Integrand size = 53, antiderivative size = 25 \[ \int \frac {e^{-x} \left (-5600+525 x+20500 x^5-750 x^6-12500 x^{10}\right )}{3675-630 x+27 x^2-10500 x^5+900 x^6+7500 x^{10}} \, dx=\frac {5 e^{-x}}{3-\frac {9 x}{5 \left (7-10 x^5\right )}} \] Output:
5/(3-9/5*x/(-10*x^5+7))/exp(x)
Time = 2.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {e^{-x} \left (-5600+525 x+20500 x^5-750 x^6-12500 x^{10}\right )}{3675-630 x+27 x^2-10500 x^5+900 x^6+7500 x^{10}} \, dx=-\frac {25}{3} e^{-x} \left (-\frac {1}{5}+\frac {3 x}{5 \left (-35+3 x+50 x^5\right )}\right ) \] Input:
Integrate[(-5600 + 525*x + 20500*x^5 - 750*x^6 - 12500*x^10)/(E^x*(3675 - 630*x + 27*x^2 - 10500*x^5 + 900*x^6 + 7500*x^10)),x]
Output:
(-25*(-1/5 + (3*x)/(5*(-35 + 3*x + 50*x^5))))/(3*E^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.
\(\displaystyle \int \frac {e^{-x} \left (-12500 x^{10}-750 x^6+20500 x^5+525 x-5600\right )}{7500 x^{10}+900 x^6-10500 x^5+27 x^2-630 x+3675} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \frac {e^{-x} \left (-12500 x^{10}-750 x^6+20500 x^5+525 x-5600\right )}{3 \left (50 x^5+3 x-35\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int -\frac {25 e^{-x} \left (500 x^{10}+30 x^6-820 x^5-21 x+224\right )}{\left (-50 x^5-3 x+35\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {25}{3} \int \frac {e^{-x} \left (500 x^{10}+30 x^6-820 x^5-21 x+224\right )}{\left (-50 x^5-3 x+35\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {25}{3} \int \left (-\frac {3 e^{-x} (x+4)}{5 \left (50 x^5+3 x-35\right )}+\frac {e^{-x}}{5}+\frac {3 e^{-x} (12 x-175)}{5 \left (50 x^5+3 x-35\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {25}{3} \left (-105 \int \frac {e^{-x}}{\left (50 x^5+3 x-35\right )^2}dx+\frac {36}{5} \int \frac {e^{-x} x}{\left (50 x^5+3 x-35\right )^2}dx-\frac {12}{5} \int \frac {e^{-x}}{50 x^5+3 x-35}dx-\frac {3}{5} \int \frac {e^{-x} x}{50 x^5+3 x-35}dx-\frac {e^{-x}}{5}\right )\) |
Input:
Int[(-5600 + 525*x + 20500*x^5 - 750*x^6 - 12500*x^10)/(E^x*(3675 - 630*x + 27*x^2 - 10500*x^5 + 900*x^6 + 7500*x^10)),x]
Output:
$Aborted
Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00
method | result | size |
norman | \(\frac {\left (-\frac {175}{3}+\frac {250 x^{5}}{3}\right ) {\mathrm e}^{-x}}{50 x^{5}+3 x -35}\) | \(25\) |
gosper | \(\frac {25 \left (10 x^{5}-7\right ) {\mathrm e}^{-x}}{3 \left (50 x^{5}+3 x -35\right )}\) | \(26\) |
risch | \(\frac {25 \left (10 x^{5}-7\right ) {\mathrm e}^{-x}}{3 \left (50 x^{5}+3 x -35\right )}\) | \(26\) |
parallelrisch | \(\frac {\left (12500 x^{5}-8750\right ) {\mathrm e}^{-x}}{7500 x^{5}+450 x -5250}\) | \(26\) |
orering | \(-\frac {\left (10 x^{5}-7\right ) \left (50 x^{5}+3 x -35\right ) \left (-12500 x^{10}-750 x^{6}+20500 x^{5}+525 x -5600\right ) {\mathrm e}^{-x}}{\left (500 x^{10}+30 x^{6}-820 x^{5}-21 x +224\right ) \left (7500 x^{10}+900 x^{6}-10500 x^{5}+27 x^{2}-630 x +3675\right )}\) | \(93\) |
