Integrand size = 51, antiderivative size = 21 \[ \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}} \left (11520+624 x-96 x^2\right )}{225 x^{17}-30 x^{18}+x^{19}} \, dx=e^{\frac {16 e^{-2 x}}{\left (-5+\frac {x}{3}\right ) x^{16}}} \] Output:
exp(16/x^16/exp(x)^2/(-5+1/3*x))
Time = 0.47 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}} \left (11520+624 x-96 x^2\right )}{225 x^{17}-30 x^{18}+x^{19}} \, dx=e^{\frac {48 e^{-2 x}}{(-15+x) x^{16}}} \] Input:
Integrate[(E^(-2*x + 48/(E^(2*x)*(-15*x^16 + x^17)))*(11520 + 624*x - 96*x ^2))/(225*x^17 - 30*x^18 + x^19),x]
Output:
E^(48/(E^(2*x)*(-15 + x)*x^16))
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.
| \(\displaystyle \int \frac {e^{\frac {48 e^{-2 x}}{x^{17}-15 x^{16}}-2 x} \left (-96 x^2+624 x+11520\right )}{x^{19}-30 x^{18}+225 x^{17}} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {e^{\frac {48 e^{-2 x}}{x^{17}-15 x^{16}}-2 x} \left (-96 x^2+624 x+11520\right )}{x^{17} \left (x^2-30 x+225\right )}dx\) |
| \(\Big \downarrow \) 7277 |
| \(\displaystyle 4 \int \frac {12 e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}} \left (-2 x^2+13 x+240\right )}{(15-x)^2 x^{17}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 48 \int \frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}} \left (-2 x^2+13 x+240\right )}{(15-x)^2 x^{17}}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 48 \int \left (\frac {2 e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{6568408355712890625 x}+\frac {31 e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{6568408355712890625 x^2}+\frac {32 e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{437893890380859375 x^3}+\frac {11 e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{9730975341796875 x^4}+\frac {34 e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{1946195068359375 x^5}+\frac {7 e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{25949267578125 x^6}+\frac {4 e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{961083984375 x^7}+\frac {37 e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{576650390625 x^8}+\frac {38 e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{38443359375 x^9}+\frac {13 e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{854296875 x^{10}}+\frac {8 e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{34171875 x^{11}}+\frac {41 e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{11390625 x^{12}}+\frac {14 e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{253125 x^{13}}+\frac {43 e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{50625 x^{14}}+\frac {44 e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{3375 x^{15}}+\frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{5 x^{16}}+\frac {16 e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{15 x^{17}}-\frac {2 e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{6568408355712890625 (x-15)}-\frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{6568408355712890625 (x-15)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 48 \left (-\frac {\int \frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{(x-15)^2}dx}{6568408355712890625}-\frac {2 \int \frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{x-15}dx}{6568408355712890625}+\frac {16}{15} \int \frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{x^{17}}dx+\frac {1}{5} \int \frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{x^{16}}dx+\frac {2 \int \frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{x}dx}{6568408355712890625}+\frac {44 \int \frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{x^{15}}dx}{3375}+\frac {43 \int \frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{x^{14}}dx}{50625}+\frac {14 \int \frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{x^{13}}dx}{253125}+\frac {41 \int \frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{x^{12}}dx}{11390625}+\frac {8 \int \frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{x^{11}}dx}{34171875}+\frac {13 \int \frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{x^{10}}dx}{854296875}+\frac {38 \int \frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{x^9}dx}{38443359375}+\frac {37 \int \frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{x^8}dx}{576650390625}+\frac {4 \int \frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{x^7}dx}{961083984375}+\frac {7 \int \frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{x^6}dx}{25949267578125}+\frac {34 \int \frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{x^5}dx}{1946195068359375}+\frac {11 \int \frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{x^4}dx}{9730975341796875}+\frac {32 \int \frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{x^3}dx}{437893890380859375}+\frac {31 \int \frac {e^{-2 x-\frac {48 e^{-2 x}}{15 x^{16}-x^{17}}}}{x^2}dx}{6568408355712890625}\right )\) |
Input:
Int[(E^(-2*x + 48/(E^(2*x)*(-15*x^16 + x^17)))*(11520 + 624*x - 96*x^2))/( 225*x^17 - 30*x^18 + x^19),x]
Output:
$Aborted
Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76
\[{\mathrm e}^{\frac {48 \,{\mathrm e}^{-2 x}}{x^{16} \left (x -15\right )}}\]
Input:
int((-96*x^2+624*x+11520)*exp(48/(x^17-15*x^16)/exp(x)^2)/(x^19-30*x^18+22 5*x^17)/exp(x)^2,x)
Output:
exp(48*exp(-2*x)/x^16/(x-15))
Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).
