Integrand size = 103, antiderivative size = 32 \[ \int \frac {400 x+800 x^2-432 x^3+48 x^4+e^x \left (-400+160 x-16 x^2\right )}{25 x^4+70 x^5+29 x^6-28 x^7+4 x^8+e^{2 x} \left (100-40 x+4 x^2\right )+e^x \left (-100 x^2-120 x^3+68 x^4-8 x^5\right )} \, dx=\frac {4}{e^x-\frac {x^2}{2}+x^2 \left (-x+\frac {x}{5-x}\right )} \] Output:
4/(exp(x)+(x/(5-x)-x)*x^2-1/2*x^2)
Time = 2.62 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {400 x+800 x^2-432 x^3+48 x^4+e^x \left (-400+160 x-16 x^2\right )}{25 x^4+70 x^5+29 x^6-28 x^7+4 x^8+e^{2 x} \left (100-40 x+4 x^2\right )+e^x \left (-100 x^2-120 x^3+68 x^4-8 x^5\right )} \, dx=-\frac {8 (-5+x)}{-2 e^x (-5+x)+x^2 \left (-5-7 x+2 x^2\right )} \] Input:
Integrate[(400*x + 800*x^2 - 432*x^3 + 48*x^4 + E^x*(-400 + 160*x - 16*x^2 ))/(25*x^4 + 70*x^5 + 29*x^6 - 28*x^7 + 4*x^8 + E^(2*x)*(100 - 40*x + 4*x^ 2) + E^x*(-100*x^2 - 120*x^3 + 68*x^4 - 8*x^5)),x]
Output:
(-8*(-5 + x))/(-2*E^x*(-5 + x) + x^2*(-5 - 7*x + 2*x^2))
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.
\(\displaystyle \int \frac {48 x^4-432 x^3+800 x^2+e^x \left (-16 x^2+160 x-400\right )+400 x}{4 x^8-28 x^7+29 x^6+70 x^5+25 x^4+e^{2 x} \left (4 x^2-40 x+100\right )+e^x \left (-8 x^5+68 x^4-120 x^3-100 x^2\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {16 x \left (3 x^3-27 x^2+50 x+25\right )-16 e^x (x-5)^2}{\left (\left (-2 x^2+7 x+5\right ) x^2+2 e^x (x-5)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {8 (x-5)}{2 x^4-7 x^3-5 x^2-2 e^x x+10 e^x}-\frac {8 x \left (2 x^4-23 x^3+84 x^2-75 x-50\right )}{\left (2 x^4-7 x^3-5 x^2-2 e^x x+10 e^x\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 40 \int \frac {1}{-2 x^4+7 x^3+5 x^2+2 e^x x-10 e^x}dx+400 \int \frac {x}{\left (2 x^4-7 x^3-5 x^2-2 e^x x+10 e^x\right )^2}dx+600 \int \frac {x^2}{\left (2 x^4-7 x^3-5 x^2-2 e^x x+10 e^x\right )^2}dx-672 \int \frac {x^3}{\left (2 x^4-7 x^3-5 x^2-2 e^x x+10 e^x\right )^2}dx+184 \int \frac {x^4}{\left (2 x^4-7 x^3-5 x^2-2 e^x x+10 e^x\right )^2}dx+8 \int \frac {x}{2 x^4-7 x^3-5 x^2-2 e^x x+10 e^x}dx-16 \int \frac {x^5}{\left (2 x^4-7 x^3-5 x^2-2 e^x x+10 e^x\right )^2}dx\) |
Input:
Int[(400*x + 800*x^2 - 432*x^3 + 48*x^4 + E^x*(-400 + 160*x - 16*x^2))/(25 *x^4 + 70*x^5 + 29*x^6 - 28*x^7 + 4*x^8 + E^(2*x)*(100 - 40*x + 4*x^2) + E ^x*(-100*x^2 - 120*x^3 + 68*x^4 - 8*x^5)),x]
