Integrand size = 190, antiderivative size = 27 \[ \int \frac {e^{20} \left (548371200000000 x^3+172947840000000 x^4+23828513280000 x^5+1873311897600 x^6+91912863744 x^7+2882018304 x^8+56398848 x^9+629760 x^{10}+3072 x^{11}\right )}{152587890625 x^8+48828125000 x^9+6835937500 x^{10}+546875000 x^{11}+27343750 x^{12}+875000 x^{13}+17500 x^{14}+200 x^{15}+x^{16}+e^{40} \left (218971048064843776+67375707096875008 x+9069806724579328 x^2+697677440352256 x^3+33542184632320 x^4+1032067219456 x^5+19847446528 x^6+218103808 x^7+1048576 x^8\right )+e^{20} \left (365580800000000 x^4+114736128000000 x^5+15753287680000 x^6+1235891404800 x^7+60596226048 x^8+1901371392 x^9+37285888 x^{10}+417792 x^{11}+2048 x^{12}\right )} \, dx=\frac {3}{4+\frac {256 e^{20} \left (6+\frac {6}{25+x}\right )^4}{81 x^4}} \]
Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {e^{20} \left (548371200000000 x^3+172947840000000 x^4+23828513280000 x^5+1873311897600 x^6+91912863744 x^7+2882018304 x^8+56398848 x^9+629760 x^{10}+3072 x^{11}\right )}{152587890625 x^8+48828125000 x^9+6835937500 x^{10}+546875000 x^{11}+27343750 x^{12}+875000 x^{13}+17500 x^{14}+200 x^{15}+x^{16}+e^{40} \left (218971048064843776+67375707096875008 x+9069806724579328 x^2+697677440352256 x^3+33542184632320 x^4+1032067219456 x^5+19847446528 x^6+218103808 x^7+1048576 x^8\right )+e^{20} \left (365580800000000 x^4+114736128000000 x^5+15753287680000 x^6+1235891404800 x^7+60596226048 x^8+1901371392 x^9+37285888 x^{10}+417792 x^{11}+2048 x^{12}\right )} \, dx=-\frac {768 e^{20} (26+x)^4}{x^4 (25+x)^4+1024 e^{20} (26+x)^4} \]
Integrate[(E^20*(548371200000000*x^3 + 172947840000000*x^4 + 2382851328000 0*x^5 + 1873311897600*x^6 + 91912863744*x^7 + 2882018304*x^8 + 56398848*x^ 9 + 629760*x^10 + 3072*x^11))/(152587890625*x^8 + 48828125000*x^9 + 683593 7500*x^10 + 546875000*x^11 + 27343750*x^12 + 875000*x^13 + 17500*x^14 + 20 0*x^15 + x^16 + E^40*(218971048064843776 + 67375707096875008*x + 906980672 4579328*x^2 + 697677440352256*x^3 + 33542184632320*x^4 + 1032067219456*x^5 + 19847446528*x^6 + 218103808*x^7 + 1048576*x^8) + E^20*(365580800000000* x^4 + 114736128000000*x^5 + 15753287680000*x^6 + 1235891404800*x^7 + 60596 226048*x^8 + 1901371392*x^9 + 37285888*x^10 + 417792*x^11 + 2048*x^12)),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.
