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Resolución de integrales/ x^3+1/ x^13/(x^5+x^3+1)^2

Integral de x^13/(x^5+x^3+1)^2 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo                  
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0
∫0∞x13((x5+x3)+1)2 dx\int\limits_{0}^{\infty} \frac{x^{13}}{\left(\left(x^{5} + x^{3}\right) + 1\right)^{2}}\, dx 
Integral(x^13/(x^5 + x^3 + 1)^2, (x, 0, oo))
Respuesta (Indefinida) [src] 
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 |      x                   2          |      5       3      2                      | 13       2*t   2918*t    3233*t    22631*t ||          |             5               3             2                                   |  2536678726146920768009914566       51415602083482733676020730820237*t    874911410055162894534919988675*t    57799106089266511847971052385*t   238165824682266762776982753785*t ||            |             5              3            2                              |290332708249646050       153545298040762276407875*t    904637403823710720545*t    8373100771536560567*t   73508719629038995551746147*t ||            |      5       3       2                |  4255200       62335027*t   105468750*t    478209195*t    6096225625*t ||            |             5              3             2                                |   958304371714520226706258       868196506949216703376421237*t    502435466831356689591754543*t    333030732113469856273732750*t    58541432542729362533762198*t||            |      5         4         3        2                       |475       100223*t    6875*t   12743*t    113155*t ||            |      5       3      2                |  1395450       1277439125*t    187442874*t    52536215*t    19763162*t||   x    -60 - 661*x - 36*x  + 100*x  + 250*x    -36 + 60*x  + 150*x  + 250*x + 625*x    2*\250 + 36*x  + 60*x + 150*x  + 661*x /   3*\625 + 36*x  + 90*x  + 150*x + 375*x /          |      5        3      2                      |  2959556        2728580874*t    43205812*t    55949997*t   124306844*t ||          |             5              3            2                                 |290332708249646050       180927480764742144109*t    8373100771536560567*t   1228362384326098211263*t    73508719629038995551746147*t ||
 | -------------- dx = C - x  - RootSum|3233*t  - 45*t  + 5*t  + 5*t - 1, t -> t*log|--- + x - --- + ------- + ------- + --------|| - RootSum|33792250337*t  - 1157772780*t  - 66179744*t  + 33414720*t - 3242336, t -> t*log|- ---------------------------- + x - ----------------------------------- - --------------------------------- + ------------------------------- + ---------------------------------|| - 3*RootSum|33792250337*t  - 110920500*t  - 1336176*t  + 324000*t + 7776, t -> t*log|------------------ + x - --------------------------- - ------------------------ + --------------------- + -----------------------------|| - 2*RootSum|3233*t  + 27*t  - 50*t  + 1, t -> t*log|- ------- + x - ---------- + ------------ + ------------ + -------------|| - 2*RootSum|33792250337*t  + 609322167*t  + 25541028*t  + 815184*t + 763263, t -> t*log|- ------------------------- + x - ------------------------------ - ------------------------------ - ------------------------------ - ----------------------------|| + 3*RootSum|3233*t  - 3233*t  + 1277*t  - 250*t  + 25*t - 1, t -> t*log|--- + x - --------- - ------ + -------- + ---------|| + 4*RootSum|3233*t  + 75*t  + 4*t  + 1, t -> t*log|- ------- + x - ------------- - ------------ - ----------- - ----------|| + -- + ------------------------------------- + ------------------------------------- - ---------------------------------------- + ---------------------------------------- + RootSum|3233*t  - 125*t  - 4*t  + 4*t - 1, t -> t*log|- -------- + x - ------------- - ----------- + ---------- + ------------|| + RootSum|33792250337*t  - 308112500*t  + 6186000*t  + 2500000*t - 100000, t -> t*log|------------------ + x - ------------------------ - --------------------- + ------------------------- + -----------------------------||
