Integrand size = 76, antiderivative size = 22 \[ \int \frac {-2500-3000 x+18800 x^2+11840 x^3+47360 x^5-300800 x^6-192000 x^7+640000 x^8}{81 x^5+432 e^5 x^5+864 e^{10} x^5+768 e^{15} x^5+256 e^{20} x^5} \, dx=\frac {\left (2+\frac {5}{x}-20 x\right )^4}{\left (3+4 e^5\right )^4} \] Output:
(5/x+2-20*x)^4/(4*exp(5)+3)^4
Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(22)=44\).
Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.36 \[ \int \frac {-2500-3000 x+18800 x^2+11840 x^3+47360 x^5-300800 x^6-192000 x^7+640000 x^8}{81 x^5+432 e^5 x^5+864 e^{10} x^5+768 e^{15} x^5+256 e^{20} x^5} \, dx=\frac {20 \left (\frac {125}{4 x^4}+\frac {50}{x^3}-\frac {470}{x^2}-\frac {592}{x}+2368 x-7520 x^2-3200 x^3+8000 x^4\right )}{\left (3+4 e^5\right )^4} \] Input:
Integrate[(-2500 - 3000*x + 18800*x^2 + 11840*x^3 + 47360*x^5 - 300800*x^6 - 192000*x^7 + 640000*x^8)/(81*x^5 + 432*E^5*x^5 + 864*E^10*x^5 + 768*E^1 5*x^5 + 256*E^20*x^5),x]
Output:
(20*(125/(4*x^4) + 50/x^3 - 470/x^2 - 592/x + 2368*x - 7520*x^2 - 3200*x^3 + 8000*x^4))/(3 + 4*E^5)^4
Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(22)=44\).
Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.36, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6, 6, 6, 6, 27, 27, 2010, 2009}
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.
| \(\displaystyle \int \frac {640000 x^8-192000 x^7-300800 x^6+47360 x^5+11840 x^3+18800 x^2-3000 x-2500}{256 e^{20} x^5+768 e^{15} x^5+864 e^{10} x^5+432 e^5 x^5+81 x^5} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {640000 x^8-192000 x^7-300800 x^6+47360 x^5+11840 x^3+18800 x^2-3000 x-2500}{\left (81+432 e^5\right ) x^5+256 e^{20} x^5+768 e^{15} x^5+864 e^{10} x^5}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {640000 x^8-192000 x^7-300800 x^6+47360 x^5+11840 x^3+18800 x^2-3000 x-2500}{\left (864 e^{10}+768 e^{15}\right ) x^5+\left (81+432 e^5\right ) x^5+256 e^{20} x^5}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {640000 x^8-192000 x^7-300800 x^6+47360 x^5+11840 x^3+18800 x^2-3000 x-2500}{\left (81+432 e^5+256 e^{20}\right ) x^5+\left (864 e^{10}+768 e^{15}\right ) x^5}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {640000 x^8-192000 x^7-300800 x^6+47360 x^5+11840 x^3+18800 x^2-3000 x-2500}{\left (81+432 e^5+864 e^{10}+768 e^{15}+256 e^{20}\right ) x^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {20 \left (-32000 x^8+9600 x^7+15040 x^6-2368 x^5-592 x^3-940 x^2+150 x+125\right )}{x^5}dx}{\left (3+4 e^5\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {20 \int \frac {-32000 x^8+9600 x^7+15040 x^6-2368 x^5-592 x^3-940 x^2+150 x+125}{x^5}dx}{\left (3+4 e^5\right )^4}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle -\frac {20 \int \left (-32000 x^3+9600 x^2+15040 x-2368-\frac {592}{x^2}-\frac {940}{x^3}+\frac {150}{x^4}+\frac {125}{x^5}\right )dx}{\left (3+4 e^5\right )^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {20 \left (-8000 x^4-\frac {125}{4 x^4}+3200 x^3-\frac {50}{x^3}+7520 x^2+\frac {470}{x^2}-2368 x+\frac {592}{x}\right )}{\left (3+4 e^5\right )^4}\) |
Input:
Int[(-2500 - 3000*x + 18800*x^2 + 11840*x^3 + 47360*x^5 - 300800*x^6 - 192 000*x^7 + 640000*x^8)/(81*x^5 + 432*E^5*x^5 + 864*E^10*x^5 + 768*E^15*x^5 + 256*E^20*x^5),x]
Output:
(-20*(-125/(4*x^4) - 50/x^3 + 470/x^2 + 592/x - 2368*x + 7520*x^2 + 3200*x ^3 - 8000*x^4))/(3 + 4*E^5)^4
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(21)=42\).
Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.00
| method | result | size |
| parallelrisch | \(\frac {160000 x^{8}-64000 x^{7}-150400 x^{6}+47360 x^{5}-11840 x^{3}-9400 x^{2}+1000 x +625}{\left (256 \,{\mathrm e}^{20}+768 \,{\mathrm e}^{15}+864 \,{\mathrm e}^{10}+432 \,{\mathrm e}^{5}+81\right ) x^{4}}\) | \(66\) |
| gosper | \(\frac {160000 x^{8}-64000 x^{7}-150400 x^{6}+47360 x^{5}-11840 x^{3}-9400 x^{2}+1000 x +625}{\left (256 \,{\mathrm e}^{20}+768 \,{\mathrm e}^{15}+864 \,{\mathrm e}^{10}+432 \,{\mathrm e}^{5}+81\right ) x^{4}}\) | \(67\) |
| default | \(\frac {160000 x^{4}-64000 x^{3}-150400 x^{2}+47360 x +\frac {625}{x^{4}}-\frac {9400}{x^{2}}+\frac {1000}{x^{3}}-\frac {11840}{x}}{256 \,{\mathrm e}^{20}+768 \,{\mathrm e}^{15}+864 \,{\mathrm e}^{10}+432 \,{\mathrm e}^{5}+81}\) | \(68\) |
| norman | \(\frac {\frac {625}{4 \,{\mathrm e}^{5}+3}+\frac {1000 x}{4 \,{\mathrm e}^{5}+3}-\frac {9400 x^{2}}{4 \,{\mathrm e}^{5}+3}-\frac {11840 x^{3}}{4 \,{\mathrm e}^{5}+3}+\frac {47360 x^{5}}{4 \,{\mathrm e}^{5}+3}-\frac {150400 x^{6}}{4 \,{\mathrm e}^{5}+3}-\frac {64000 x^{7}}{4 \,{\mathrm e}^{5}+3}+\frac {160000 x^{8}}{4 \,{\mathrm e}^{5}+3}}{x^{4} \left (4 \,{\mathrm e}^{5}+3\right )^{3}}\) | \(113\) |
| orering | \(\frac {\left (32000 x^{8}-12800 x^{7}-30080 x^{6}+9472 x^{5}-2368 x^{3}-1880 x^{2}+200 x +125\right ) x \left (640000 x^{8}-192000 x^{7}-300800 x^{6}+47360 x^{5}+11840 x^{3}+18800 x^{2}-3000 x -2500\right )}{4 \left (4 x^{2}+1\right ) \left (20 x^{2}-2 x -5\right )^{3} \left (256 x^{5} {\mathrm e}^{20}+768 x^{5} {\mathrm e}^{15}+864 x^{5} {\mathrm e}^{10}+432 x^{5} {\mathrm e}^{5}+81 x^{5}\right )}\) | \(137\) |
| risch | \(\frac {160000 x^{4}}{256 \,{\mathrm e}^{20}+768 \,{\mathrm e}^{15}+864 \,{\mathrm e}^{10}+432 \,{\mathrm e}^{5}+81}-\frac {64000 x^{3}}{256 \,{\mathrm e}^{20}+768 \,{\mathrm e}^{15}+864 \,{\mathrm e}^{10}+432 \,{\mathrm e}^{5}+81}-\frac {150400 x^{2}}{256 \,{\mathrm e}^{20}+768 \,{\mathrm e}^{15}+864 \,{\mathrm e}^{10}+432 \,{\mathrm e}^{5}+81}+\frac {47360 x}{256 \,{\mathrm e}^{20}+768 \,{\mathrm e}^{15}+864 \,{\mathrm e}^{10}+432 \,{\mathrm e}^{5}+81}+\frac {256 \left (-11840 \,{\mathrm e}^{20}-35520 \,{\mathrm e}^{15}-39960 \,{\mathrm e}^{10}-19980 \,{\mathrm e}^{5}-\frac {14985}{4}\right ) x^{3}+256 \left (-9400 \,{\mathrm e}^{20}-28200 \,{\mathrm e}^{15}-31725 \,{\mathrm e}^{10}-\frac {31725 \,{\mathrm e}^{5}}{2}-\frac {95175}{32}\right ) x^{2}+256 \left (1000 \,{\mathrm e}^{20}+3000 \,{\mathrm e}^{15}+3375 \,{\mathrm e}^{10}+\frac {3375 \,{\mathrm e}^{5}}{2}+\frac {10125}{32}\right ) x +160000 \,{\mathrm e}^{20}+480000 \,{\mathrm e}^{15}+540000 \,{\mathrm e}^{10}+270000 \,{\mathrm e}^{5}+50625}{\left (256 \,{\mathrm e}^{20}+768 \,{\mathrm e}^{15}+864 \,{\mathrm e}^{10}+432 \,{\mathrm e}^{5}+81\right )^{2} x^{4}}\) | \(207\) |
