Integrand size = 113, antiderivative size = 23 \[ \int \frac {3750000000 x^{16}+3250000000 x^{17}+1050000000 x^{18}+150000000 x^{19}+8000000 x^{20}+e^x \left (2500000000 x^{15}+1500000000 x^{16}+175000000 x^{17}-55000000 x^{18}-15000000 x^{19}-1000000 x^{20}\right )}{81 e^{5 x}+810 e^{4 x} x+3240 e^{3 x} x^2+6480 e^{2 x} x^3+6480 e^x x^4+2592 x^5} \, dx=\frac {250000 x^{12} (5+x)^4}{81 \left (2+\frac {e^x}{x}\right )^4} \]
Time = 2.66 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {3750000000 x^{16}+3250000000 x^{17}+1050000000 x^{18}+150000000 x^{19}+8000000 x^{20}+e^x \left (2500000000 x^{15}+1500000000 x^{16}+175000000 x^{17}-55000000 x^{18}-15000000 x^{19}-1000000 x^{20}\right )}{81 e^{5 x}+810 e^{4 x} x+3240 e^{3 x} x^2+6480 e^{2 x} x^3+6480 e^x x^4+2592 x^5} \, dx=\frac {250000 x^{16} (5+x)^4}{81 \left (e^x+2 x\right )^4} \]
Integrate[(3750000000*x^16 + 3250000000*x^17 + 1050000000*x^18 + 150000000 *x^19 + 8000000*x^20 + E^x*(2500000000*x^15 + 1500000000*x^16 + 175000000* x^17 - 55000000*x^18 - 15000000*x^19 - 1000000*x^20))/(81*E^(5*x) + 810*E^ (4*x)*x + 3240*E^(3*x)*x^2 + 6480*E^(2*x)*x^3 + 6480*E^x*x^4 + 2592*x^5),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.
\(\displaystyle \int \frac {8000000 x^{20}+150000000 x^{19}+1050000000 x^{18}+3250000000 x^{17}+3750000000 x^{16}+e^x \left (-1000000 x^{20}-15000000 x^{19}-55000000 x^{18}+175000000 x^{17}+1500000000 x^{16}+2500000000 x^{15}\right )}{2592 x^5+6480 e^x x^4+6480 e^{2 x} x^3+3240 e^{3 x} x^2+810 e^{4 x} x+81 e^{5 x}} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {1000000 x^{15} (x+5)^3 \left (2 x (4 x+15)-e^x \left (x^2-20\right )\right )}{81 \left (2 x+e^x\right )^5}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1000000}{81} \int \frac {x^{15} (x+5)^3 \left (2 x (4 x+15)+e^x \left (20-x^2\right )\right )}{\left (2 x+e^x\right )^5}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1000000}{81} \int \left (\frac {2 (x-1) x^{16} (x+5)^4}{\left (2 x+e^x\right )^5}-\frac {x^{15} (x+5)^3 \left (x^2-20\right )}{\left (2 x+e^x\right )^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1000000}{81} \left (2 \int \frac {x^{21}}{\left (2 x+e^x\right )^5}dx+38 \int \frac {x^{20}}{\left (2 x+e^x\right )^5}dx-\int \frac {x^{20}}{\left (2 x+e^x\right )^4}dx+260 \int \frac {x^{19}}{\left (2 x+e^x\right )^5}dx-15 \int \frac {x^{19}}{\left (2 x+e^x\right )^4}dx+700 \int \frac {x^{18}}{\left (2 x+e^x\right )^5}dx-55 \int \frac {x^{18}}{\left (2 x+e^x\right )^4}dx+250 \int \frac {x^{17}}{\left (2 x+e^x\right )^5}dx+175 \int \frac {x^{17}}{\left (2 x+e^x\right )^4}dx-1250 \int \frac {x^{16}}{\left (2 x+e^x\right )^5}dx+1500 \int \frac {x^{16}}{\left (2 x+e^x\right )^4}dx+2500 \int \frac {x^{15}}{\left (2 x+e^x\right )^4}dx\right )\) |
Int[(3750000000*x^16 + 3250000000*x^17 + 1050000000*x^18 + 150000000*x^19 + 8000000*x^20 + E^x*(2500000000*x^15 + 1500000000*x^16 + 175000000*x^17 - 55000000*x^18 - 15000000*x^19 - 1000000*x^20))/(81*E^(5*x) + 810*E^(4*x)* x + 3240*E^(3*x)*x^2 + 6480*E^(2*x)*x^3 + 6480*E^x*x^4 + 2592*x^5),x]
