\(\int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}} (11520+624 x-96 x^2)}{225 x^{17}-30 x^{18}+x^{19}} \, dx\) [8413]

Optimal result

Integrand size = 51, antiderivative size = 21 \[ \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}} \left (11520+624 x-96 x^2\right )}{225 x^{17}-30 x^{18}+x^{19}} \, dx=e^{\frac {16 e^{-2 x}}{\left (-5+\frac {x}{3}\right ) x^{16}}} \]

[Out]

exp(16/x^16/exp(x)^2/(-5+1/3*x))

Rubi [F]

\[ \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}} \left (11520+624 x-96 x^2\right )}{225 x^{17}-30 x^{18}+x^{19}} \, dx=\int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}} \left (11520+624 x-96 x^2\right )}{225 x^{17}-30 x^{18}+x^{19}} \, dx \]

[In]

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]

[Out]

(-16*Defer[Int][E^(-2*x + 48/(E^(2*x)*(-15*x^16 + x^17)))/(-15 + x)^2, x])/2189469451904296875 - (32*Defer[Int
][E^(-2*x + 48/(E^(2*x)*(-15*x^16 + x^17)))/(-15 + x), x])/2189469451904296875 + (256*Defer[Int][E^(-2*x + 48/
(E^(2*x)*(-15*x^16 + x^17)))/x^17, x])/5 + (48*Defer[Int][E^(-2*x + 48/(E^(2*x)*(-15*x^16 + x^17)))/x^16, x])/
5 + (704*Defer[Int][E^(-2*x + 48/(E^(2*x)*(-15*x^16 + x^17)))/x^15, x])/1125 + (688*Defer[Int][E^(-2*x + 48/(E
^(2*x)*(-15*x^16 + x^17)))/x^14, x])/16875 + (224*Defer[Int][E^(-2*x + 48/(E^(2*x)*(-15*x^16 + x^17)))/x^13, x
])/84375 + (656*Defer[Int][E^(-2*x + 48/(E^(2*x)*(-15*x^16 + x^17)))/x^12, x])/3796875 + (128*Defer[Int][E^(-2
*x + 48/(E^(2*x)*(-15*x^16 + x^17)))/x^11, x])/11390625 + (208*Defer[Int][E^(-2*x + 48/(E^(2*x)*(-15*x^16 + x^
17)))/x^10, x])/284765625 + (608*Defer[Int][E^(-2*x + 48/(E^(2*x)*(-15*x^16 + x^17)))/x^9, x])/12814453125 + (
592*Defer[Int][E^(-2*x + 48/(E^(2*x)*(-15*x^16 + x^17)))/x^8, x])/192216796875 + (64*Defer[Int][E^(-2*x + 48/(
E^(2*x)*(-15*x^16 + x^17)))/x^7, x])/320361328125 + (112*Defer[Int][E^(-2*x + 48/(E^(2*x)*(-15*x^16 + x^17)))/
x^6, x])/8649755859375 + (544*Defer[Int][E^(-2*x + 48/(E^(2*x)*(-15*x^16 + x^17)))/x^5, x])/648731689453125 +
(176*Defer[Int][E^(-2*x + 48/(E^(2*x)*(-15*x^16 + x^17)))/x^4, x])/3243658447265625 + (512*Defer[Int][E^(-2*x
+ 48/(E^(2*x)*(-15*x^16 + x^17)))/x^3, x])/145964630126953125 + (496*Defer[Int][E^(-2*x + 48/(E^(2*x)*(-15*x^1
6 + x^17)))/x^2, x])/2189469451904296875 + (32*Defer[Int][E^(-2*x + 48/(E^(2*x)*(-15*x^16 + x^17)))/x, x])/218
9469451904296875

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}} \left (11520+624 x-96 x^2\right )}{x^{17} \left (225-30 x+x^2\right )} \, dx \\ & = \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}} \left (11520+624 x-96 x^2\right )}{(-15+x)^2 x^{17}} \, dx \\ & = \int \left (-\frac {16 e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{2189469451904296875 (-15+x)^2}-\frac {32 e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{2189469451904296875 (-15+x)}+\frac {256 e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{5 x^{17}}+\frac {48 e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{5 x^{16}}+\frac {704 e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{1125 x^{15}}+\frac {688 e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{16875 x^{14}}+\frac {224 e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{84375 x^{13}}+\frac {656 e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{3796875 x^{12}}+\frac {128 e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{11390625 x^{11}}+\frac {208 e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{284765625 x^{10}}+\frac {608 e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{12814453125 x^9}+\frac {592 e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{192216796875 x^8}+\frac {64 e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{320361328125 x^7}+\frac {112 e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{8649755859375 x^6}+\frac {544 e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{648731689453125 x^5}+\frac {176 e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{3243658447265625 x^4}+\frac {512 e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{145964630126953125 x^3}+\frac {496 e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{2189469451904296875 x^2}+\frac {32 e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{2189469451904296875 x}\right ) \, dx \\ & = -\frac {16 \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{(-15+x)^2} \, dx}{2189469451904296875}-\frac {32 \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{-15+x} \, dx}{2189469451904296875}+\frac {32 \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{x} \, dx}{2189469451904296875}+\frac {496 \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{x^2} \, dx}{2189469451904296875}+\frac {512 \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{x^3} \, dx}{145964630126953125}+\frac {176 \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{x^4} \, dx}{3243658447265625}+\frac {544 \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{x^5} \, dx}{648731689453125}+\frac {112 \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{x^6} \, dx}{8649755859375}+\frac {64 \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{x^7} \, dx}{320361328125}+\frac {592 \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{x^8} \, dx}{192216796875}+\frac {608 \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{x^9} \, dx}{12814453125}+\frac {208 \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{x^{10}} \, dx}{284765625}+\frac {128 \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{x^{11}} \, dx}{11390625}+\frac {656 \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{x^{12}} \, dx}{3796875}+\frac {224 \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{x^{13}} \, dx}{84375}+\frac {688 \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{x^{14}} \, dx}{16875}+\frac {704 \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{x^{15}} \, dx}{1125}+\frac {48}{5} \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{x^{16}} \, dx+\frac {256}{5} \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}}}{x^{17}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}} \left (11520+624 x-96 x^2\right )}{225 x^{17}-30 x^{18}+x^{19}} \, dx=e^{\frac {48 e^{-2 x}}{(-15+x) x^{16}}} \]