Time = 0.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57 \[ \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}} \left (11520+624 x-96 x^2\right )}{225 x^{17}-30 x^{18}+x^{19}} \, dx=e^{\left (2 \, x - \frac {2 \, {\left (x^{18} - 15 \, x^{17} - 24 \, e^{\left (-2 \, x\right )}\right )}}{x^{17} - 15 \, x^{16}}\right )} \] Input:
integrate((-96*x^2+624*x+11520)*exp(48/(x^17-15*x^16)/exp(x)^2)/(x^19-30*x ^18+225*x^17)/exp(x)^2,x, algorithm="fricas")
Output:
e^(2*x - 2*(x^18 - 15*x^17 - 24*e^(-2*x))/(x^17 - 15*x^16))
Time = 0.16 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}} \left (11520+624 x-96 x^2\right )}{225 x^{17}-30 x^{18}+x^{19}} \, dx=e^{\frac {48 e^{- 2 x}}{x^{17} - 15 x^{16}}} \] Input:
integrate((-96*x**2+624*x+11520)*exp(48/(x**17-15*x**16)/exp(x)**2)/(x**19 -30*x**18+225*x**17)/exp(x)**2,x)
Output:
exp(48*exp(-2*x)/(x**17 - 15*x**16))
Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (15) = 30\).
Time = 0.34 (sec) , antiderivative size = 157, normalized size of antiderivative = 7.48 \[ \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}} \left (11520+624 x-96 x^2\right )}{225 x^{17}-30 x^{18}+x^{19}} \, dx=e^{\left (\frac {16 \, e^{\left (-2 \, x\right )}}{2189469451904296875 \, {\left (x - 15\right )}} - \frac {16 \, e^{\left (-2 \, x\right )}}{2189469451904296875 \, x} - \frac {16 \, e^{\left (-2 \, x\right )}}{145964630126953125 \, x^{2}} - \frac {16 \, e^{\left (-2 \, x\right )}}{9730975341796875 \, x^{3}} - \frac {16 \, e^{\left (-2 \, x\right )}}{648731689453125 \, x^{4}} - \frac {16 \, e^{\left (-2 \, x\right )}}{43248779296875 \, x^{5}} - \frac {16 \, e^{\left (-2 \, x\right )}}{2883251953125 \, x^{6}} - \frac {16 \, e^{\left (-2 \, x\right )}}{192216796875 \, x^{7}} - \frac {16 \, e^{\left (-2 \, x\right )}}{12814453125 \, x^{8}} - \frac {16 \, e^{\left (-2 \, x\right )}}{854296875 \, x^{9}} - \frac {16 \, e^{\left (-2 \, x\right )}}{56953125 \, x^{10}} - \frac {16 \, e^{\left (-2 \, x\right )}}{3796875 \, x^{11}} - \frac {16 \, e^{\left (-2 \, x\right )}}{253125 \, x^{12}} - \frac {16 \, e^{\left (-2 \, x\right )}}{16875 \, x^{13}} - \frac {16 \, e^{\left (-2 \, x\right )}}{1125 \, x^{14}} - \frac {16 \, e^{\left (-2 \, x\right )}}{75 \, x^{15}} - \frac {16 \, e^{\left (-2 \, x\right )}}{5 \, x^{16}}\right )} \] Input:
integrate((-96*x^2+624*x+11520)*exp(48/(x^17-15*x^16)/exp(x)^2)/(x^19-30*x ^18+225*x^17)/exp(x)^2,x, algorithm="maxima")
Output:
