3.1.49 \(\int \frac {1}{(3-2 x)^{41/2} (1+x+2 x^2)^{20}} \, dx\) [49]
Optimal. Leaf size=1058 \[ \text {result too large to display} \]
[Out]
-143401467550777247627940437025/73985542663511997461099839851260280832/(3-2*x)^(9/2)-4611053278117143010907562
317585/7250583181024175751187784305423507521536/(3-2*x)^(7/2)-405965372440630510720926890227/20715951945783359
28910795515835287863296/(3-2*x)^(5/2)-4986681479187781853417316522775/8700699817229010901425341166508209025843
2/(3-2*x)^(3/2)+173441368149804378661935869705/896508488907352010051592447177261056/(3-2*x)^(19/2)-22724090823
469905152713519545/1604278348571050965355481221264572416/(3-2*x)^(17/2)-101190274412779618678573275245/3963511
214116714149701777134888943616/(3-2*x)^(15/2)-460503190416958283087439337135/343504305223448559640820685023708
44672/(3-2*x)^(13/2)-2211619588790911794826342607495/406920484649315986036049119181931544576/(3-2*x)^(11/2)-13
056959628363355534285785425/106924014357253562723941220352/(3-2*x)^(39/2)-3948194343291401740321996415/2028814
63139404195937734623232/(3-2*x)^(37/2)-304688229262620222736480811/537361713180043545997243056128/(3-2*x)^(35/
2)+2124315846756567455653862925/1688851098565851144562763890688/(3-2*x)^(33/2)+47657515074514118796095929535/6
6632852434325399703658138959872/(3-2*x)^(31/2)+34911619993974714062172751985/124667917457770102671360389021696
/(3-2*x)^(29/2)+149066309808794760843017404825/1624981820656451683095663001731072/(3-2*x)^(27/2)+1584861396416
9066543734380171/601845118761648771516912222863360/(3-2*x)^(25/2)+11155168222970774232376891145/16851663325326
16560247354224017408/(3-2*x)^(23/2)+14011818498091020272474956375/10110997995195699361484125344104448/(3-2*x)^
(21/2)+115/3248261265098830736532127368829731369648128*ln(3-2*x+14^(1/2)-(3-2*x)^(1/2)*(7+2*14^(1/2))^(1/2))*(
30297118912219360725028693061-8061110911143276053983022787*14^(1/2))*(-14+4*14^(1/2))^(1/2)-115/32482612650988
30736532127368829731369648128*ln(3-2*x+14^(1/2)+(3-2*x)^(1/2)*(7+2*14^(1/2))^(1/2))*(3029711891221936072502869
3061-8061110911143276053983022787*14^(1/2))*(-14+4*14^(1/2))^(1/2)+115/162413063254941536826606368441486568482
4064*arctan((-2*(3-2*x)^(1/2)+(7+2*14^(1/2))^(1/2))/(-7+2*14^(1/2))^(1/2))*(30297118912219360725028693061+8061
110911143276053983022787*14^(1/2))*(14+4*14^(1/2))^(1/2)-115/1624130632549415368266063684414865684824064*arcta
n((2*(3-2*x)^(1/2)+(7+2*14^(1/2))^(1/2))/(-7+2*14^(1/2))^(1/2))*(30297118912219360725028693061+806111091114327
6053983022787*14^(1/2))*(14+4*14^(1/2))^(1/2)-927027754781476746208047620505/580046654481934060095022744433880
60172288/(3-2*x)^(1/2)+5/595601664*(751303+1831285*x)/(3-2*x)^(39/2)/(2*x^2+x+1)^16+1/25015269888*(184959785+4
29411497*x)/(3-2*x)^(39/2)/(2*x^2+x+1)^15+1/4902992898048*(41652915209+92630823167*x)/(3-2*x)^(39/2)/(2*x^2+x+
1)^14+1/297448235814912*(2871555518177+6100156355517*x)/(3-2*x)^(39/2)/(2*x^2+x+1)^13+1/7138757659557888*(7755
9130805859+156274047129113*x)/(3-2*x)^(39/2)/(2*x^2+x+1)^12+5/1099368679571914752*(2656658801194921+5020880176
134289*x)/(3-2*x)^(39/2)/(2*x^2+x+1)^11+1/3420258114223734784*(45187921585208601+78752911037377255*x)/(3-2*x)^
(39/2)/(2*x^2+x+1)^10+1/430952522392190582784*(6063974149878048635+9477172618423641847*x)/(3-2*x)^(39/2)/(2*x^
2+x+1)^9+1/48266682507925345271808*(691833601144925854831+919498192874055581221*x)/(3-2*x)^(39/2)/(2*x^2+x+1)^
8+23/1576711628592227945545728*(919498192874055581221+908287136092467468517*x)/(3-2*x)^(39/2)/(2*x^2+x+1)^7+11
5/10187982830903626725064704*(908287136092467468517+298281884944522225747*x)/(3-2*x)^(39/2)/(2*x^2+x+1)^6+23/2
0375965661807253450129408*(2599313568802265110081-10426142448623187379187*x)/(3-2*x)^(39/2)/(2*x^2+x+1)^5-23/2