 |              2                      \                                            \125        25     125       125       125   //          \                                                                               \  3051877583154404909829226613          48830041330470478557267625808        24415020665235239278633812904       6103755166308809819658453226       6103755166308809819658453226  //            \                                                                        \280175042313434077           30258904569850880316        5043150761641813386        840525126940302231        363106854838210563792    //            \                                       \  7363759        7363759       7363759        7363759         7363759   //            \                                                                           \  2299388666307718707816583         1171386679062422737944297        20694497996769468370349247        390462226354140912648099       6898165998923156123449749  //            \                                                           \ 81           81        81        27          81   //            \                                      \  9712831          9712831        9712831        9712831      9712831  //   4                        3         5                             3         5                              3         5                                3         5                  \                                             \  10177721          10177721       10177721     10177721      10177721  //          \                                                                           \280175042313434077         2801750423134340770       1400875211567170385       1120700169253736308          2801750423134340770000   //
 | / 5    3    \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      3233 + 3233*x  + 3233*x                 3233 + 3233*x  + 3233*x                  3233 + 3233*x  + 3233*x                    3233 + 3233*x  + 3233*x                                                                                                                                                                                                                                                                                                                                                                          
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∫x13((x5+x3)+1)2 dx=C+x44−x2−2RootSum⁡(3233t5+27t3−50t2+1,(t↦tlog⁡(6096225625t47363759+478209195t37363759+105468750t27363759−62335027t7363759+x−42552007363759)))+4RootSum⁡(3233t5+75t3+4t2+1,(t↦tlog⁡(−187442874t49712831−1277439125t39712831−52536215t29712831−19763162t9712831+x−13954509712831)))−RootSum⁡(3233t5−45t3+5t2+5t−1,(t↦tlog⁡(22631t4125+3233t3125+2918t2125−2t25+x+13125)))+RootSum⁡(3233t5−125t3−4t2+4t−1,(t↦tlog⁡(−2728580874t410177721−43205812t310177721+124306844t210177721+55949997t10177721+x−295955610177721)))−3RootSum⁡(33792250337t5−110920500t3−1336176t2+324000t+7776,(t↦tlog⁡(73508719629038995551746147t4363106854838210563792−153545298040762276407875t330258904569850880316−904637403823710720545t25043150761641813386+8373100771536560567t840525126940302231+x+290332708249646050280175042313434077)))+RootSum⁡(33792250337t5−308112500t3+6186000t2+2500000t−100000,(t↦tlog⁡(73508719629038995551746147t42801750423134340770000+1228362384326098211263t31120700169253736308−180927480764742144109t22801750423134340770−8373100771536560567t1400875211567170385+x+290332708249646050280175042313434077)))−2RootSum⁡(33792250337t5+609322167t3+25541028t2+815184t+763263,(t↦tlog⁡(−333030732113469856273732750t4390462226354140912648099−868196506949216703376421237t31171386679062422737944297−502435466831356689591754543t220694497996769468370349247−58541432542729362533762198t6898165998923156123449749+x−9583043717145202267062582299388666307718707816583)))−RootSum⁡(33792250337t5−1157772780t3−66179744t2+33414720t−3242336,(t↦tlog⁡(−51415602083482733676020730820237t448830041330470478557267625808−874911410055162894534919988675t324415020665235239278633812904+238165824682266762776982753785t26103755166308809819658453226+57799106089266511847971052385t6103755166308809819658453226+x−25366787261469207680099145663051877583154404909829226613)))+3RootSum⁡(3233t5−3233t4+1277t3−250t2+25t−1,(t↦tlog⁡(113155t481−100223t381+12743t227−6875t81+x+47581)))−2(36x4+150x3+661x2+60x+250)3233x5+3233x3+3233+3(90x4+375x3+36x2+150x+625)3233x5+3233x3+3233+150x4+625x3+60x2+250x−363233x5+3233x3+3233+250x4−36x3+100x2−661x−603233x5+3233x3+3233\int \frac{x^{13}}{\left(\left(x^{5} + x^{3}\right) + 1\right)^{2}}\, dx = C + \frac{x^{4}}{4} - x^{2} - 