| \(\displaystyle \int \frac {e^{20} \left (3072 x^{11}+629760 x^{10}+56398848 x^9+2882018304 x^8+91912863744 x^7+1873311897600 x^6+23828513280000 x^5+172947840000000 x^4+548371200000000 x^3\right )}{x^{16}+200 x^{15}+17500 x^{14}+875000 x^{13}+27343750 x^{12}+546875000 x^{11}+6835937500 x^{10}+48828125000 x^9+152587890625 x^8+e^{40} \left (1048576 x^8+218103808 x^7+19847446528 x^6+1032067219456 x^5+33542184632320 x^4+697677440352256 x^3+9069806724579328 x^2+67375707096875008 x+218971048064843776\right )+e^{20} \left (2048 x^{12}+417792 x^{11}+37285888 x^{10}+1901371392 x^9+60596226048 x^8+1235891404800 x^7+15753287680000 x^6+114736128000000 x^5+365580800000000 x^4\right )} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle e^{20} \int \frac {3072 \left (x^{11}+205 x^{10}+18359 x^9+938157 x^8+29919552 x^7+609802050 x^6+7756677500 x^5+56298125000 x^4+178506250000 x^3\right )}{x^{16}+200 x^{15}+17500 x^{14}+875000 x^{13}+27343750 x^{12}+546875000 x^{11}+6835937500 x^{10}+48828125000 x^9+152587890625 x^8+1048576 e^{40} \left (x^8+208 x^7+18928 x^6+984256 x^5+31988320 x^4+665357056 x^3+8649641728 x^2+64254481408 x+208827064576\right )+2048 e^{20} \left (x^{12}+204 x^{11}+18206 x^{10}+928404 x^9+29588001 x^8+603462600 x^7+7692035000 x^6+56023500000 x^5+178506250000 x^4\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3072 e^{20} \int \frac {x^{11}+205 x^{10}+18359 x^9+938157 x^8+29919552 x^7+609802050 x^6+7756677500 x^5+56298125000 x^4+178506250000 x^3}{x^{16}+200 x^{15}+17500 x^{14}+875000 x^{13}+27343750 x^{12}+546875000 x^{11}+6835937500 x^{10}+48828125000 x^9+152587890625 x^8+1048576 e^{40} \left (x^8+208 x^7+18928 x^6+984256 x^5+31988320 x^4+665357056 x^3+8649641728 x^2+64254481408 x+208827064576\right )+2048 e^{20} \left (x^{12}+204 x^{11}+18206 x^{10}+928404 x^9+29588001 x^8+603462600 x^7+7692035000 x^6+56023500000 x^5+178506250000 x^4\right )}dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle 3072 e^{20} \int \left (\frac {-x^2-4 \left (13-2 e^5\right ) x-2 \left (325-212 e^5-32 e^{10}\right )}{2048 e^{15} \left (x^4+2 \left (25-4 e^5\right ) x^3+\left (625-408 e^5+32 e^{10}\right ) x^2-208 e^5 \left (25-8 e^5\right ) x+21632 e^{10}\right )}+\frac {x^2+4 \left (13+2 e^5\right ) x+2 \left (325+212 e^5-32 e^{10}\right )}{2048 e^{15} \left (x^4+2 \left (25+4 e^5\right ) x^3+\left (625+408 e^5+32 e^{10}\right ) x^2+208 e^5 \left (25+8 e^5\right ) x+21632 e^{10}\right )}+\frac {\left (677+432 e^5+32 e^{10}\right ) x^3+\left (33825+21824 e^5-96 e^{10}-256 e^{15}\right ) x^2+52 \left (8125+5300 e^5-1312 e^{10}-256 e^{15}\right ) x-21632 e^{10} \left (53+8 e^5\right )}{256 e^{10} \left (x^4+2 \left (25-4 e^5\right ) x^3+\left (625-408 e^5+32 e^{10}\right ) x^2-208 e^5 \left (25-8 e^5\right ) x+21632 e^{10}\right )^2}+\frac {\left (677-432 e^5+32 e^{10}\right ) x^3+\left (33825-21824 e^5-96 e^{10}+256 e^{15}\right ) x^2+52 \left (8125-5300 e^5-1312 e^{10}+256 e^{15}\right ) x-21632 e^{10} \left (53-8 e^5\right )}{256 e^{10} \left (x^4+2 \left (25+4 e^5\right ) x^3+\left (625+408 e^5+32 e^{10}\right ) x^2+208 e^5 \left (25+8 e^5\right ) x+21632 e^{10}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3072 e^{20} \left (\frac {\int \frac {x^2}{-x^4-2 \left (25-4 e^5\right ) x^3-\left (625-408 e^5+32 e^{10}\right ) x^2+208 e^5 \left (25-8 e^5\right ) x-21632 e^{10}}dx}{2048 e^{15}}+\frac {13 \left (16925-16664 e^5-5984 e^{10}-256 e^{15}\right ) \int \frac {1}{\left (x^4+2 \left (25-4 e^5\right ) x^3+\left (625-408 e^5+32 e^{10}\right ) x^2-208 e^5 \left (25-8 e^5\right ) x+21632 e^{10}\right )^2}dx}{64 e^5}+\frac {\left (421875+557416 e^5-1856 e^{10}-27392 e^{15}-1024 e^{20}\right ) \int \frac {x}{\left (x^4+2 \left (25-4 e^5\right ) x^3+\left (625-408 e^5+32 e^{10}\right ) x^2-208 e^5 \left (25-8 e^5\right ) x+21632 e^{10}\right )^2}dx}{512 e^{10}}+\frac {\left (16875+19372 e^5+2592 e^{10}-128 e^{15}\right ) \int \frac {x^2}{\left (x^4+2 \left (25-4 e^5\right ) x^3+\left (625-408 e^5+32 e^{10}\right ) x^2-208 e^5 \left (25-8 e^5\right ) x+21632 e^{10}\right )^2}dx}{512 e^{10}}-\frac {\left (325-212 e^5-32 e^{10}\right ) \int \frac {1}{x^4+2 \left (25-4 e^5\right ) x^3+\left (625-408 e^5+32 e^{10}\right ) x^2-208 e^5 \left (25-8 e^5\right ) x+21632 e^{10}}dx}{1024 e^{15}}-\frac {\left (13-2 e^5\right ) \int \frac {x}{x^4+2 \left (25-4 e^5\right ) x^3+\left (625-408 e^5+32 e^{10}\right ) x^2-208 e^5 \left (25-8 e^5\right ) x+21632 e^{10}}dx}{512 e^{15}}-\frac {13 \left (16925+16664 e^5-5984 e^{10}+256 e^{15}\right ) \int \frac {1}{\left (x^4+2 \left (25+4 e^5\right ) x^3+\left (625+408 e^5+32 e^{10}\right ) x^2+208 e^5 \left (25+8 e^5\right ) x+21632 e^{10}\right )^2}dx}{64 e^5}+\frac {\left (421875-557416 e^5-1856 e^{10}+27392 e^{15}-1024 e^{20}\right ) \int \frac {x}{\left (x^4+2 \left (25+4 e^5\right ) x^3+\left (625+408 e^5+32 e^{10}\right ) x^2+208 e^5 \left (25+8 e^5\right ) x+21632 e^{10}\right )^2}dx}{512 e^{10}}+\frac {\left (16875-19372 e^5+2592 e^{10}+128 e^{15}\right ) \int \frac {x^2}{\left (x^4+2 \left (25+4 e^5\right ) x^3+\left (625+408 e^5+32 e^{10}\right ) x^2+208 e^5 \left (25+8 e^5\right ) x+21632 e^{10}\right )^2}dx}{512 e^{10}}+\frac {\left (325+212 e^5-32 e^{10}\right ) \int \frac {1}{x^4+2 \left (25+4 e^5\right ) x^3+\left (625+408 e^5+32 e^{10}\right ) x^2+208 e^5 \left (25+8 e^5\right ) x+21632 e^{10}}dx}{1024 e^{15}}+\frac {\left (13+2 e^5\right ) \int \frac {x}{x^4+2 \left (25+4 e^5\right ) x^3+\left (625+408 e^5+32 e^{10}\right ) x^2+208 e^5 \left (25+8 e^5\right ) x+21632 e^{10}}dx}{512 e^{15}}+\frac {\int \frac {x^2}{x^4+2 \left (25+4 e^5\right ) x^3+\left (625+408 e^5+32 e^{10}\right ) x^2+208 e^5 \left (25+8 e^5\right ) x+21632 e^{10}}dx}{2048 e^{15}}-\frac {677+432 e^5+32 e^{10}}{1024 e^{10} \left (x^4+2 \left (25-4 e^5\right ) x^3+\left (625-408 e^5+32 e^{10}\right ) x^2-208 e^5 \left (25-8 e^5\right ) x+21632 e^{10}\right )}-\frac {677-432 e^5+32 e^{10}}{1024 e^{10} \left (x^4+2 \left (25+4 e^5\right ) x^3+\left (625+408 e^5+32 e^{10}\right ) x^2+208 e^5 \left (25+8 e^5\right ) x+21632 e^{10}\right )}\right )\) |
Int[(E^20*(548371200000000*x^3 + 172947840000000*x^4 + 23828513280000*x^5 + 1873311897600*x^6 + 91912863744*x^7 + 2882018304*x^8 + 56398848*x^9 + 62 9760*x^10 + 3072*x^11))/(152587890625*x^8 + 48828125000*x^9 + 6835937500*x ^10 + 546875000*x^11 + 27343750*x^12 + 875000*x^13 + 17500*x^14 + 200*x^15 + x^16 + E^40*(218971048064843776 + 67375707096875008*x + 906980672457932 8*x^2 + 697677440352256*x^3 + 33542184632320*x^4 + 1032067219456*x^5 + 198 47446528*x^6 + 218103808*x^7 + 1048576*x^8) + E^20*(365580800000000*x^4 + 114736128000000*x^5 + 15753287680000*x^6 + 1235891404800*x^7 + 60596226048 *x^8 + 1901371392*x^9 + 37285888*x^10 + 417792*x^11 + 2048*x^12)),x]
3.7.58.3.1 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_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(80\) vs. \(2(26)=52\).
