Integrand size = 168, antiderivative size = 24 \[ \int \frac {\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5+e^{x+\frac {100}{\frac {e^{2 x}}{9}+\frac {2 e^x x}{3}+x^2}} \left (\frac {1}{27} e^{3 x} (1-x)+200 x+x^3-x^4+\frac {1}{9} e^{2 x} \left (3 x-3 x^2\right )+\frac {1}{3} e^x \left (200 x+3 x^2-3 x^3\right )\right )}{\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5} \, dx=-\frac {e^{x+\frac {100}{\left (\frac {e^x}{3}+x\right )^2}}}{x}+x \]
Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5+e^{x+\frac {100}{\frac {e^{2 x}}{9}+\frac {2 e^x x}{3}+x^2}} \left (\frac {1}{27} e^{3 x} (1-x)+200 x+x^3-x^4+\frac {1}{9} e^{2 x} \left (3 x-3 x^2\right )+\frac {1}{3} e^x \left (200 x+3 x^2-3 x^3\right )\right )}{\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5} \, dx=27 \left (-\frac {e^{x+\frac {900}{\left (e^x+3 x\right )^2}}}{27 x}+\frac {x}{27}\right ) \]
Integrate[((E^(3*x)*x^2)/27 + (E^(2*x)*x^3)/3 + E^x*x^4 + x^5 + E^(x + 100 /(E^(2*x)/9 + (2*E^x*x)/3 + x^2))*((E^(3*x)*(1 - x))/27 + 200*x + x^3 - x^ 4 + (E^(2*x)*(3*x - 3*x^2))/9 + (E^x*(200*x + 3*x^2 - 3*x^3))/3))/((E^(3*x )*x^2)/27 + (E^(2*x)*x^3)/3 + E^x*x^4 + x^5),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.
| \(\displaystyle \int \frac {x^5+e^x x^4+\frac {1}{3} e^{2 x} x^3+\frac {1}{27} e^{3 x} x^2+e^{\frac {100}{x^2+\frac {2 e^x x}{3}+\frac {e^{2 x}}{9}}+x} \left (-x^4+x^3+\frac {1}{9} e^{2 x} \left (3 x-3 x^2\right )+\frac {1}{3} e^x \left (-3 x^3+3 x^2+200 x\right )+200 x+\frac {1}{27} e^{3 x} (1-x)\right )}{x^5+e^x x^4+\frac {1}{3} e^{2 x} x^3+\frac {1}{27} e^{3 x} x^2} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {27 \left (x^5+e^x x^4+\frac {1}{3} e^{2 x} x^3+\frac {1}{27} e^{3 x} x^2+e^{\frac {100}{x^2+\frac {2 e^x x}{3}+\frac {e^{2 x}}{9}}+x} \left (-x^4+x^3+\frac {1}{9} e^{2 x} \left (3 x-3 x^2\right )+\frac {1}{3} e^x \left (-3 x^3+3 x^2+200 x\right )+200 x+\frac {1}{27} e^{3 x} (1-x)\right )\right )}{x^2 \left (3 x+e^x\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 27 \int \frac {27 x^5+27 e^x x^4+9 e^{2 x} x^3+e^{3 x} x^2+e^{x+\frac {900}{9 x^2+6 e^x x+e^{2 x}}} \left (-27 x^4+27 x^3+5400 x+e^{3 x} (1-x)+9 e^{2 x} \left (x-x^2\right )+9 e^x \left (-3 x^3+3 x^2+200 x\right )\right )}{27 x^2 \left (3 x+e^x\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {27 x^5+27 e^x x^4+9 e^{2 x} x^3+e^{3 x} x^2+e^{\frac {900}{9 x^2+6 e^x x+e^{2 x}}+x} \left (-27 x^4+27 x^3+9 e^{2 x} \left (x-x^2\right )+9 e^x \left (-3 x^3+3 x^2+200 x\right )+5400 x+e^{3 x} (1-x)\right )}{x^2 \left (3 x+e^x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {27 x^3}{\left (3 x+e^x\right )^3}+\frac {27 e^x x^2}{\left (3 