default | \(\frac {5600 \,{\mathrm e}^{-x} \left (216000 x^{4}+3150000 x^{3}+45937500 x^{2}+669921875 x +10368\right )}{351709077687 \left (50 x^{5}+3 x -35\right )}-\frac {5600 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (50 \textit {\_Z}^{5}+3 \textit {\_Z} -35\right )}{\sum }\frac {\left (216000 \textit {\_R1}^{4}+3366000 \textit {\_R1}^{3}+52237500 \textit {\_R1}^{2}+807734375 \textit {\_R1} +2679697868\right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {expIntegral}_{1}\left (x -\textit {\_R1} \right )}{250 \textit {\_R1}^{4}+3}\right )}{351709077687}-\frac {175 \,{\mathrm e}^{-x} \left (3150000 x^{4}+45937500 x^{3}+669921875 x^{2}-2592 x +151200\right )}{117236359229 \left (50 x^{5}+3 x -35\right )}+\frac {175 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (50 \textit {\_Z}^{5}+3 \textit {\_Z} -35\right )}{\sum }\frac {\left (3150000 \textit {\_R1}^{4}+49087500 \textit {\_R1}^{3}+761796875 \textit {\_R1}^{2}+2009763033 \textit {\_R1} +140832\right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {expIntegral}_{1}\left (x -\textit {\_R1} \right )}{250 \textit {\_R1}^{4}+3}\right )}{117236359229}+\frac {410 \,{\mathrm e}^{-x} \left (1890000 x^{4}+27562500 x^{3}+401953125 x^{2}-23447273401 x +90720\right )}{351709077687 \left (50 x^{5}+3 x -35\right )}-\frac {410 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (50 \textit {\_Z}^{5}+3 \textit {\_Z} -35\right )}{\sum }\frac {\left (1890000 \textit {\_R1}^{4}+29452500 \textit {\_R1}^{3}+457078125 \textit {\_R1}^{2}-22241414026 \textit {\_R1} +23447356345\right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {expIntegral}_{1}\left (x -\textit {\_R1} \right )}{250 \textit {\_R1}^{4}+3}\right )}{351709077687}-\frac {5 \,{\mathrm e}^{-x} \left (27562500 x^{4}+401953125 x^{3}-23447273401 x^{2}-22680 x +1323000\right )}{117236359229 \left (50 x^{5}+3 x -35\right )}+\frac {5 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (50 \textit {\_Z}^{5}+3 \textit {\_Z} -35\right )}{\sum }\frac {\left (27562500 \textit {\_R1}^{4}+429515625 \textit {\_R1}^{3}-22643367151 \textit {\_R1}^{2}+46894516346 \textit {\_R1} +1232280\right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {expIntegral}_{1}\left (x -\textit {\_R1} \right )}{250 \textit {\_R1}^{4}+3}\right )}{117236359229}+\frac {5 \,{\mathrm e}^{-x}}{3}+\frac {5 \,{\mathrm e}^{-x} \left (16537500 x^{4}+241171875 x^{3}-84410179578 x^{2}+820654500995 x +793800\right )}{351709077687 \left (50 x^{5}+3 x -35\right )}-\frac {5 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (50 \textit {\_Z}^{5}+3 \textit {\_Z} -35\right )}{\sum }\frac {\left (16537500 \textit {\_R1}^{4}+257709375 \textit {\_R1}^{3}-83927835828 \textit {\_R1}^{2}+1270842117635 \textit {\_R1} -4923926348250\right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {expIntegral}_{1}\left (x -\textit {\_R1} \right )}{250 \textit {\_R1}^{4}+3}\right )}{351709077687}\) | \(483\) |
Input:
