
Time = 0.29 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.44 \[ \int \frac {e^{\frac {1}{18} \left (225-30 x+x^2\right )} \left (e^{3 x} (60-4 x)+15 x^3-x^4+e^{2 x} \left (36+84 x-8 x^2\right )+e^x \left (18 x+57 x^2-5 x^3\right )\right )}{225 e^{3 x}+675 e^{2 x} x+675 e^x x^2+225 x^3} \, dx=-\frac {{\left (x^{2} e^{\frac {25}{2}} + 4 \, x e^{\left (x + \frac {25}{2}\right )} + 4 \, e^{\left (2 \, x + \frac {25}{2}\right )}\right )} e^{\left (\frac {1}{18} \, x^{2}\right )}}{25 \, {\left (x^{2} e^{\left (\frac {5}{3} \, x\right )} + 2 \, x e^{\left (\frac {8}{3} \, x\right )} + e^{\left (\frac {11}{3} \, x\right )}\right )}} \]
integrate(((-4*x+60)*exp(x)^3+(-8*x^2+84*x+36)*exp(x)^2+(-5*x^3+57*x^2+18* x)*exp(x)-x^4+15*x^3)*exp(1/36*x^2-5/6*x+25/4)^2/(225*exp(x)^3+675*x*exp(x )^2+675*exp(x)*x^2+225*x^3),x, algorithm=\
-1/25*(x^2*e^(25/2) + 4*x*e^(x + 25/2) + 4*e^(2*x + 25/2))*e^(1/18*x^2)/(x ^2*e^(5/3*x) + 2*x*e^(8/3*x) + e^(11/3*x))
\[ \int \frac {e^{\frac {1}{18} \left (225-30 x+x^2\right )} \left (e^{3 x} (60-4 x)+15 x^3-x^4+e^{2 x} \left (36+84 x-8 x^2\right )+e^x \left (18 x+57 x^2-5 x^3\right )\right )}{225 e^{3 x}+675 e^{2 x} x+675 e^x x^2+225 x^3} \, dx=\int { -\frac {{\left (x^{4} - 15 \, x^{3} + 4 \, {\left (x - 15\right )} e^{\left (3 \, x\right )} + 4 \, {\left (2 \, x^{2} - 21 \, x - 9\right )} e^{\left (2 \, x\right )} + {\left (5 \, x^{3} - 57 \, x^{2} - 18 \, x\right )} e^{x}\right )} e^{\left (\frac {1}{18} \, x^{2} - \frac {5}{3} \, x + \frac {25}{2}\right )}}{225 \, {\left (x^{3} + 3 \, x^{2} e^{x} + 3 \, x e^{\left (2 \, x\right )} + e^{\left (3 \, x\right )}\right )}} \,d x } \]
integrate(((-4*x+60)*exp(x)^3+(-8*x^2+84*x+36)*exp(x)^2+(-5*x^3+57*x^2+18* x)*exp(x)-x^4+15*x^3)*exp(1/36*x^2-5/6*x+25/4)^2/(225*exp(x)^3+675*x*exp(x )^2+675*exp(x)*x^2+225*x^3),x, algorithm=\
Timed out. \[ \int \frac {e^{\frac {1}{18} \left (225-30 x+x^2\right )} \left (e^{3 x} (60-4 x)+15 x^3-x^4+e^{2 x} \left (36+84 x-8 x^2\right )+e^x \left (18 x+57 x^2-5 x^3\right )\right )}{225 e^{3 x}+675 e^{2 x} x+675 e^x x^2+225 x^3} \, dx=\int \frac {{\mathrm {e}}^{\frac {x^2}{18}-\frac {5\,x}{3}+\frac {25}{2}}\,\left ({\mathrm {e}}^{2\,x}\,\left (-8\,x^2+84\,x+36\right )-{\mathrm {e}}^{3\,x}\,\left (4\,x-60\right )+15\,x^3-x^4+{\mathrm {e}}^x\,\left (-5\,x^3+57\,x^2+18\,x\right )\right )}{225\,{\mathrm {e}}^{3\,x}+675\,x\,{\mathrm {e}}^{2\,x}+675\,x^2\,{\mathrm {e}}^x+225\,x^3} \,d x \]
int((exp(x^2/18 - (5*x)/3 + 25/2)*(exp(2*x)*(84*x - 8*x^2 + 36) - exp(3*x) *(4*x - 60) + 15*x^3 - x^4 + exp(x)*(18*x + 57*x^2 - 5*x^3)))/(225*exp(3*x ) + 675*x*exp(2*x) + 675*x^2*exp(x) + 225*x^3),x)