2 \operatorname{RootSum} {\left(3233 t^{5} + 27 t^{3} - 50 t^{2} + 1, \left( t \mapsto t \log{\left(\frac{6096225625 t^{4}}{7363759} + \frac{478209195 t^{3}}{7363759} + \frac{105468750 t^{2}}{7363759} - \frac{62335027 t}{7363759} + x - \frac{4255200}{7363759} \right)} \right)\right)} + 4 \operatorname{RootSum} {\left(3233 t^{5} + 75 t^{3} + 4 t^{2} + 1, \left( t \mapsto t \log{\left(- \frac{187442874 t^{4}}{9712831} - \frac{1277439125 t^{3}}{9712831} - \frac{52536215 t^{2}}{9712831} - \frac{19763162 t}{9712831} + x - \frac{1395450}{9712831} \right)} \right)\right)} - \operatorname{RootSum} {\left(3233 t^{5} - 45 t^{3} + 5 t^{2} + 5 t - 1, \left( t \mapsto t \log{\left(\frac{22631 t^{4}}{125} + \frac{3233 t^{3}}{125} + \frac{2918 t^{2}}{125} - \frac{2 t}{25} + x + \frac{13}{125} \right)} \right)\right)} + \operatorname{RootSum} {\left(3233 t^{5} - 125 t^{3} - 4 t^{2} + 4 t - 1, \left( t \mapsto t \log{\left(- \frac{2728580874 t^{4}}{10177721} - \frac{43205812 t^{3}}{10177721} + \frac{124306844 t^{2}}{10177721} + \frac{55949997 t}{10177721} + x - \frac{2959556}{10177721} \right)} \right)\right)} - 3 \operatorname{RootSum} {\left(33792250337 t^{5} - 110920500 t^{3} - 1336176 t^{2} + 324000 t + 7776, \left( t \mapsto t \log{\left(\frac{73508719629038995551746147 t^{4}}{363106854838210563792} - \frac{153545298040762276407875 t^{3}}{30258904569850880316} - \frac{904637403823710720545 t^{2}}{5043150761641813386} + \frac{8373100771536560567 t}{840525126940302231} + x + \frac{290332708249646050}{280175042313434077} \right)} \right)\right)} + \operatorname{RootSum} {\left(33792250337 t^{5} - 308112500 t^{3} + 6186000 t^{2} + 2500000 t - 100000, \left( t \mapsto t \log{\left(\frac{73508719629038995551746147 t^{4}}{2801750423134340770000} + \frac{1228362384326098211263 t^{3}}{1120700169253736308} - \frac{180927480764742144109 t^{2}}{2801750423134340770} - \frac{8373100771536560567 t}{1400875211567170385} + x + \frac{290332708249646050}{280175042313434077} \right)} \right)\right)} - 2 \operatorname{RootSum} {\left(33792250337 t^{5} + 609322167 t^{3} + 25541028 t^{2} + 815184 t + 763263, \left( t \mapsto t \log{\left(- \frac{333030732113469856273732750 t^{4}}{390462226354140912648099} - \frac{868196506949216703376421237 t^{3}}{1171386679062422737944297} - \frac{502435466831356689591754543 t^{2}}{20694497996769468370349247} - \frac{58541432542729362533762198 t}{6898165998923156123449749} + x - \frac{958304371714520226706258}{2299388666307718707816583} \right)} \right)\right)} - \operatorname{RootSum} {\left(33792250337 t^{5} - 1157772780 t^{3} - 66179744 t^{2} + 33414720 t - 3242336, \left( t \mapsto t \log{\left(- \frac{51415602083482733676020730820237 t^{4}}{48830041330470478557267625808} - \frac{874911410055162894534919988675 t^{3}}{24415020665235239278633812904} + \frac{238165824682266762776982753785 t^{2}}{6103755166308809819658453226} + \frac{57799106089266511847971052385 t}{6103755166308809819658453226} + x - \frac{2536678726146920768009914566}{3051877583154404909829226613} \right)} \right)\right)} + 3 \operatorname{RootSum} {\left(3233 t^{5} - 3233 t^{4} + 1277 t^{3} - 250 t^{2} + 25 t - 1, \left( t \mapsto t \log{\left(\frac{113155 t^{4}}{81} - \frac{100223 t^{3}}{81} + \frac{12743 t^{2}}{27} - \frac{6875 t}{81} + x + \frac{475}{81} \right)} \right)\right)} - \frac{2 \left(36 x^{4} + 150 x^{3} + 661 x^{2} + 60 x + 250\right)}{3233 x^{5} + 3233 x^{3} + 3233} + \frac{3 \left(90 x^{4} + 375 x^{3} + 36 x^{2} + 150 x + 625\right)}{3233 x^{5} + 3233 x^{3} + 3233} + \frac{150 x^{4} + 625 x^{3} + 60 x^{2} + 250 x - 36}{3233 x^{5} + 3233 x^{3} + 3233} + \frac{250 x^{4} - 36 x^{3} + 100 x^{2} - 661 x - 60}{3233 x^{5} + 3233 x^{3} + 3233}
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Respuesta [src] 
 oo                  
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 |        13         
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0
∫0∞x13(x5+x3+1)2 dx\int\limits_{0}^{\infty} \frac{x^{13}}{\left(x^{5} + x^{3} + 1\right)^{2}}\, dx 
=
= 
 oo                  
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0
∫0∞x13(x5+x3+1)2 dx\int\limits_{0}^{\infty} \frac{x^{13}}{\left(x^{5} + x^{3} + 1\right)^{2}}\, dx 
Integral(x^13/(1 + x^3 + x^5)^2, (x, 0, oo))

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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