int((-12500*x^10-750*x^6+20500*x^5+525*x-5600)/(7500*x^10+900*x^6-10500*x^ 5+27*x^2-630*x+3675)/exp(x),x,method=_RETURNVERBOSE)
Output:
(-175/3+250/3*x^5)/(50*x^5+3*x-35)/exp(x)
Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-x} \left (-5600+525 x+20500 x^5-750 x^6-12500 x^{10}\right )}{3675-630 x+27 x^2-10500 x^5+900 x^6+7500 x^{10}} \, dx=\frac {25 \, {\left (10 \, x^{5} - 7\right )} e^{\left (-x\right )}}{3 \, {\left (50 \, x^{5} + 3 \, x - 35\right )}} \] Input:
integrate((-12500*x^10-750*x^6+20500*x^5+525*x-5600)/(7500*x^10+900*x^6-10 500*x^5+27*x^2-630*x+3675)/exp(x),x, algorithm="fricas")
Output:
25/3*(10*x^5 - 7)*e^(-x)/(50*x^5 + 3*x - 35)
Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {e^{-x} \left (-5600+525 x+20500 x^5-750 x^6-12500 x^{10}\right )}{3675-630 x+27 x^2-10500 x^5+900 x^6+7500 x^{10}} \, dx=\frac {\left (250 x^{5} - 175\right ) e^{- x}}{150 x^{5} + 9 x - 105} \] Input:
integrate((-12500*x**10-750*x**6+20500*x**5+525*x-5600)/(7500*x**10+900*x* *6-10500*x**5+27*x**2-630*x+3675)/exp(x),x)
Output:
(250*x**5 - 175)*exp(-x)/(150*x**5 + 9*x - 105)
Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-x} \left (-5600+525 x+20500 x^5-750 x^6-12500 x^{10}\right )}{3675-630 x+27 x^2-10500 x^5+900 x^6+7500 x^{10}} \, dx=\frac {25 \, {\left (10 \, x^{5} - 7\right )} e^{\left (-x\right )}}{3 \, {\left (50 \, x^{5} + 3 \, x - 35\right )}} \] Input:
integrate((-12500*x^10-750*x^6+20500*x^5+525*x-5600)/(7500*x^10+900*x^6-10 500*x^5+27*x^2-630*x+3675)/exp(x),x, algorithm="maxima")
Output:
25/3*(10*x^5 - 7)*e^(-x)/(50*x^5 + 3*x - 35)
Time = 0.12 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {e^{-x} \left (-5600+525 x+20500 x^5-750 x^6-12500 x^{10}\right )}{3675-630 x+27 x^2-10500 x^5+900 x^6+7500 x^{10}} \, dx=\frac {25 \, {\left (10 \, x^{5} e^{\left (-x\right )} - 7 \, e^{\left (-x\right )}\right )}}{3 \, {\left (50 \, x^{5} + 3 \, x - 35\right )}} \] Input:
integrate((-12500*x^10-750*x^6+20500*x^5+525*x-5600)/(7500*x^10+900*x^6-10 500*x^5+27*x^2-630*x+3675)/exp(x),x, algorithm="giac")
Output:
25/3*(10*x^5*e^(-x) - 7*e^(-x))/(50*x^5 + 3*x - 35)
Time = 3.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-x} \left (-5600+525 x+20500 x^5-750 x^6-12500 x^{10}\right )}{3675-630 x+27 x^2-10500 x^5+900 x^6+7500 x^{10}} \, dx=\frac {{\mathrm {e}}^{-x}\,\left (\frac {5\,x^5}{3}-\frac {7}{6}\right )}{x^5+\frac {3\,x}{50}-\frac {7}{10}} \] Input:
int(-(exp(-x)*(750*x^6 - 20500*x^5 - 525*x + 12500*x^10 + 5600))/(27*x^2 - 630*x - 10500*x^5 + 900*x^6 + 7500*x^10 + 3675),x)
Output:
(exp(-x)*((5*x^5)/3 - 7/6))/((3*x)/50 + x^5 - 7/10)
Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-x} \left (-5600+525 x+20500 x^5-750 x^6-12500 x^{10}\right )}{3675-630 x+27 x^2-10500 x^5+900 x^6+7500 x^{10}} \, dx=\frac {\frac {250 x^{5}}{3}-\frac {175}{3}}{e^{x} \left (50 x^{5}+3 x -35\right )} \] Input:
int((-12500*x^10-750*x^6+20500*x^5+525*x-5600)/(7500*x^10+900*x^6-10500*x^ 5+27*x^2-630*x+3675)/exp(x),x)
Output:
(25*(10*x**5 - 7))/(3*e**x*(50*x**5 + 3*x - 35))