x+e^x\right )^3}-\frac {e^{x+\frac {900}{\left (3 x+e^x\right )^2}} \left (27 x^4+27 e^x x^3-27 x^3-27 e^x x^2+9 e^{2 x} x^2-1800 e^x x-9 e^{2 x} x+e^{3 x} x-5400 x-e^{3 x}\right )}{\left (3 x+e^x\right )^3 x^2}+\frac {9 e^{2 x} x}{\left (3 x+e^x\right )^3}+\frac {e^{3 x}}{\left (3 x+e^x\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {27 x^3}{\left (3 x+e^x\right )^3}+\frac {27 e^x x^2}{\left (3 x+e^x\right )^3}-\frac {e^{x+\frac {900}{\left (3 x+e^x\right )^2}} \left (27 x^4+27 e^x x^3-27 x^3-27 e^x x^2+9 e^{2 x} x^2-1800 e^x x-9 e^{2 x} x+e^{3 x} x-5400 x-e^{3 x}\right )}{\left (3 x+e^x\right )^3 x^2}+\frac {9 e^{2 x} x}{\left (3 x+e^x\right )^3}+\frac {e^{3 x}}{\left (3 x+e^x\right )^3}\right )dx\) |
Int[((E^(3*x)*x^2)/27 + (E^(2*x)*x^3)/3 + E^x*x^4 + x^5 + E^(x + 100/(E^(2 *x)/9 + (2*E^x*x)/3 + x^2))*((E^(3*x)*(1 - x))/27 + 200*x + x^3 - x^4 + (E ^(2*x)*(3*x - 3*x^2))/9 + (E^x*(200*x + 3*x^2 - 3*x^3))/3))/((E^(3*x)*x^2) /27 + (E^(2*x)*x^3)/3 + E^x*x^4 + x^5),x]
3.24.70.3.1 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 = 3.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83
| method | result | size |
| parallelrisch | \(\frac {15 x^{2}-15 \,{\mathrm e}^{x} {\mathrm e}^{\frac {100}{\frac {{\mathrm e}^{2 x}}{9}+2 x \,{\mathrm e}^{-\ln \left (3\right )+x}+x^{2}}}}{15 x}\) | \(44\) |
| risch | \(x -\frac {{\mathrm e}^{\frac {6 \,{\mathrm e}^{x} x^{2}+9 x^{3}+x \,{\mathrm e}^{2 x}+900}{6 \,{\mathrm e}^{x} x +9 x^{2}+{\mathrm e}^{2 x}}}}{x}\) | \(47\) |
int((((1-x)*exp(-ln(3)+x)^3+(-3*x^2+3*x)*exp(-ln(3)+x)^2+(-3*x^3+3*x^2+200 *x)*exp(-ln(3)+x)-x^4+x^3+200*x)*exp(x)*exp(100/(exp(-ln(3)+x)^2+2*x*exp(- ln(3)+x)+x^2))+x^2*exp(-ln(3)+x)^3+3*x^3*exp(-ln(3)+x)^2+3*x^4*exp(-ln(3)+ x)+x^5)/(x^2*exp(-ln(3)+x)^3+3*x^3*exp(-ln(3)+x)^2+3*x^4*exp(-ln(3)+x)+x^5 ),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (23) = 46\).
Time = 0.26 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.71 \[ \int \frac {\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5+e^{x+\frac {100}{\frac {e^{2 x}}{9}+\frac {2 e^x x}{3}+x^2}} \left (\frac {1}{27} e^{3 x} (1-x)+200 x+x^3-x^4+\frac {1}{9} e^{2 x} \left (3 x-3 x^2\right )+\frac {1}{3} e^x \left (200 x+3 x^2-3 x^3\right )\right )}{\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5} \, dx=\frac {x^{2} - e^{\left (\frac {x^{3} + 2 \, x^{2} e^{\left (x - \log \left (3\right )\right )} + x e^{\left (2 \, x - 2 \, \log \left (3\right )\right )} + 100}{x^{2} + 2 \, x e^{\left (x - \log \left (3\right )\right )} + e^{\left (2 \, x - 2 \, \log \left (3\right )\right )}}\right )}}{x} \]