Output:
$Aborted
Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03
method | result | size |
risch | \(-\frac {8 \left (-5+x \right )}{2 x^{4}-7 x^{3}-5 x^{2}-2 \,{\mathrm e}^{x} x +10 \,{\mathrm e}^{x}}\) | \(33\) |
norman | \(\frac {40-8 x}{2 x^{4}-7 x^{3}-5 x^{2}-2 \,{\mathrm e}^{x} x +10 \,{\mathrm e}^{x}}\) | \(34\) |
parallelrisch | \(-\frac {-80+16 x}{2 \left (2 x^{4}-7 x^{3}-5 x^{2}-2 \,{\mathrm e}^{x} x +10 \,{\mathrm e}^{x}\right )}\) | \(35\) |
Input:
int(((-16*x^2+160*x-400)*exp(x)+48*x^4-432*x^3+800*x^2+400*x)/((4*x^2-40*x +100)*exp(x)^2+(-8*x^5+68*x^4-120*x^3-100*x^2)*exp(x)+4*x^8-28*x^7+29*x^6+ 70*x^5+25*x^4),x,method=_RETURNVERBOSE)
Output:
-8*(-5+x)/(2*x^4-7*x^3-5*x^2-2*exp(x)*x+10*exp(x))
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {400 x+800 x^2-432 x^3+48 x^4+e^x \left (-400+160 x-16 x^2\right )}{25 x^4+70 x^5+29 x^6-28 x^7+4 x^8+e^{2 x} \left (100-40 x+4 x^2\right )+e^x \left (-100 x^2-120 x^3+68 x^4-8 x^5\right )} \, dx=-\frac {8 \, {\left (x - 5\right )}}{2 \, x^{4} - 7 \, x^{3} - 5 \, x^{2} - 2 \, {\left (x - 5\right )} e^{x}} \] Input:
integrate(((-16*x^2+160*x-400)*exp(x)+48*x^4-432*x^3+800*x^2+400*x)/((4*x^ 2-40*x+100)*exp(x)^2+(-8*x^5+68*x^4-120*x^3-100*x^2)*exp(x)+4*x^8-28*x^7+2 9*x^6+70*x^5+25*x^4),x, algorithm="fricas")
Output:
-8*(x - 5)/(2*x^4 - 7*x^3 - 5*x^2 - 2*(x - 5)*e^x)
Time = 0.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {400 x+800 x^2-432 x^3+48 x^4+e^x \left (-400+160 x-16 x^2\right )}{25 x^4+70 x^5+29 x^6-28 x^7+4 x^8+e^{2 x} \left (100-40 x+4 x^2\right )+e^x \left (-100 x^2-120 x^3+68 x^4-8 x^5\right )} \, dx=\frac {8 x - 40}{- 2 x^{4} + 7 x^{3} + 5 x^{2} + \left (2 x - 10\right ) e^{x}} \] Input:
integrate(((-16*x**2+160*x-400)*exp(x)+48*x**4-432*x**3+800*x**2+400*x)/(( 4*x**2-40*x+100)*exp(x)**2+(-8*x**5+68*x**4-120*x**3-100*x**2)*exp(x)+4*x* *8-28*x**7+29*x**6+70*x**5+25*x**4),x)
Output:
(8*x - 40)/(-2*x**4 + 7*x**3 + 5*x**2 + (2*x - 10)*exp(x))
Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {400 x+800 x^2-432 x^3+48 x^4+e^x \left (-400+160 x-16 x^2\right )}{25 x^4+70 x^5+29 x^6-28 x^7+4 x^8+e^{2 x} \left (100-40 x+4 x^2\right )+e^x \left (-100 x^2-120 x^3+68 x^4-8 x^5\right )} \, dx=-\frac {8 \, {\left (x - 5\right )}}{2 \, x^{4} - 7 \, x^{3} - 5 \, x^{2} - 2 \, {\left (x - 5\right )} e^{x}} \] Input:
