Integrand size = 66, antiderivative size = 27 \[ \int \frac {-62181 x^2-64827 x^3+e^x (-882+1764 x)}{4 e^{3 x}+e^{2 x} \left (24 x-588 x^2\right )+e^x \left (36 x^2-1764 x^3+21609 x^4\right )} \, dx=\frac {3 e^{-x}}{x-\frac {2 \left (\frac {e^x}{3}+x\right )}{49 x}} \] Output:
3/exp(x)/(x-2/49*(x+1/3*exp(x))/x)
Time = 1.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {-62181 x^2-64827 x^3+e^x (-882+1764 x)}{4 e^{3 x}+e^{2 x} \left (24 x-588 x^2\right )+e^x \left (36 x^2-1764 x^3+21609 x^4\right )} \, dx=-\frac {441 e^{-x} x}{2 e^x+3 (2-49 x) x} \] Input:
Integrate[(-62181*x^2 - 64827*x^3 + E^x*(-882 + 1764*x))/(4*E^(3*x) + E^(2 *x)*(24*x - 588*x^2) + E^x*(36*x^2 - 1764*x^3 + 21609*x^4)),x]
Output:
(-441*x)/(E^x*(2*E^x + 3*(2 - 49*x)*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.
\(\displaystyle \int \frac {-64827 x^3-62181 x^2+e^x (1764 x-882)}{e^{2 x} \left (24 x-588 x^2\right )+e^x \left (21609 x^4-1764 x^3+36 x^2\right )+4 e^{3 x}} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{-x} \left (-64827 x^3-62181 x^2+e^x (1764 x-882)\right )}{\left (-147 x^2+6 x+2 e^x\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {1323 e^{-x} x \left (49 x^2-100 x+2\right )}{\left (147 x^2-6 x-2 e^x\right )^2}-\frac {441 e^{-x} (2 x-1)}{147 x^2-6 x-2 e^x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -441 \int \frac {e^{-x}}{-147 x^2+6 x+2 e^x}dx+2646 \int \frac {e^{-x} x}{\left (147 x^2-6 x-2 e^x\right )^2}dx-132300 \int \frac {e^{-x} x^2}{\left (147 x^2-6 x-2 e^x\right )^2}dx-882 \int \frac {e^{-x} x}{147 x^2-6 x-2 e^x}dx+64827 \int \frac {e^{-x} x^3}{\left (147 x^2-6 x-2 e^x\right )^2}dx\) |
Input:
Int[(-62181*x^2 - 64827*x^3 + E^x*(-882 + 1764*x))/(4*E^(3*x) + E^(2*x)*(2 4*x - 588*x^2) + E^x*(36*x^2 - 1764*x^3 + 21609*x^4)),x]
Output:
$Aborted
Time = 0.12 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85
method | result | size |
norman | \(\frac {441 x \,{\mathrm e}^{-x}}{147 x^{2}-6 x -2 \,{\mathrm e}^{x}}\) | \(23\) |
parallelrisch | \(\frac {441 x \,{\mathrm e}^{-x}}{147 x^{2}-6 x -2 \,{\mathrm e}^{x}}\) | \(23\) |
risch | \(\frac {147 \,{\mathrm e}^{-x}}{49 x -2}+\frac {294}{\left (49 x -2\right ) \left (147 x^{2}-6 x -2 \,{\mathrm e}^{x}\right )}\) | \(39\) |
Input:
int(((1764*x-882)*exp(x)-64827*x^3-62181*x^2)/(4*exp(x)^3+(-588*x^2+24*x)* exp(x)^2+(21609*x^4-1764*x^3+36*x^2)*exp(x)),x,method=_RETURNVERBOSE)
Output:
441*x/exp(x)/(147*x^2-6*x-2*exp(x))
Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {-62181 x^2-64827 x^3+e^x (-882+1764 x)}{4 e^{3 x}+e^{2 x} \left (24 x-588 x^2\right )+e^x \left (36 x^2-1764 x^3+21609 x^4\right )} \, dx=\frac {441 \, x}{3 \, {\left (49 \, x^{2} - 2 \, x\right )} e^{x} - 2 \, e^{\left (2 \, x\right )}} \] Input:
integrate(((1764*x-882)*exp(x)-64827*x^3-62181*x^2)/(4*exp(x)^3+(-588*x^2+ 24*x)*exp(x)^2+(21609*x^4-1764*x^3+36*x^2)*exp(x)),x, algorithm="fricas")