e^(16/2189469451904296875*e^(-2*x)/(x - 15) - 16/2189469451904296875*e^(-2 *x)/x - 16/145964630126953125*e^(-2*x)/x^2 - 16/9730975341796875*e^(-2*x)/ x^3 - 16/648731689453125*e^(-2*x)/x^4 - 16/43248779296875*e^(-2*x)/x^5 - 1 6/2883251953125*e^(-2*x)/x^6 - 16/192216796875*e^(-2*x)/x^7 - 16/128144531 25*e^(-2*x)/x^8 - 16/854296875*e^(-2*x)/x^9 - 16/56953125*e^(-2*x)/x^10 - 16/3796875*e^(-2*x)/x^11 - 16/253125*e^(-2*x)/x^12 - 16/16875*e^(-2*x)/x^1 3 - 16/1125*e^(-2*x)/x^14 - 16/75*e^(-2*x)/x^15 - 16/5*e^(-2*x)/x^16)
\[ \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}} \left (11520+624 x-96 x^2\right )}{225 x^{17}-30 x^{18}+x^{19}} \, dx=\int { -\frac {48 \, {\left (2 \, x^{2} - 13 \, x - 240\right )} e^{\left (-2 \, x + \frac {48 \, e^{\left (-2 \, x\right )}}{x^{17} - 15 \, x^{16}}\right )}}{x^{19} - 30 \, x^{18} + 225 \, x^{17}} \,d x } \] Input:
integrate((-96*x^2+624*x+11520)*exp(48/(x^17-15*x^16)/exp(x)^2)/(x^19-30*x ^18+225*x^17)/exp(x)^2,x, algorithm="giac")
Output:
integrate(-48*(2*x^2 - 13*x - 240)*e^(-2*x + 48*e^(-2*x)/(x^17 - 15*x^16)) /(x^19 - 30*x^18 + 225*x^17), x)
Time = 4.42 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}} \left (11520+624 x-96 x^2\right )}{225 x^{17}-30 x^{18}+x^{19}} \, dx={\mathrm {e}}^{-\frac {48\,{\mathrm {e}}^{-2\,x}}{15\,x^{16}-x^{17}}} \] Input:
int((exp(-(48*exp(-2*x))/(15*x^16 - x^17))*exp(-2*x)*(624*x - 96*x^2 + 115 20))/(225*x^17 - 30*x^18 + x^19),x)
Output:
exp(-(48*exp(-2*x))/(15*x^16 - x^17))
\[ \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}} \left (11520+624 x-96 x^2\right )}{225 x^{17}-30 x^{18}+x^{19}} \, dx=11520 \left (\int \frac {e^{\frac {48}{e^{2 x} x^{17}-15 e^{2 x} x^{16}}}}{e^{2 x} x^{19}-30 e^{2 x} x^{18}+225 e^{2 x} x^{17}}d x \right )+624 \left (\int \frac {e^{\frac {48}{e^{2 x} x^{17}-15 e^{2 x} x^{16}}}}{e^{2 x} x^{18}-30 e^{2 x} x^{17}+225 e^{2 x} x^{16}}d x \right )-96 \left (\int \frac {e^{\frac {48}{e^{2 x} x^{17}-15 e^{2 x} x^{16}}}}{e^{2 x} x^{17}-30 e^{2 x} x^{16}+225 e^{2 x} x^{15}}d x \right ) \] Input:
int((-96*x^2+624*x+11520)*exp(48/(x^17-15*x^16)/exp(x)^2)/(x^19-30*x^18+22 5*x^17)/exp(x)^2,x)
Output:
48*(240*int(e**(48/(e**(2*x)*x**17 - 15*e**(2*x)*x**16))/(e**(2*x)*x**19 - 30*e**(2*x)*x**18 + 225*e**(2*x)*x**17),x) + 13*int(e**(48/(e**(2*x)*x**1 7 - 15*e**(2*x)*x**16))/(e**(2*x)*x**18 - 30*e**(2*x)*x**17 + 225*e**(2*x) *x**16),x) - 2*int(e**(48/(e**(2*x)*x**17 - 15*e**(2*x)*x**16))/(e**(2*x)* x**17 - 30*e**(2*x)*x**16 + 225*e**(2*x)*x**15),x))