integrate(((-16*x^2+160*x-400)*exp(x)+48*x^4-432*x^3+800*x^2+400*x)/((4*x^ 2-40*x+100)*exp(x)^2+(-8*x^5+68*x^4-120*x^3-100*x^2)*exp(x)+4*x^8-28*x^7+2 9*x^6+70*x^5+25*x^4),x, algorithm="maxima")
Output:
-8*(x - 5)/(2*x^4 - 7*x^3 - 5*x^2 - 2*(x - 5)*e^x)
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {400 x+800 x^2-432 x^3+48 x^4+e^x \left (-400+160 x-16 x^2\right )}{25 x^4+70 x^5+29 x^6-28 x^7+4 x^8+e^{2 x} \left (100-40 x+4 x^2\right )+e^x \left (-100 x^2-120 x^3+68 x^4-8 x^5\right )} \, dx=-\frac {8 \, {\left (x - 5\right )}}{2 \, x^{4} - 7 \, x^{3} - 5 \, x^{2} - 2 \, x e^{x} + 10 \, e^{x}} \] Input:
integrate(((-16*x^2+160*x-400)*exp(x)+48*x^4-432*x^3+800*x^2+400*x)/((4*x^ 2-40*x+100)*exp(x)^2+(-8*x^5+68*x^4-120*x^3-100*x^2)*exp(x)+4*x^8-28*x^7+2 9*x^6+70*x^5+25*x^4),x, algorithm="giac")
Output:
-8*(x - 5)/(2*x^4 - 7*x^3 - 5*x^2 - 2*x*e^x + 10*e^x)
Timed out. \[ \int \frac {400 x+800 x^2-432 x^3+48 x^4+e^x \left (-400+160 x-16 x^2\right )}{25 x^4+70 x^5+29 x^6-28 x^7+4 x^8+e^{2 x} \left (100-40 x+4 x^2\right )+e^x \left (-100 x^2-120 x^3+68 x^4-8 x^5\right )} \, dx=\int \frac {400\,x-{\mathrm {e}}^x\,\left (16\,x^2-160\,x+400\right )+800\,x^2-432\,x^3+48\,x^4}{{\mathrm {e}}^{2\,x}\,\left (4\,x^2-40\,x+100\right )-{\mathrm {e}}^x\,\left (8\,x^5-68\,x^4+120\,x^3+100\,x^2\right )+25\,x^4+70\,x^5+29\,x^6-28\,x^7+4\,x^8} \,d x \] Input:
int((400*x - exp(x)*(16*x^2 - 160*x + 400) + 800*x^2 - 432*x^3 + 48*x^4)/( exp(2*x)*(4*x^2 - 40*x + 100) - exp(x)*(100*x^2 + 120*x^3 - 68*x^4 + 8*x^5 ) + 25*x^4 + 70*x^5 + 29*x^6 - 28*x^7 + 4*x^8),x)
Output:
int((400*x - exp(x)*(16*x^2 - 160*x + 400) + 800*x^2 - 432*x^3 + 48*x^4)/( exp(2*x)*(4*x^2 - 40*x + 100) - exp(x)*(100*x^2 + 120*x^3 - 68*x^4 + 8*x^5 ) + 25*x^4 + 70*x^5 + 29*x^6 - 28*x^7 + 4*x^8), x)
Time = 0.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {400 x+800 x^2-432 x^3+48 x^4+e^x \left (-400+160 x-16 x^2\right )}{25 x^4+70 x^5+29 x^6-28 x^7+4 x^8+e^{2 x} \left (100-40 x+4 x^2\right )+e^x \left (-100 x^2-120 x^3+68 x^4-8 x^5\right )} \, dx=\frac {-40+8 x}{2 e^{x} x -10 e^{x}-2 x^{4}+7 x^{3}+5 x^{2}} \] Input:
int(((-16*x^2+160*x-400)*exp(x)+48*x^4-432*x^3+800*x^2+400*x)/((4*x^2-40*x +100)*exp(x)^2+(-8*x^5+68*x^4-120*x^3-100*x^2)*exp(x)+4*x^8-28*x^7+29*x^6+ 70*x^5+25*x^4),x)
Output:
(8*(x - 5))/(2*e**x*x - 10*e**x - 2*x**4 + 7*x**3 + 5*x**2)