integrate((((1-x)*exp(-log(3)+x)^3+(-3*x^2+3*x)*exp(-log(3)+x)^2+(-3*x^3+3 *x^2+200*x)*exp(-log(3)+x)-x^4+x^3+200*x)*exp(x)*exp(100/(exp(-log(3)+x)^2 +2*x*exp(-log(3)+x)+x^2))+x^2*exp(-log(3)+x)^3+3*x^3*exp(-log(3)+x)^2+3*x^ 4*exp(-log(3)+x)+x^5)/(x^2*exp(-log(3)+x)^3+3*x^3*exp(-log(3)+x)^2+3*x^4*e xp(-log(3)+x)+x^5),x, algorithm=\
(x^2 - e^((x^3 + 2*x^2*e^(x - log(3)) + x*e^(2*x - 2*log(3)) + 100)/(x^2 + 2*x*e^(x - log(3)) + e^(2*x - 2*log(3)))))/x
Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5+e^{x+\frac {100}{\frac {e^{2 x}}{9}+\frac {2 e^x x}{3}+x^2}} \left (\frac {1}{27} e^{3 x} (1-x)+200 x+x^3-x^4+\frac {1}{9} e^{2 x} \left (3 x-3 x^2\right )+\frac {1}{3} e^x \left (200 x+3 x^2-3 x^3\right )\right )}{\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5} \, dx=x - \frac {e^{x} e^{\frac {100}{x^{2} + \frac {2 x e^{x}}{3} + \frac {e^{2 x}}{9}}}}{x} \]
integrate((((1-x)*exp(-ln(3)+x)**3+(-3*x**2+3*x)*exp(-ln(3)+x)**2+(-3*x**3 +3*x**2+200*x)*exp(-ln(3)+x)-x**4+x**3+200*x)*exp(x)*exp(100/(exp(-ln(3)+x )**2+2*x*exp(-ln(3)+x)+x**2))+x**2*exp(-ln(3)+x)**3+3*x**3*exp(-ln(3)+x)** 2+3*x**4*exp(-ln(3)+x)+x**5)/(x**2*exp(-ln(3)+x)**3+3*x**3*exp(-ln(3)+x)** 2+3*x**4*exp(-ln(3)+x)+x**5),x)
\[ \int \frac {\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5+e^{x+\frac {100}{\frac {e^{2 x}}{9}+\frac {2 e^x x}{3}+x^2}} \left (\frac {1}{27} e^{3 x} (1-x)+200 x+x^3-x^4+\frac {1}{9} e^{2 x} \left (3 x-3 x^2\right )+\frac {1}{3} e^x \left (200 x+3 x^2-3 x^3\right )\right )}{\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5} \, dx=\int { \frac {x^{5} + 3 \, x^{4} e^{\left (x - \log \left (3\right )\right )} + 3 \, x^{3} e^{\left (2 \, x - 2 \, \log \left (3\right )\right )} + x^{2} e^{\left (3 \, x - 3 \, \log \left (3\right )\right )} - {\left (x^{4} - x^{3} + {\left (x - 1\right )} e^{\left (3 \, x - 3 \, \log \left (3\right )\right )} + 3 \, {\left (x^{2} - x\right )} e^{\left (2 \, x - 2 \, \log \left (3\right )\right )} + {\left (3 \, x^{3} - 3 \, x^{2} - 200 \, x\right )} e^{\left (x - \log \left (3\right )\right )} - 200 \, x\right )} e^{\left (x + \frac {100}{x^{2} + 2 \, x e^{\left (x - \log \left (3\right )\right )} + e^{\left (2 \, x - 2 \, \log \left (3\right )\right )}}\right )}}{x^{5} + 3 \, x^{4} e^{\left (x - \log \left (3\right )\right )} + 3 \, x^{3} e^{\left (2 \, x - 2 \, \log \left (3\right )\right )} + x^{2} e^{\left (3 \, x - 3 \, \log \left (3\right )\right )}} \,d x } \]
integrate((((1-x)*exp(-log(3)+x)^3+(-3*x^2+3*x)*exp(-log(3)+x)^2+(-3*x^3+3 *x^2+200*x)*exp(-log(3)+x)-x^4+x^3+200*x)*exp(x)*exp(100/(exp(-log(3)+x)^2 +2*x*exp(-log(3)+x)+x^2))+x^2*exp(-log(3)+x)^3+3*x^3*exp(-log(3)+x)^2+3*x^ 4*exp(-log(3)+x)+x^5)/(x^2*exp(-log(3)+x)^3+3*x^3*exp(-log(3)+x)^2+3*x^4*e xp(-log(3)+x)+x^5),x, algorithm=\