Output:
441*x/(3*(49*x^2 - 2*x)*e^x - 2*e^(2*x))
Time = 0.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {-62181 x^2-64827 x^3+e^x (-882+1764 x)}{4 e^{3 x}+e^{2 x} \left (24 x-588 x^2\right )+e^x \left (36 x^2-1764 x^3+21609 x^4\right )} \, dx=- \frac {7203}{- \frac {352947 x^{3}}{2} + 14406 x^{2} - 294 x + \left (2401 x - 98\right ) e^{x}} + \frac {147 e^{- x}}{49 x - 2} \] Input:
integrate(((1764*x-882)*exp(x)-64827*x**3-62181*x**2)/(4*exp(x)**3+(-588*x **2+24*x)*exp(x)**2+(21609*x**4-1764*x**3+36*x**2)*exp(x)),x)
Output:
-7203/(-352947*x**3/2 + 14406*x**2 - 294*x + (2401*x - 98)*exp(x)) + 147*e xp(-x)/(49*x - 2)
\[ \int \frac {-62181 x^2-64827 x^3+e^x (-882+1764 x)}{4 e^{3 x}+e^{2 x} \left (24 x-588 x^2\right )+e^x \left (36 x^2-1764 x^3+21609 x^4\right )} \, dx=\int { \frac {441 \, {\left (147 \, x^{3} + 141 \, x^{2} - 2 \, {\left (2 \, x - 1\right )} e^{x}\right )}}{12 \, {\left (49 \, x^{2} - 2 \, x\right )} e^{\left (2 \, x\right )} - 9 \, {\left (2401 \, x^{4} - 196 \, x^{3} + 4 \, x^{2}\right )} e^{x} - 4 \, e^{\left (3 \, x\right )}} \,d x } \] Input:
integrate(((1764*x-882)*exp(x)-64827*x^3-62181*x^2)/(4*exp(x)^3+(-588*x^2+ 24*x)*exp(x)^2+(21609*x^4-1764*x^3+36*x^2)*exp(x)),x, algorithm="maxima")
Output:
441*integrate((147*x^3 + 141*x^2 - 2*(2*x - 1)*e^x)/(12*(49*x^2 - 2*x)*e^( 2*x) - 9*(2401*x^4 - 196*x^3 + 4*x^2)*e^x - 4*e^(3*x)), x)
Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {-62181 x^2-64827 x^3+e^x (-882+1764 x)}{4 e^{3 x}+e^{2 x} \left (24 x-588 x^2\right )+e^x \left (36 x^2-1764 x^3+21609 x^4\right )} \, dx=\frac {441 \, x}{147 \, x^{2} e^{x} - 6 \, x e^{x} - 2 \, e^{\left (2 \, x\right )}} \] Input:
integrate(((1764*x-882)*exp(x)-64827*x^3-62181*x^2)/(4*exp(x)^3+(-588*x^2+ 24*x)*exp(x)^2+(21609*x^4-1764*x^3+36*x^2)*exp(x)),x, algorithm="giac")
Output:
441*x/(147*x^2*e^x - 6*x*e^x - 2*e^(2*x))
Time = 7.41 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {-62181 x^2-64827 x^3+e^x (-882+1764 x)}{4 e^{3 x}+e^{2 x} \left (24 x-588 x^2\right )+e^x \left (36 x^2-1764 x^3+21609 x^4\right )} \, dx=-\frac {441\,x}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (6\,x-147\,x^2\right )} \] Input:
int(-(62181*x^2 - exp(x)*(1764*x - 882) + 64827*x^3)/(4*exp(3*x) + exp(2*x )*(24*x - 588*x^2) + exp(x)*(36*x^2 - 1764*x^3 + 21609*x^4)),x)
Output:
-(441*x)/(2*exp(2*x) + exp(x)*(6*x - 147*x^2))
Time = 0.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {-62181 x^2-64827 x^3+e^x (-882+1764 x)}{4 e^{3 x}+e^{2 x} \left (24 x-588 x^2\right )+e^x \left (36 x^2-1764 x^3+21609 x^4\right )} \, dx=-\frac {441 x}{e^{x} \left (2 e^{x}-147 x^{2}+6 x \right )} \] Input:
int(((1764*x-882)*exp(x)-64827*x^3-62181*x^2)/(4*exp(x)^3+(-588*x^2+24*x)* exp(x)^2+(21609*x^4-1764*x^3+36*x^2)*exp(x)),x)
Output:
( - 441*x)/(e**x*(2*e**x - 147*x**2 + 6*x))