3.27.62.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.36 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39
method | result | size |
risch | \(\frac {250000 \left (x^{4}+20 x^{3}+150 x^{2}+500 x +625\right ) x^{16}}{81 \left ({\mathrm e}^{x}+2 x \right )^{4}}\) | \(32\) |
parallelrisch | \(\frac {6000000 x^{20}+120000000 x^{19}+900000000 x^{18}+3000000000 x^{17}+3750000000 x^{16}}{31104 x^{4}+62208 \,{\mathrm e}^{x} x^{3}+46656 \,{\mathrm e}^{2 x} x^{2}+15552 x \,{\mathrm e}^{3 x}+1944 \,{\mathrm e}^{4 x}}\) | \(64\) |
int(((-1000000*x^20-15000000*x^19-55000000*x^18+175000000*x^17+1500000000* x^16+2500000000*x^15)*exp(x)+8000000*x^20+150000000*x^19+1050000000*x^18+3 250000000*x^17+3750000000*x^16)/(81*exp(x)^5+810*x*exp(x)^4+3240*x^2*exp(x )^3+6480*exp(x)^2*x^3+6480*exp(x)*x^4+2592*x^5),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.65 \[ \int \frac {3750000000 x^{16}+3250000000 x^{17}+1050000000 x^{18}+150000000 x^{19}+8000000 x^{20}+e^x \left (2500000000 x^{15}+1500000000 x^{16}+175000000 x^{17}-55000000 x^{18}-15000000 x^{19}-1000000 x^{20}\right )}{81 e^{5 x}+810 e^{4 x} x+3240 e^{3 x} x^2+6480 e^{2 x} x^3+6480 e^x x^4+2592 x^5} \, dx=\frac {250000 \, {\left (x^{20} + 20 \, x^{19} + 150 \, x^{18} + 500 \, x^{17} + 625 \, x^{16}\right )}}{81 \, {\left (16 \, x^{4} + 32 \, x^{3} e^{x} + 24 \, x^{2} e^{\left (2 \, x\right )} + 8 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}\right )}} \]
integrate(((-1000000*x^20-15000000*x^19-55000000*x^18+175000000*x^17+15000 00000*x^16+2500000000*x^15)*exp(x)+8000000*x^20+150000000*x^19+1050000000* x^18+3250000000*x^17+3750000000*x^16)/(81*exp(x)^5+810*x*exp(x)^4+3240*x^2 *exp(x)^3+6480*exp(x)^2*x^3+6480*exp(x)*x^4+2592*x^5),x, algorithm=\
250000/81*(x^20 + 20*x^19 + 150*x^18 + 500*x^17 + 625*x^16)/(16*x^4 + 32*x ^3*e^x + 24*x^2*e^(2*x) + 8*x*e^(3*x) + e^(4*x))
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (19) = 38\).
Time = 0.14 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.74 \[ \int \frac {3750000000 x^{16}+3250000000 x^{17}+1050000000 x^{18}+150000000 x^{19}+8000000 x^{20}+e^x \left (2500000000 x^{15}+1500000000 x^{16}+175000000 x^{17}-55000000 x^{18}-15000000 x^{19}-1000000 x^{20}\right )}{81 e^{5 x}+810 e^{4 x} x+3240 e^{3 x} x^2+6480 e^{2 x} x^3+6480 e^x x^4+2592 x^5} \, dx=\frac {250000 x^{20} + 5000000 x^{19} + 37500000 x^{18} + 125000000 x^{17} + 156250000 x^{16}}{1296 x^{4} + 2592 x^{3} e^{x} + 1944 x^{2} e^{2 x} + 648 x e^{3 x} + 81 e^{4 x}} \]
integrate(((-1000000*x**20-15000000*x**19-55000000*x**18+175000000*x**17+1 500000000*x**16+2500000000*x**15)*exp(x)+8000000*x**20+150000000*x**19+105 0000000*x**18+3250000000*x**17+3750000000*x**16)/(81*exp(x)**5+810*x*exp(x )**4+3240*x**2*exp(x)**3+6480*exp(x)**2*x**3+6480*exp(x)*x**4+2592*x**5),x )
(250000*x**20 + 5000000*x**19 + 37500000*x**18 + 125000000*x**17 + 1562500 00*x**16)/(1296*x**4 + 2592*x**3*exp(x) + 1944*x**2*exp(2*x) + 648*x*exp(3 *x) + 81*exp(4*x))
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (20) = 40\).
Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.65 \[ \int \frac {3750000000 x^{16}+3250000000 x^{17}+1050000000 x^{18}+150000000 x^{19}+8000000 x^{20}+e^x \left (2500000000 x^{15}+1500000000 x^{16}+175000000 x^{17}-55000000 x^{18}-15000000 x^{19}-1000000 x^{20}\right )}{81 e^{5 x}+810 e^{4 x} x+3240 e^{3 x} x^2+6480 e^{2 x} x^3+6480 e^x x^4+2592 x^5} \, dx=\frac {250000 \, {\left (x^{20} + 20 \, x^{19} + 150 \, x^{18} + 500 \, x^{17} + 625 \, x^{16}\right )}}{81 \, {\left (16 \, x^{4} + 32 \, x^{3} e^{x} + 24 \, x^{2} e^{\left (2 \, x\right )} + 8 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}\right )}} \]
integrate(((-1000000*x^20-15000000*x^19-55000000*x^18+175000000*x^17+15000 00000*x^16+2500000000*x^15)*exp(x)+8000000*x^20+150000000*x^19+1050000000* x^18+3250000000*x^17+3750000000*x^16)/(81*exp(x)^5+810*x*exp(x)^4+3240*x^2 *exp(x)^3+6480*exp(x)^2*x^3+6480*exp(x)*x^4+2592*x^5),x, algorithm=\
250000/81*(x^20 + 20*x^19 + 150*x^18 + 500*x^17 + 625*x^16)/(16*x^4 + 32*x ^3*e^x + 24*x^2*e^(2*x) + 8*x*e^(3*x) + e^(4*x))
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.65 \[ \int \frac {3750000000 x^{16}+3250000000 x^{17}+1050000000 x^{18}+150000000 x^{19}+8000000 x^{20}+e^x \left (2500000000 x^{15}+1500000000 x^{16}+175000000 x^{17}-55000000 x^{18}-15000000 x^{19}-1000000 x^{20}\right )}{81 e^{5 x}+810 e^{4 x} x+3240 e^{3 x} x^2+6480 e^{2 x} x^3+6480 e^x x^4+2592 x^5} \, dx=\frac {250000 \, {\left (x^{20} + 20 \, x^{19} + 150 \, x^{18} + 500 \, x^{17} + 625 \, x^{16}\right )}}{81 \, {\left (16 \, x^{4} + 32 \, x^{3} e^{x} + 24 \, x^{2} e^{\left (2 \, x\right )} + 8 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}\right )}} \]
integrate(((-1000000*x^20-15000000*x^19-55000000*x^18+175000000*x^17+15000 00000*x^16+2500000000*x^15)*exp(x)+8000000*x^20+150000000*x^19+1050000000* x^18+3250000000*x^17+3750000000*x^16)/(81*exp(x)^5+810*x*exp(x)^4+3240*x^2 *exp(x)^3+6480*exp(x)^2*x^3+6480*exp(x)*x^4+2592*x^5),x, algorithm=\
250000/81*(x^20 + 20*x^19 + 150*x^18 + 500*x^17 + 625*x^16)/(16*x^4 + 32*x ^3*e^x + 24*x^2*e^(2*x) + 8*x*e^(3*x) + e^(4*x))
Time = 14.33 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.09 \[ \int \frac {3750000000 x^{16}+3250000000 x^{17}+1050000000 x^{18}+150000000 x^{19}+8000000 x^{20}+e^x \left (2500000000 x^{15}+1500000000 x^{16}+175000000 x^{17}-55000000 x^{18}-15000000 x^{19}-1000000 x^{20}\right )}{81 e^{5 x}+810 e^{4 x} x+3240 e^{3 x} x^2+6480 e^{2 x} x^3+6480 e^x x^4+2592 x^5} \, dx=\frac {250000\,\left (x^{21}+19\,x^{20}+130\,x^{19}+350\,x^{18}+125\,x^{17}-625\,x^{16}\right )}{81\,\left (x-1\right )\,\left ({\mathrm {e}}^{4\,x}+8\,x\,{\mathrm {e}}^{3\,x}+32\,x^3\,{\mathrm {e}}^x+24\,x^2\,{\mathrm {e}}^{2\,x}+16\,x^4\right )} \]
int((3750000000*x^16 + 3250000000*x^17 + 1050000000*x^18 + 150000000*x^19 + 8000000*x^20 + exp(x)*(2500000000*x^15 + 1500000000*x^16 + 175000000*x^1 7 - 55000000*x^18 - 15000000*x^19 - 1000000*x^20))/(81*exp(5*x) + 810*x*ex p(4*x) + 6480*x^4*exp(x) + 3240*x^2*exp(3*x) + 6480*x^3*exp(2*x) + 2592*x^ 5),x)