0018492580021161284337664*(10426142448623187379187+27513723463194262383705*x)/(3-2*x)^(39/2)/(2*x^2+x+1)^4-115
/76434244396444433994743808*(26513224428169016478843+30673415406553789342019*x)/(3-2*x)^(39/2)/(2*x^2+x+1)^3-1
15/125891696652967303050166272*(88411609113007981044643-5712269536245152162963*x)/(3-2*x)^(39/2)/(2*x^2+x+1)^2
+115/195831528126838026966925312*(28561347681225760814815+965934812839019490346107*x)/(3-2*x)^(39/2)/(2*x^2+x+
1)+1/133*x/(3-2*x)^(39/2)/(2*x^2+x+1)^19+1/33516*(113+373*x)/(3-2*x)^(39/2)/(2*x^2+x+1)^18+1/7976808*(40657+10
7329*x)/(3-2*x)^(39/2)/(2*x^2+x+1)^17
________________________________________________________________________________________
Rubi [A]
time = 1.63, antiderivative size = 1058, normalized size of antiderivative = 1.00, number of steps
used = 49, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {754, 836, 842,
840, 1183, 648, 632, 210, 642} \begin {gather*} \frac {23 (2599313568802265110081-10426142448623187379187 x)}{20375965661807253450129408 (3-2 x)^{39/2} \left (2 x^2+x+1\right )^5}+\frac {115 \sqrt {\frac {1}{2} \left (7+2 \sqrt {14}\right )} \left (30297118912219360725028693061+8061110911143276053983022787 \sqrt {14}\right ) \text {ArcTan}\left (\frac {\sqrt {7+2 \sqrt {14}}-2 \sqrt {3-2 x}}{\sqrt {-7+2 \sqrt {14}}}\right )}{812065316274707684133031842207432842412032}-\frac {115 \sqrt {\frac {1}{2} \left (7+2 \sqrt {14}\right )} \left (30297118912219360725028693061+8061110911143276053983022787 \sqrt {14}\right ) \text {ArcTan}\left (\frac {2 \sqrt {3-2 x}+\sqrt {7+2 \sqrt {14}}}{\sqrt {-7+2 \sqrt {14}}}\right )}{812065316274707684133031842207432842412032}+\frac {115 \left (30297118912219360725028693061-8061110911143276053983022787 \sqrt {14}\right ) \sqrt {\frac {1}{2} \left (-7+2 \sqrt {14}\right )} \log \left (-2 x-\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3\right )}{1624130632549415368266063684414865684824064}-\frac {115 \left (30297118912219360725028693061-8061110911143276053983022787 \sqrt {14}\right ) \sqrt {\frac {1}{2} \left (-7+2 \sqrt {14}\right )} \log \left (-2 x+\sqrt {7+2 \sqrt {14}} \sqrt {3-2 x}+\sqrt {14}+3\right )}{1624130632549415368266063684414865684824064}-\frac {927027754781476746208047620505}{58004665448193406009502274443388060172288 \sqrt {3-2 x}}+\frac {115 (965934812839019490346107 x+28561347681225760814815)}{195831528126838026966925312 (3-2 x)^{39/2} \left (2 x^2+x+1\right )}-\frac {4986681479187781853417316522775}{87006998172290109014253411665082090258432 (3-2 x)^{3/2}}-\frac {115 (88411609113007981044643-5712269536245152162963 x)}{125891696652967303050166272 (3-2 x)^{39/2} \left (2 x^2+x+1\right )^2}-\frac {405965372440630510720926890227}{2071595194578335928910795515835287863296 (3-2 x)^{5/2}}-\frac {115 (30673415406553789342019 x+26513224428169016478843)}{76434244396444433994743808 (3-2 x)^{39/2} \left (2 x^2+x+1\right )^3}-\frac {4611053278117143010907562317585}{7250583181024175751187784305423507521536 (3-2 x)^{7/2}}-\frac {23 (27513723463194262383705 x+10426142448623187379187)}{20018492580021161284337664 (3-2 x)^{39/2} \left (2 x^2+x+1\right )^4}-\frac {143401467550777247627940437025}{73985542663511997461099839851260280832 (3-2 x)^{9/2}}-\frac {2211619588790911794826342607495}{406920484649315986036049119181931544576 (3-2 x)^{11/2}}+\frac {115 (298281884944522225747 x+908287136092467468517)}{10187982830903626725064704 (3-2 x)^{39/2} \left (2 x^2+x+1\right )^6}-\frac {460503190416958283087439337135}{34350430522344855964082068502370844672 (3-2 x)^{13/2}}+\frac {23 (908287136092467468517 x+919498192874055581221)}{1576711628592227945545728 (3-2 x)^{39/2} \left (2 x^2+x+1\right )^7}-\frac {101190274412779618678573275245}{3963511214116714149701777134888943616 (3-2 x)^{15/2}}+\frac {919498192874055581221 x+691833601144925854831}{48266682507925345271808 (3-2 x)^{39/2} \left (2 x^2+x+1\right )^8}-\frac {22724090823469905152713519545}{1604278348571050965355481221264572416 (3-2 x)^{17/2}}+\frac {9477172618423641847 x+6063974149878048635}{430952522392190582784 (3-2 x)^{39/2} \left (2 x^2+x+1\right )^9}+\frac {173441368149804378661935869705}{896508488907352010051592447177261056 (3-2 x)^{19/2}}+\frac {78752911037377255 x+45187921585208601}{3420258114223734784 (3-2 x)^{39/2} \left (2 x^2+x+1\right )^{10}}+\frac {14011818498091020272474956375}{10110997995195699361484125344104448 (3-2 x)^{21/2}}+\frac {5 (5020880176134289 x+2656658801194921)}{1099368679571914752 (3-2 x)^{39/2} \left (2 x^2+x+1\right )^{11}}+\frac {11155168222970774232376891145}{1685166332532616560247354224017408 (3-2 x)^{23/2}}+\frac {156274047129113 x+77559130805859}{7138757659557888 (3-2 x)^{39/2} \left (2 x^2+x+1\right )^{12}}+\frac {15848613964169066543734380171}{601845118761648771516912222863360 (3-2 x)^{25/2}}+\frac {6100156355517 x+2871555518177}{297448235814912 (3-2 x)^{39/2} \left (2 x^2+x+1\right )^{13}}+\frac {149066309808794760843017404825}{1624981820656451683095663001731072 (3-2 x)^{27/2}}+\frac {92630823167 x+41652915209}{4902992898048 (3-2 x)^{39/2} \left (2 x^2+x+1\right )^{14}}+\frac {34911619993974714062172751985}{124667917457770102671360389021696 (3-2 x)^{29/2}}+\frac {429411497 x+184959785}{25015269888 (3-2 x)^{39/2} \left (2 x^2+x+1\right )^{15}}+\frac {47657515074514118796095929535}{66632852434325399703658138959872 (3-2 x)^{31/2}}+\frac {5 (1831285 x+751303)}{595601664 (3-2 x)^{39/2} \left (2 x^2+x+1\right )^{16}}+\frac {2124315846756567455653862925}{1688851098565851144562763890688 (3-2 x)^{33/2}}+\frac {107329 x+40657}{7976808 (3-2 x)^{39/2} \left (2 x^2+x+1\right )^{17}}-\frac {304688229262620222736480811}{537361713180043545997243056128 (3-2 x)^{35/2}}+\frac {373 x+113}{33516 (3-2 x)^{39/2} \left (2 x^2+x+1\right )^{18}}-\frac {3948194343291401740321996415}{202881463139404195937734623232 (3-2 x)^{37/2}}+\frac {x}{133 (3-2 x)^{39/2} \left (2 x^2+x+1\right )^{19}}-\frac {13056959628363355534285785425}{106924014357253562723941220352 (3-2 x)^{39/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((3 - 2*x)^(41/2)*(1 + x + 2*x^2)^20),x]
[Out]
-13056959628363355534285785425/(106924014357253562723941220352*(3 - 2*x)^(39/2)) - 394819434329140174032199641
5/(202881463139404195937734623232*(3 - 2*x)^(37/2)) - 304688229262620222736480811/(537361713180043545997243056
128*(3 - 2*x)^(35/2)) + 2124315846756567455653862925/(1688851098565851144562763890688*(3 - 2*x)^(33/2)) + 4765
7515074514118796095929535/(66632852434325399703658138959872*(3 - 2*x)^(31/2)) + 34911619993974714062172751985/
(124667917457770102671360389021696*(3 - 2*x)^(29/2)) + 149066309808794760843017404825/(16249818206564516830956
63001731072*(3 - 2*x)^(27/2)) + 15848613964169066543734380171/(601845118761648771516912222863360*(3 - 2*x)^(25
/2)) + 11155168222970774232376891145/(1685166332532616560247354224017408*(3 - 2*x)^(23/2)) + 14011818498091020
272474956375/(10110997995195699361484125344104448*(3 - 2*x)^(21/2)) + 173441368149804378661935869705/(89650848
8907352010051592447177261056*(3 - 2*x)^(19/2)) - 22724090823469905152713519545/(160427834857105096535548122126
4572416*(3 - 2*x)^(17/2)) - 101190274412779618678573275245/(3963511214116714149701777134888943616*(3 - 2*x)^(1
5/2)) - 460503190416958283087439337135/(34350430522344855964082068502370844672*(3 - 2*x)^(13/2)) - 22116195887