[In]

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]

[Out]

E^(48/(E^(2*x)*(-15 + x)*x^16))

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76

\[{\mathrm e}^{\frac {48 \,{\mathrm e}^{-2 x}}{x^{16} \left (x -15\right )}}\]

[In]

int((-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)

[Out]

exp(48*exp(-2*x)/x^16/(x-15))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).

Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57 \[ \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}} \left (11520+624 x-96 x^2\right )}{225 x^{17}-30 x^{18}+x^{19}} \, dx=e^{\left (2 \, x - \frac {2 \, {\left (x^{18} - 15 \, x^{17} - 24 \, e^{\left (-2 \, x\right )}\right )}}{x^{17} - 15 \, x^{16}}\right )} \]

[In]

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")

[Out]

e^(2*x - 2*(x^18 - 15*x^17 - 24*e^(-2*x))/(x^17 - 15*x^16))

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}} \left (11520+624 x-96 x^2\right )}{225 x^{17}-30 x^{18}+x^{19}} \, dx=e^{\frac {48 e^{- 2 x}}{x^{17} - 15 x^{16}}} \]

[In]

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)

[Out]

exp(48*exp(-2*x)/(x**17 - 15*x**16))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (15) = 30\).

Time = 0.50 (sec) , antiderivative size = 157, normalized size of antiderivative = 7.48 \[ \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}} \left (11520+624 x-96 x^2\right )}{225 x^{17}-30 x^{18}+x^{19}} \, dx=e^{\left (\frac {16 \, e^{\left (-2 \, x\right )}}{2189469451904296875 \, {\left (x - 15\right )}} - \frac {16 \, e^{\left (-2 \, x\right )}}{2189469451904296875 \, x} - \frac {16 \, e^{\left (-2 \, x\right )}}{145964630126953125 \, x^{2}} - \frac {16 \, e^{\left (-2 \, x\right )}}{9730975341796875 \, x^{3}} - \frac {16 \, e^{\left (-2 \, x\right )}}{648731689453125 \, x^{4}} - \frac {16 \, e^{\left (-2 \, x\right )}}{43248779296875 \, x^{5}} - \frac {16 \, e^{\left (-2 \, x\right )}}{2883251953125 \, x^{6}} - \frac {16 \, e^{\left (-2 \, x\right )}}{192216796875 \, x^{7}} - \frac {16 \, e^{\left (-2 \, x\right )}}{12814453125 \, x^{8}} - \frac {16 \, e^{\left (-2 \, x\right )}}{854296875 \, x^{9}} - \frac {16 \, e^{\left (-2 \, x\right )}}{56953125 \, x^{10}} - \frac {16 \, e^{\left (-2 \, x\right )}}{3796875 \, x^{11}} - \frac {16 \, e^{\left (-2 \, x\right )}}{253125 \, x^{12}} - \frac {16 \, e^{\left (-2 \, x\right )}}{16875 \, x^{13}} - \frac {16 \, e^{\left (-2 \, x\right )}}{1125 \, x^{14}} - \frac {16 \, e^{\left (-2 \, x\right )}}{75 \, x^{15}} - \frac {16 \, e^{\left (-2 \, x\right )}}{5 \, x^{16}}\right )} \]

[In]

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")

[Out]

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 -
16/2883251953125*e^(-2*x)/x^6 - 16/192216796875*e^(-2*x)/x^7 - 16/12814453125*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^13 - 16/1125*e^(-2*x)/x^14 - 16/75*e^(-2*x)/x^15 - 16/5*e^(-2*x)/x^16)

Giac [F]

\[ \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}} \left (11520+624 x-96 x^2\right )}{225 x^{17}-30 x^{18}+x^{19}} \, dx=\int { -\frac {48 \, {\left (2 \, x^{2} - 13 \, x - 240\right )} e^{\left (-2 \, x + \frac {48 \, e^{\left (-2 \, x\right )}}{x^{17} - 15 \, x^{16}}\right )}}{x^{19} - 30 \, x^{18} + 225 \, x^{17}} \,d x } \]

[In]

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")

[Out]

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)

Mupad [B] (verification not implemented)

Time = 14.42 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {e^{-2 x+\frac {48 e^{-2 x}}{-15 x^{16}+x^{17}}} \left (11520+624 x-96 x^2\right )}{225 x^{17}-30 x^{18}+x^{19}} \, dx={\mathrm {e}}^{-\frac {48\,{\mathrm {e}}^{-2\,x}}{15\,x^{16}-x^{17}}} \]

[In]

int((exp(-(48*exp(-2*x))/(15*x^16 - x^17))*exp(-2*x)*(624*x - 96*x^2 + 11520))/(225*x^17 - 30*x^18 + x^19),x)

[Out]

exp(-(48*exp(-2*x))/(15*x^16 - x^17))