Input:
int((640000*x^8-192000*x^7-300800*x^6+47360*x^5+11840*x^3+18800*x^2-3000*x -2500)/(256*x^5*exp(5)^4+768*x^5*exp(5)^3+864*x^5*exp(5)^2+432*x^5*exp(5)+ 81*x^5),x,method=_RETURNVERBOSE)
Output:
(160000*x^8-64000*x^7-150400*x^6+47360*x^5-11840*x^3-9400*x^2+1000*x+625)/ (256*exp(5)^4+768*exp(5)^3+864*exp(5)^2+432*exp(5)+81)/x^4
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (21) = 42\).
Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.32 \[ \int \frac {-2500-3000 x+18800 x^2+11840 x^3+47360 x^5-300800 x^6-192000 x^7+640000 x^8}{81 x^5+432 e^5 x^5+864 e^{10} x^5+768 e^{15} x^5+256 e^{20} x^5} \, dx=\frac {5 \, {\left (32000 \, x^{8} - 12800 \, x^{7} - 30080 \, x^{6} + 9472 \, x^{5} - 2368 \, x^{3} - 1880 \, x^{2} + 200 \, x + 125\right )}}{256 \, x^{4} e^{20} + 768 \, x^{4} e^{15} + 864 \, x^{4} e^{10} + 432 \, x^{4} e^{5} + 81 \, x^{4}} \] Input:
integrate((640000*x^8-192000*x^7-300800*x^6+47360*x^5+11840*x^3+18800*x^2- 3000*x-2500)/(256*x^5*exp(5)^4+768*x^5*exp(5)^3+864*x^5*exp(5)^2+432*x^5*e xp(5)+81*x^5),x, algorithm="fricas")
Output:
5*(32000*x^8 - 12800*x^7 - 30080*x^6 + 9472*x^5 - 2368*x^3 - 1880*x^2 + 20 0*x + 125)/(256*x^4*e^20 + 768*x^4*e^15 + 864*x^4*e^10 + 432*x^4*e^5 + 81* x^4)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (17) = 34\).
Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.64 \[ \int \frac {-2500-3000 x+18800 x^2+11840 x^3+47360 x^5-300800 x^6-192000 x^7+640000 x^8}{81 x^5+432 e^5 x^5+864 e^{10} x^5+768 e^{15} x^5+256 e^{20} x^5} \, dx=\frac {160000 x^{4} - 64000 x^{3} - 150400 x^{2} + 47360 x + \frac {- 11840 x^{3} - 9400 x^{2} + 1000 x + 625}{x^{4}}}{81 + 432 e^{5} + 864 e^{10} + 768 e^{15} + 256 e^{20}} \] Input:
integrate((640000*x**8-192000*x**7-300800*x**6+47360*x**5+11840*x**3+18800 *x**2-3000*x-2500)/(256*x**5*exp(5)**4+768*x**5*exp(5)**3+864*x**5*exp(5)* *2+432*x**5*exp(5)+81*x**5),x)
Output:
(160000*x**4 - 64000*x**3 - 150400*x**2 + 47360*x + (-11840*x**3 - 9400*x* *2 + 1000*x + 625)/x**4)/(81 + 432*exp(5) + 864*exp(10) + 768*exp(15) + 25 6*exp(20))
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (21) = 42\).
Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.73 \[ \int \frac {-2500-3000 x+18800 x^2+11840 x^3+47360 x^5-300800 x^6-192000 x^7+640000 x^8}{81 x^5+432 e^5 x^5+864 e^{10} x^5+768 e^{15} x^5+256 e^{20} x^5} \, dx=\frac {640 \, {\left (250 \, x^{4} - 100 \, x^{3} - 235 \, x^{2} + 74 \, x\right )}}{256 \, e^{20} + 768 \, e^{15} + 864 \, e^{10} + 432 \, e^{5} + 81} - \frac {5 \, {\left (2368 \, x^{3} + 1880 \, x^{2} - 200 \, x - 125\right )}}{x^{4} {\left (256 \, e^{20} + 768 \, e^{15} + 864 \, e^{10} + 432 \, e^{5} + 81\right )}} \] Input:
integrate((640000*x^8-192000*x^7-300800*x^6+47360*x^5+11840*x^3+18800*x^2- 3000*x-2500)/(256*x^5*exp(5)^4+768*x^5*exp(5)^3+864*x^5*exp(5)^2+432*x^5*e xp(5)+81*x^5),x, algorithm="maxima")
Output:
640*(250*x^4 - 100*x^3 - 235*x^2 + 74*x)/(256*e^20 + 768*e^15 + 864*e^10 + 432*e^5 + 81) - 5*(2368*x^3 + 1880*x^2 - 200*x - 125)/(x^4*(256*e^20 + 76 8*e^15 + 864*e^10 + 432*e^5 + 81))
Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (21) = 42\).
Time = 0.11 (sec) , antiderivative size = 442, normalized size of antiderivative = 20.09 \[ \int \frac {-2500-3000 x+18800 x^2+11840 x^3+47360 x^5-300800 x^6-192000 x^7+640000 x^8}{81 x^5+432 e^5 x^5+864 e^{10} x^5+768 e^{15} x^5+256 e^{20} x^5} \, dx =\text {Too large to display} \] Input:
integrate((640000*x^8-192000*x^7-300800*x^6+47360*x^5+11840*x^3+18800*x^2- 3000*x-2500)/(256*x^5*exp(5)^4+768*x^5*exp(5)^3+864*x^5*exp(5)^2+432*x^5*e xp(5)+81*x^5),x, algorithm="giac")
Output:
640*(4194304000*x^4*e^60 + 37748736000*x^4*e^55 + 155713536000*x^4*e^50 + 389283840000*x^4*e^45 + 656916480000*x^4*e^40 + 788299776000*x^4*e^35 + 68 9762304000*x^4*e^30 + 443418624000*x^4*e^25 + 207852480000*x^4*e^20 + 6928 4160000*x^4*e^15 + 15588936000*x^4*e^10 + 2125764000*x^4*e^5 + 132860250*x ^4 - 1677721600*x^3*e^60 - 15099494400*x^3*e^55 - 62285414400*x^3*e^50 - 1 55713536000*x^3*e^45 - 262766592000*x^3*e^40 - 315319910400*x^3*e^35 - 275 904921600*x^3*e^30 - 177367449600*x^3*e^25 - 83140992000*x^3*e^20 - 277136 64000*x^3*e^15 - 6235574400*x^3*e^10 - 850305600*x^3*e^5 - 53144100*x^3 - 3942645760*x^2*e^60 - 35483811840*x^2*e^55 - 146370723840*x^2*e^50 - 36592 6809600*x^2*e^45 - 617501491200*x^2*e^40 - 741001789440*x^2*e^35 - 6483765 65760*x^2*e^30 - 416813506560*x^2*e^25 - 195381331200*x^2*e^20 - 651271104 00*x^2*e^15 - 14653599840*x^2*e^10 - 1998218160*x^2*e^5 - 124888635*x^2 + 1241513984*x*e^60 + 11173625856*x*e^55 + 46091206656*x*e^50 + 115228016640 *x*e^45 + 194447278080*x*e^40 + 233336733696*x*e^35 + 204169641984*x*e^30 + 131251912704*x*e^25 + 61524334080*x*e^20 + 20508111360*x*e^15 + 46143250 56*x*e^10 + 629226144*x*e^5 + 39326634*x)/(4294967296*e^80 + 51539607552*e ^75 + 289910292480*e^70 + 1014686023680*e^65 + 2473297182720*e^60 + 445193 4928896*e^55 + 6121410527232*e^50 + 6558654136320*e^45 + 5533864427520*e^4 0 + 3689242951680*e^35 + 1936852549632*e^30 + 792348770304*e^25 + 24760899 0720*e^20 + 57140536320*e^15 + 9183300480*e^10 + 918330048*e^5 + 430467...