90911794826342607495/(406920484649315986036049119181931544576*(3 - 2*x)^(11/2)) - 1434014675507772476279404370
25/(73985542663511997461099839851260280832*(3 - 2*x)^(9/2)) - 4611053278117143010907562317585/(725058318102417
5751187784305423507521536*(3 - 2*x)^(7/2)) - 405965372440630510720926890227/(207159519457833592891079551583528
7863296*(3 - 2*x)^(5/2)) - 4986681479187781853417316522775/(87006998172290109014253411665082090258432*(3 - 2*x
)^(3/2)) - 927027754781476746208047620505/(58004665448193406009502274443388060172288*Sqrt[3 - 2*x]) + x/(133*(
3 - 2*x)^(39/2)*(1 + x + 2*x^2)^19) + (113 + 373*x)/(33516*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^18) + (40657 + 107
329*x)/(7976808*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^17) + (5*(751303 + 1831285*x))/(595601664*(3 - 2*x)^(39/2)*(1
+ x + 2*x^2)^16) + (184959785 + 429411497*x)/(25015269888*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^15) + (41652915209
+ 92630823167*x)/(4902992898048*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^14) + (2871555518177 + 6100156355517*x)/(297
448235814912*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^13) + (77559130805859 + 156274047129113*x)/(7138757659557888*(3
- 2*x)^(39/2)*(1 + x + 2*x^2)^12) + (5*(2656658801194921 + 5020880176134289*x))/(1099368679571914752*(3 - 2*x)
^(39/2)*(1 + x + 2*x^2)^11) + (45187921585208601 + 78752911037377255*x)/(3420258114223734784*(3 - 2*x)^(39/2)*
(1 + x + 2*x^2)^10) + (6063974149878048635 + 9477172618423641847*x)/(430952522392190582784*(3 - 2*x)^(39/2)*(1
+ x + 2*x^2)^9) + (691833601144925854831 + 919498192874055581221*x)/(48266682507925345271808*(3 - 2*x)^(39/2)
*(1 + x + 2*x^2)^8) + (23*(919498192874055581221 + 908287136092467468517*x))/(1576711628592227945545728*(3 - 2
*x)^(39/2)*(1 + x + 2*x^2)^7) + (115*(908287136092467468517 + 298281884944522225747*x))/(101879828309036267250
64704*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^6) + (23*(2599313568802265110081 - 10426142448623187379187*x))/(2037596
5661807253450129408*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^5) - (23*(10426142448623187379187 + 275137234631942623837
05*x))/(20018492580021161284337664*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^4) - (115*(26513224428169016478843 + 30673
415406553789342019*x))/(76434244396444433994743808*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^3) - (115*(884116091130079
81044643 - 5712269536245152162963*x))/(125891696652967303050166272*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^2) + (115*
(28561347681225760814815 + 965934812839019490346107*x))/(195831528126838026966925312*(3 - 2*x)^(39/2)*(1 + x +
2*x^2)) + (115*Sqrt[(7 + 2*Sqrt[14])/2]*(30297118912219360725028693061 + 8061110911143276053983022787*Sqrt[14
])*ArcTan[(Sqrt[7 + 2*Sqrt[14]] - 2*Sqrt[3 - 2*x])/Sqrt[-7 + 2*Sqrt[14]]])/81206531627470768413303184220743284
2412032 - (115*Sqrt[(7 + 2*Sqrt[14])/2]*(30297118912219360725028693061 + 8061110911143276053983022787*Sqrt[14]
)*ArcTan[(Sqrt[7 + 2*Sqrt[14]] + 2*Sqrt[3 - 2*x])/Sqrt[-7 + 2*Sqrt[14]]])/812065316274707684133031842207432842
412032 + (115*(30297118912219360725028693061 - 8061110911143276053983022787*Sqrt[14])*Sqrt[(-7 + 2*Sqrt[14])/2
]*Log[3 + Sqrt[14] - Sqrt[7 + 2*Sqrt[14]]*Sqrt[3 - 2*x] - 2*x])/1624130632549415368266063684414865684824064 -
(115*(30297118912219360725028693061 - 8061110911143276053983022787*Sqrt[14])*Sqrt[(-7 + 2*Sqrt[14])/2]*Log[3 +
Sqrt[14] + Sqrt[7 + 2*Sqrt[14]]*Sqrt[3 - 2*x] - 2*x])/1624130632549415368266063684414865684824064
Rule 210
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 632
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 642
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rule 648