Time = 6.73 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.00
| method | result | size |
| risch | \(\frac {{\mathrm e}^{20} \left (-\frac {3}{4} x^{4}-78 x^{3}-3042 x^{2}-52728 x -342732\right )}{{\mathrm e}^{20} x^{4}+\frac {x^{8}}{1024}+104 x^{3} {\mathrm e}^{20}+\frac {25 x^{7}}{256}+4056 x^{2} {\mathrm e}^{20}+\frac {1875 x^{6}}{512}+70304 x \,{\mathrm e}^{20}+\frac {15625 x^{5}}{256}+456976 \,{\mathrm e}^{20}+\frac {390625 x^{4}}{1024}}\) | \(81\) |
| gosper | \(-\frac {768 \left (x +26\right ) \left (x^{3}+78 x^{2}+2028 x +17576\right ) {\mathrm e}^{20}}{1024 \,{\mathrm e}^{20} x^{4}+x^{8}+106496 x^{3} {\mathrm e}^{20}+100 x^{7}+4153344 x^{2} {\mathrm e}^{20}+3750 x^{6}+71991296 x \,{\mathrm e}^{20}+62500 x^{5}+467943424 \,{\mathrm e}^{20}+390625 x^{4}}\) | \(89\) |
| parallelrisch | \(\frac {{\mathrm e}^{20} \left (-768 x^{4}-79872 x^{3}-3115008 x^{2}-53993472 x -350957568\right )}{1024 \,{\mathrm e}^{20} x^{4}+x^{8}+106496 x^{3} {\mathrm e}^{20}+100 x^{7}+4153344 x^{2} {\mathrm e}^{20}+3750 x^{6}+71991296 x \,{\mathrm e}^{20}+62500 x^{5}+467943424 \,{\mathrm e}^{20}+390625 x^{4}}\) | \(92\) |
| norman | \(\frac {-53993472 x \,{\mathrm e}^{20}-3115008 x^{2} {\mathrm e}^{20}-79872 x^{3} {\mathrm e}^{20}-768 \,{\mathrm e}^{20} x^{4}-350957568 \,{\mathrm e}^{20}}{1024 \,{\mathrm e}^{20} x^{4}+x^{8}+106496 x^{3} {\mathrm e}^{20}+100 x^{7}+4153344 x^{2} {\mathrm e}^{20}+3750 x^{6}+71991296 x \,{\mathrm e}^{20}+62500 x^{5}+467943424 \,{\mathrm e}^{20}+390625 x^{4}}\) | \(109\) |
| default | \(384 \,{\mathrm e}^{20} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}+200 \textit {\_Z}^{15}+17500 \textit {\_Z}^{14}+875000 \textit {\_Z}^{13}+\left (2048 \,{\mathrm e}^{20}+27343750\right ) \textit {\_Z}^{12}+\left (417792 \,{\mathrm e}^{20}+546875000\right ) \textit {\_Z}^{11}+\left (37285888 \,{\mathrm e}^{20}+6835937500\right ) \textit {\_Z}^{10}+\left (1901371392 \,{\mathrm e}^{20}+48828125000\right ) \textit {\_Z}^{9}+\left (60596226048 \,{\mathrm e}^{20}+1048576 \,{\mathrm e}^{40}+152587890625\right ) \textit {\_Z}^{8}+\left (1235891404800 \,{\mathrm e}^{20}+218103808 \,{\mathrm e}^{40}\right ) \textit {\_Z}^{7}+\left (15753287680000 \,{\mathrm e}^{20}+19847446528 \,{\mathrm e}^{40}\right ) \textit {\_Z}^{6}+\left (114736128000000 \,{\mathrm e}^{20}+1032067219456 \,{\mathrm e}^{40}\right ) \textit {\_Z}^{5}+\left (365580800000000 \,{\mathrm e}^{20}+33542184632320 \,{\mathrm e}^{40}\right ) \textit {\_Z}^{4}+697677440352256 \,{\mathrm e}^{40} \textit {\_Z}^{3}+9069806724579328 \,{\mathrm e}^{40} \textit {\_Z}^{2}+67375707096875008 \,{\mathrm e}^{40} \textit {\_Z} +218971048064843776 \,{\mathrm e}^{40}\right )}{\sum }\frac {\left (\textit {\_R}^{11}+205 \textit {\_R}^{10}+18359 \textit {\_R}^{9}+938157 \textit {\_R}^{8}+29919552 \textit {\_R}^{7}+609802050 \textit {\_R}^{6}+7756677500 \textit {\_R}^{5}+56298125000 \textit {\_R}^{4}+178506250000 \textit {\_R}^{3}\right ) \ln \left (x -\textit {\_R} \right )}{54931640625 \textit {\_R}^{8}+8544921875 \textit {\_R}^{9}+8421963387109376 \,{\mathrm e}^{40}+152587890625 \textit {\_R}^{7}+41015625 \textit {\_R}^{11}+1421875 \textit {\_R}^{12}+30625 \textit {\_R}^{13}+375 \textit {\_R}^{14}+2 \textit {\_R}^{15}+751953125 \textit {\_R}^{10}+46607360 \,{\mathrm e}^{20} \textit {\_R}^{9}+1081404979200 \,{\mathrm e}^{20} \textit {\_R}^{6}+3072 \,{\mathrm e}^{20} \textit {\_R}^{11}+574464 \,{\mathrm e}^{20} \textit {\_R}^{10}+60596226048 \,{\mathrm e}^{20} \textit {\_R}^{7}+2139042816 \,{\mathrm e}^{20} \textit {\_R}^{8}+71710080000000 \,{\mathrm e}^{20} \textit {\_R}^{4}+190840832 \,{\mathrm e}^{40} \textit {\_R}^{6}+182790400000000 \textit {\_R}^{3} {\mathrm e}^{20}+1048576 \,{\mathrm e}^{40} \textit {\_R}^{7}+16771092316160 \,{\mathrm e}^{40} \textit {\_R}^{3}+261629040132096 \,{\mathrm e}^{40} \textit {\_R}^{2}+11814965760000 \textit {\_R}^{5} {\mathrm e}^{20}+2267451681144832 \,{\mathrm e}^{40} \textit {\_R} +645042012160 \,{\mathrm e}^{40} \textit {\_R}^{4}+14885584896 \,{\mathrm e}^{40} \textit {\_R}^{5}}\right )\) | \(372\) |
int((3072*x^11+629760*x^10+56398848*x^9+2882018304*x^8+91912863744*x^7+187 3311897600*x^6+23828513280000*x^5+172947840000000*x^4+548371200000000*x^3) *exp(5)^4/((1048576*x^8+218103808*x^7+19847446528*x^6+1032067219456*x^5+33 542184632320*x^4+697677440352256*x^3+9069806724579328*x^2+6737570709687500 8*x+218971048064843776)*exp(5)^8+(2048*x^12+417792*x^11+37285888*x^10+1901 371392*x^9+60596226048*x^8+1235891404800*x^7+15753287680000*x^6+1147361280 00000*x^5+365580800000000*x^4)*exp(5)^4+x^16+200*x^15+17500*x^14+875000*x^ 13+27343750*x^12+546875000*x^11+6835937500*x^10+48828125000*x^9+1525878906 25*x^8),x,method=_RETURNVERBOSE)
exp(20)*(-3/4*x^4-78*x^3-3042*x^2-52728*x-342732)/(exp(20)*x^4+1/1024*x^8+ 104*x^3*exp(20)+25/256*x^7+4056*x^2*exp(20)+1875/512*x^6+70304*x*exp(20)+1 5625/256*x^5+456976*exp(20)+390625/1024*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (22) = 44\).
Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.59 \[ \int \frac {e^{20} \left (548371200000000 x^3+172947840000000 x^4+23828513280000 x^5+1873311897600 x^6+91912863744 x^7+2882018304 x^8+56398848 x^9+629760 x^{10}+3072 x^{11}\right )}{152587890625 x^8+48828125000 x^9+6835937500 x^{10}+546875000 x^{11}+27343750 x^{12}+875000 x^{13}+17500 x^{14}+200 x^{15}+x^{16}+e^{40} \left (218971048064843776+67375707096875008 x+9069806724579328 x^2+697677440352256 x^3+33542184632320 x^4+1032067219456 x^5+19847446528 x^6+218103808 x^7+1048576 x^8\right )+e^{20} \left (365580800000000 x^4+114736128000000 x^5+15753287680000 x^6+1235891404800 x^7+60596226048 x^8+1901371392 x^9+37285888 x^{10}+417792 x^{11}+2048 x^{12}\right )} \, dx=-\frac {768 \, {\left (x^{4} + 104 \, x^{3} + 4056 \, x^{2} + 70304 \, x + 456976\right )} e^{20}}{x^{8} + 100 \, x^{7} + 3750 \, x^{6} + 62500 \, x^{5} + 390625 \, x^{4} + 1024 \, {\left (x^{4} + 104 \, x^{3} + 4056 \, x^{2} + 70304 \, x + 456976\right )} e^{20}} \]
integrate((3072*x^11+629760*x^10+56398848*x^9+2882018304*x^8+91912863744*x ^7+1873311897600*x^6+23828513280000*x^5+172947840000000*x^4+54837120000000 0*x^3)*exp(5)^4/((1048576*x^8+218103808*x^7+19847446528*x^6+1032067219456* x^5+33542184632320*x^4+697677440352256*x^3+9069806724579328*x^2+6737570709 6875008*x+218971048064843776)*exp(5)^8+(2048*x^12+417792*x^11+37285888*x^1 0+1901371392*x^9+60596226048*x^8+1235891404800*x^7+15753287680000*x^6+1147 36128000000*x^5+365580800000000*x^4)*exp(5)^4+x^16+200*x^15+17500*x^14+875 000*x^13+27343750*x^12+546875000*x^11+6835937500*x^10+48828125000*x^9+1525 87890625*x^8),x, algorithm=\
-768*(x^4 + 104*x^3 + 4056*x^2 + 70304*x + 456976)*e^20/(x^8 + 100*x^7 + 3 750*x^6 + 62500*x^5 + 390625*x^4 + 1024*(x^4 + 104*x^3 + 4056*x^2 + 70304* x + 456976)*e^20)
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (20) = 40\).
Time = 13.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.52 \[ \int \frac {e^{20} \left (548371200000000 x^3+172947840000000 x^4+23828513280000 x^5+1873311897600 x^6+91912863744 x^7+2882018304 x^8+56398848 x^9+629760 x^{10}+3072 x^{11}\right )}{152587890625 x^8+48828125000 x^9+6835937500 x^{10}+546875000 x^{11}+27343750 x^{12}+875000 x^{13}+17500 x^{14}+200 x^{15}+x^{16}+e^{40} \left (218971048064843776+67375707096875008 x+9069806724579328 x^2+697677440352256 x^3+33542184632320 x^4+1032067219456 x^5+19847446528 x^6+218103808 x^7+1048576 x^8\right )+e^{20} \left (365580800000000 x^4+114736128000000 x^5+15753287680000 x^6+1235891404800 x^7+60596226048 x^8+1901371392 x^9+37285888 x^{10}+417792 x^{11}+2048 x^{12}\right )} \, dx=\frac {- 768 x^{4} e^{20} - 79872 x^{3} e^{20} - 3115008 x^{2} e^{20} - 53993472 x e^{20} - 350957568 e^{20}}{x^{8} + 100 x^{7} + 3750 x^{6} + 62500 x^{5} + x^{4} \cdot \left (390625 + 1024 e^{20}\right ) + 106496 x^{3} e^{20} + 4153344 x^{2} e^{20} + 71991296 x e^{20} + 467943424 e^{20}} \]
integrate((3072*x**11+629760*x**10+56398848*x**9+2882018304*x**8+919128637 44*x**7+1873311897600*x**6+23828513280000*x**5+172947840000000*x**4+548371 200000000*x**3)*exp(5)**4/((1048576*x**8+218103808*x**7+19847446528*x**6+1 032067219456*x**5+33542184632320*x**4+697677440352256*x**3+906980672457932 8*x**2+67375707096875008*x+218971048064843776)*exp(5)**8+(2048*x**12+41779 2*x**11+37285888*x**10+1901371392*x**9+60596226048*x**8+1235891404800*x**7 +15753287680000*x**6+114736128000000*x**5+365580800000000*x**4)*exp(5)**4+ x**16+200*x**15+17500*x**14+875000*x**13+27343750*x**12+546875000*x**11+68 35937500*x**10+48828125000*x**9+152587890625*x**8),x)
(-768*x**4*exp(20) - 79872*x**3*exp(20) - 3115008*x**2*exp(20) - 53993472* x*exp(20) - 350957568*exp(20))/(x**8 + 100*x**7 + 3750*x**6 + 62500*x**5 + x**4*(390625 + 1024*exp(20)) + 106496*x**3*exp(20) + 4153344*x**2*exp(20) + 71991296*x*exp(20) + 467943424*exp(20))
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (22) = 44\).
Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.81 \[ \int \frac {e^{20} \left (548371200000000 x^3+172947840000000 x^4+23828513280000 x^5+1873311897600 x^6+91912863744 x^7+2882018304 x^8+56398848 x^9+629760 x^{10}+3072 x^{11}\right )}{152587890625 x^8+48828125000 x^9+6835937500 x^{10}+546875000 x^{11}+27343750 x^{12}+875000 x^{13}+17500 x^{14}+200 x^{15}+x^{16}+e^{40} \left (218971048064843776+67375707096875008 x+9069806724579328 x^2+697677440352256 x^3+33542184632320 x^4+1032067219456 x^5+19847446528 x^6+218103808 x^7+1048576 x^8\right )+e^{20} \left (365580800000000 x^4+114736128000000 x^5+15753287680000 x^6+1235891404800 x^7+60596226048 x^8+1901371392 x^9+37285888 x^{10}+417792 x^{11}+2048 x^{12}\right )} \, dx=-\frac {768 \, {\left (x^{4} + 104 \, x^{3} + 4056 \, x^{2} + 70304 \, x + 456976\right )} e^{20}}{x^{8} + 100 \, x^{7} + 3750 \, x^{6} + 62500 \, x^{5} + x^{4} {\left (1024 \, e^{20} + 390625\right )} + 106496 \, x^{3} e^{20} + 4153344 \, x^{2} e^{20} + 71991296 \, x e^{20} + 467943424 \, e^{20}} \]
integrate((3072*x^11+629760*x^10+56398848*x^9+2882018304*x^8+91912863744*x ^7+1873311897600*x^6+23828513280000*x^5+172947840000000*x^4+54837120000000 0*x^3)*exp(5)^4/((1048576*x^8+218103808*x^7+19847446528*x^6+1032067219456* x^5+33542184632320*x^4+697677440352256*x^3+9069806724579328*x^2+6737570709 6875008*x+218971048064843776)*exp(5)^8+(2048*x^12+417792*x^11+37285888*x^1 0+1901371392*x^9+60596226048*x^8+1235891404800*x^7+15753287680000*x^6+1147 36128000000*x^5+365580800000000*x^4)*exp(5)^4+x^16+200*x^15+17500*x^14+875 000*x^13+27343750*x^12+546875000*x^11+6835937500*x^10+48828125000*x^9+1525 87890625*x^8),x, algorithm=\
-768*(x^4 + 104*x^3 + 4056*x^2 + 70304*x + 456976)*e^20/(x^8 + 100*x^7 + 3 750*x^6 + 62500*x^5 + x^4*(1024*e^20 + 390625) + 106496*x^3*e^20 + 4153344 *x^2*e^20 + 71991296*x*e^20 + 467943424*e^20)
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (22) = 44\).