Time = 0.46 (sec) , antiderivative size = 90, normalized size of antiderivative = 4.09 \[ \int \frac {-2500-3000 x+18800 x^2+11840 x^3+47360 x^5-300800 x^6-192000 x^7+640000 x^8}{81 x^5+432 e^5 x^5+864 e^{10} x^5+768 e^{15} x^5+256 e^{20} x^5} \, dx=\frac {47360\,x}{{\left (4\,{\mathrm {e}}^5+3\right )}^4}-\frac {150400\,x^2}{{\left (4\,{\mathrm {e}}^5+3\right )}^4}-\frac {64000\,x^3}{{\left (4\,{\mathrm {e}}^5+3\right )}^4}+\frac {160000\,x^4}{{\left (4\,{\mathrm {e}}^5+3\right )}^4}+\frac {-11840\,x^3-9400\,x^2+1000\,x+625}{x^4\,\left (432\,{\mathrm {e}}^5+864\,{\mathrm {e}}^{10}+768\,{\mathrm {e}}^{15}+256\,{\mathrm {e}}^{20}+81\right )} \] Input:
int(-(3000*x - 18800*x^2 - 11840*x^3 - 47360*x^5 + 300800*x^6 + 192000*x^7 - 640000*x^8 + 2500)/(432*x^5*exp(5) + 864*x^5*exp(10) + 768*x^5*exp(15) + 256*x^5*exp(20) + 81*x^5),x)
Output:
(47360*x)/(4*exp(5) + 3)^4 - (150400*x^2)/(4*exp(5) + 3)^4 - (64000*x^3)/( 4*exp(5) + 3)^4 + (160000*x^4)/(4*exp(5) + 3)^4 + (1000*x - 9400*x^2 - 118 40*x^3 + 625)/(x^4*(432*exp(5) + 864*exp(10) + 768*exp(15) + 256*exp(20) + 81))
Time = 0.25 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.91 \[ \int \frac {-2500-3000 x+18800 x^2+11840 x^3+47360 x^5-300800 x^6-192000 x^7+640000 x^8}{81 x^5+432 e^5 x^5+864 e^{10} x^5+768 e^{15} x^5+256 e^{20} x^5} \, dx=\frac {160000 x^{8}-64000 x^{7}-150400 x^{6}+47360 x^{5}-11840 x^{3}-9400 x^{2}+1000 x +625}{x^{4} \left (256 e^{20}+768 e^{15}+864 e^{10}+432 e^{5}+81\right )} \] Input:
int((640000*x^8-192000*x^7-300800*x^6+47360*x^5+11840*x^3+18800*x^2-3000*x -2500)/(256*x^5*exp(5)^4+768*x^5*exp(5)^3+864*x^5*exp(5)^2+432*x^5*exp(5)+ 81*x^5),x)
Output:
(5*(32000*x**8 - 12800*x**7 - 30080*x**6 + 9472*x**5 - 2368*x**3 - 1880*x* *2 + 200*x + 125))/(x**4*(256*e**20 + 768*e**15 + 864*e**10 + 432*e**5 + 8 1))