x - integrate(((x - 1)*e^(4*x) + 9*(x^2 - x)*e^(3*x) + 9*(3*x^3 - 3*x^2 - 200*x)*e^(2*x) + 27*(x^4 - x^3 - 200*x)*e^x)*e^(900/(9*x^2 + 6*x*e^x + e^( 2*x)))/(27*x^5 + 27*x^4*e^x + 9*x^3*e^(2*x) + x^2*e^(3*x)), x)
\[ \int \frac {\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5+e^{x+\frac {100}{\frac {e^{2 x}}{9}+\frac {2 e^x x}{3}+x^2}} \left (\frac {1}{27} e^{3 x} (1-x)+200 x+x^3-x^4+\frac {1}{9} e^{2 x} \left (3 x-3 x^2\right )+\frac {1}{3} e^x \left (200 x+3 x^2-3 x^3\right )\right )}{\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5} \, dx=\int { \frac {x^{5} + 3 \, x^{4} e^{\left (x - \log \left (3\right )\right )} + 3 \, x^{3} e^{\left (2 \, x - 2 \, \log \left (3\right )\right )} + x^{2} e^{\left (3 \, x - 3 \, \log \left (3\right )\right )} - {\left (x^{4} - x^{3} + {\left (x - 1\right )} e^{\left (3 \, x - 3 \, \log \left (3\right )\right )} + 3 \, {\left (x^{2} - x\right )} e^{\left (2 \, x - 2 \, \log \left (3\right )\right )} + {\left (3 \, x^{3} - 3 \, x^{2} - 200 \, x\right )} e^{\left (x - \log \left (3\right )\right )} - 200 \, x\right )} e^{\left (x + \frac {100}{x^{2} + 2 \, x e^{\left (x - \log \left (3\right )\right )} + e^{\left (2 \, x - 2 \, \log \left (3\right )\right )}}\right )}}{x^{5} + 3 \, x^{4} e^{\left (x - \log \left (3\right )\right )} + 3 \, x^{3} e^{\left (2 \, x - 2 \, \log \left (3\right )\right )} + x^{2} e^{\left (3 \, x - 3 \, \log \left (3\right )\right )}} \,d x } \]
integrate((((1-x)*exp(-log(3)+x)^3+(-3*x^2+3*x)*exp(-log(3)+x)^2+(-3*x^3+3 *x^2+200*x)*exp(-log(3)+x)-x^4+x^3+200*x)*exp(x)*exp(100/(exp(-log(3)+x)^2 +2*x*exp(-log(3)+x)+x^2))+x^2*exp(-log(3)+x)^3+3*x^3*exp(-log(3)+x)^2+3*x^ 4*exp(-log(3)+x)+x^5)/(x^2*exp(-log(3)+x)^3+3*x^3*exp(-log(3)+x)^2+3*x^4*e xp(-log(3)+x)+x^5),x, algorithm=\
Time = 10.73 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5+e^{x+\frac {100}{\frac {e^{2 x}}{9}+\frac {2 e^x x}{3}+x^2}} \left (\frac {1}{27} e^{3 x} (1-x)+200 x+x^3-x^4+\frac {1}{9} e^{2 x} \left (3 x-3 x^2\right )+\frac {1}{3} e^x \left (200 x+3 x^2-3 x^3\right )\right )}{\frac {1}{27} e^{3 x} x^2+\frac {1}{3} e^{2 x} x^3+e^x x^4+x^5} \, dx=x-\frac {{\mathrm {e}}^{\frac {100}{\frac {{\mathrm {e}}^{2\,x}}{9}+\frac {2\,x\,{\mathrm {e}}^x}{3}+x^2}}\,{\mathrm {e}}^x}{x} \]
int((3*x^4*exp(x - log(3)) + 3*x^3*exp(2*x - 2*log(3)) + x^2*exp(3*x - 3*l og(3)) + x^5 + exp(100/(exp(2*x - 2*log(3)) + x^2 + 2*x*exp(x - log(3))))* exp(x)*(200*x + exp(x - log(3))*(200*x + 3*x^2 - 3*x^3) + exp(2*x - 2*log( 3))*(3*x - 3*x^2) - exp(3*x - 3*log(3))*(x - 1) + x^3 - x^4))/(3*x^4*exp(x - log(3)) + 3*x^3*exp(2*x - 2*log(3)) + x^2*exp(3*x - 3*log(3)) + x^5),x)