Time = 0.37 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.89 \[ \int \frac {e^{20} \left (548371200000000 x^3+172947840000000 x^4+23828513280000 x^5+1873311897600 x^6+91912863744 x^7+2882018304 x^8+56398848 x^9+629760 x^{10}+3072 x^{11}\right )}{152587890625 x^8+48828125000 x^9+6835937500 x^{10}+546875000 x^{11}+27343750 x^{12}+875000 x^{13}+17500 x^{14}+200 x^{15}+x^{16}+e^{40} \left (218971048064843776+67375707096875008 x+9069806724579328 x^2+697677440352256 x^3+33542184632320 x^4+1032067219456 x^5+19847446528 x^6+218103808 x^7+1048576 x^8\right )+e^{20} \left (365580800000000 x^4+114736128000000 x^5+15753287680000 x^6+1235891404800 x^7+60596226048 x^8+1901371392 x^9+37285888 x^{10}+417792 x^{11}+2048 x^{12}\right )} \, dx=-\frac {768 \, {\left (x^{4} + 104 \, x^{3} + 4056 \, x^{2} + 70304 \, x + 456976\right )} e^{20}}{x^{8} + 100 \, x^{7} + 3750 \, x^{6} + 62500 \, x^{5} + 1024 \, x^{4} e^{20} + 390625 \, x^{4} + 106496 \, x^{3} e^{20} + 4153344 \, x^{2} e^{20} + 71991296 \, x e^{20} + 467943424 \, e^{20}} \]
integrate((3072*x^11+629760*x^10+56398848*x^9+2882018304*x^8+91912863744*x ^7+1873311897600*x^6+23828513280000*x^5+172947840000000*x^4+54837120000000 0*x^3)*exp(5)^4/((1048576*x^8+218103808*x^7+19847446528*x^6+1032067219456* x^5+33542184632320*x^4+697677440352256*x^3+9069806724579328*x^2+6737570709 6875008*x+218971048064843776)*exp(5)^8+(2048*x^12+417792*x^11+37285888*x^1 0+1901371392*x^9+60596226048*x^8+1235891404800*x^7+15753287680000*x^6+1147 36128000000*x^5+365580800000000*x^4)*exp(5)^4+x^16+200*x^15+17500*x^14+875 000*x^13+27343750*x^12+546875000*x^11+6835937500*x^10+48828125000*x^9+1525 87890625*x^8),x, algorithm=\
-768*(x^4 + 104*x^3 + 4056*x^2 + 70304*x + 456976)*e^20/(x^8 + 100*x^7 + 3 750*x^6 + 62500*x^5 + 1024*x^4*e^20 + 390625*x^4 + 106496*x^3*e^20 + 41533 44*x^2*e^20 + 71991296*x*e^20 + 467943424*e^20)
Time = 9.54 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.33 \[ \int \frac {e^{20} \left (548371200000000 x^3+172947840000000 x^4+23828513280000 x^5+1873311897600 x^6+91912863744 x^7+2882018304 x^8+56398848 x^9+629760 x^{10}+3072 x^{11}\right )}{152587890625 x^8+48828125000 x^9+6835937500 x^{10}+546875000 x^{11}+27343750 x^{12}+875000 x^{13}+17500 x^{14}+200 x^{15}+x^{16}+e^{40} \left (218971048064843776+67375707096875008 x+9069806724579328 x^2+697677440352256 x^3+33542184632320 x^4+1032067219456 x^5+19847446528 x^6+218103808 x^7+1048576 x^8\right )+e^{20} \left (365580800000000 x^4+114736128000000 x^5+15753287680000 x^6+1235891404800 x^7+60596226048 x^8+1901371392 x^9+37285888 x^{10}+417792 x^{11}+2048 x^{12}\right )} \, dx=-\frac {768\,{\mathrm {e}}^{20}\,{\left (x+26\right )}^4}{x^8+100\,x^7+3750\,x^6+62500\,x^5+\left (1024\,{\mathrm {e}}^{20}+390625\right )\,x^4+106496\,{\mathrm {e}}^{20}\,x^3+4153344\,{\mathrm {e}}^{20}\,x^2+71991296\,{\mathrm {e}}^{20}\,x+467943424\,{\mathrm {e}}^{20}} \]
int((exp(20)*(548371200000000*x^3 + 172947840000000*x^4 + 23828513280000*x ^5 + 1873311897600*x^6 + 91912863744*x^7 + 2882018304*x^8 + 56398848*x^9 + 629760*x^10 + 3072*x^11))/(exp(40)*(67375707096875008*x + 906980672457932 8*x^2 + 697677440352256*x^3 + 33542184632320*x^4 + 1032067219456*x^5 + 198 47446528*x^6 + 218103808*x^7 + 1048576*x^8 + 218971048064843776) + exp(20) *(365580800000000*x^4 + 114736128000000*x^5 + 15753287680000*x^6 + 1235891 404800*x^7 + 60596226048*x^8 + 1901371392*x^9 + 37285888*x^10 + 417792*x^1 1 + 2048*x^12) + 152587890625*x^8 + 48828125000*x^9 + 6835937500*x^10 + 54 6875000*x^11 + 27343750*x^12 + 875000*x^13 + 17500*x^14 + 200*x^15 + x^16) ,x)