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && !NiceSqrtQ[b^2 - 4*a*c]
Rule 754
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(b
*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Rule 836
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])
Rule 840
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
Rule 842
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e
*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(d +
e*x)^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]
Rule 1183
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Rubi steps
\begin {align*} \int \frac {1}{(3-2 x)^{41/2} \left (1+x+2 x^2\right )^{20}} \, dx &=\frac {x}{133 (3-2 x)^{39/2} \left (1+x+2 x^2\right )^{19}}+\frac {\int \frac {3640-3164 x}{(3-2 x)^{41/2} \left (1+x+2 x^2\right )^{19}} \, dx}{3724} \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 16.12, size = 1100, normalized size = 1.04 \begin {gather*} \text {Too large to display} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((3 - 2*x)^(41/2)*(1 + x + 2*x^2)^20),x]
[Out]
x/(133*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^19) + ((44296 + 146216*x)/(3528*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^18) +
((223125616 + 589021552*x)/(3332*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^17) + ((865861681440 + 2110519336800*x)/(31
36*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^16) + ((2984274342235200 + 6928434268875840*x)/(2940*(3 - 2*x)^(39/2)*(1 +
x + 2*x^2)^15) + ((9408813737133390720 + 20924013532366815360*x)/(2744*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^14) +
((27243065619141593598720 + 57873497074462503141120*x)/(2548*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^13) + ((7211037
7354780278913835520 + 145295342948683106164016640*x)/(2352*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^12) + ((1729014581
08932896335179801600 + 326770416680301421681066214400*x)/(2156*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^11) + ((370557
652515461812186329087129600 + 645802967231886306826540424448000*x)/(1960*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^10)
+ ((696175598675973438759010577554944000 + 1088028437838790621809440473088716800*x)/(1764*(3 - 2*x)^(39/2)*(1
+ x + 2*x^2)^9) + ((1111965063471244015489248163496668569600 + 1477884081820868038735185945420330393600*x)/(15
68*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^8) + ((1427636023038958525418189623276039160217600 + 141022945428029359210
8580217248432347955200*x)/(1372*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^7) + ((12833088033950671688188079976960734366
39232000 + 421439161286999121770135584246204836237312000*x)/(1176*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^6) + ((3599
09043739097249991695788946258930146664448000 - 1443636121324398194831693460992758930913796096000*x)/(980*(3 -
2*x)^(39/2)*(1 + x + 2*x^2)^5) + ((-1152021624816869759475691381872221626869209284608000 - 3040089329780519199
031170166260953381570260254720000*x)/(784*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^4) + ((-225574628269714524568112826
3365627409125133109002240000 - 2609695511325529255410382651665073470845732989009920000*x)/(588*(3 - 2*x)^(39/2
)*(1 + x + 2*x^2)^3) + ((-1790251120769313069211522499042240401000172830460805120000 + 11566803321414359689429
5804604678509924267822733393920000*x)/(392*(3 - 2*x)^(39/2)*(1 + x + 2*x^2)^2) + ((728708609249104660434063569
00947461252288728322038169600000 + 2464467090087282692969213073458776810025190662610343034880000*x)/(196*(3 -
2*x)^(39/2)*(1 + x + 2*x^2)) + (-530550566665897087493026465460148012491929957574880460800000/(3 - 2*x)^(39/2)
+ (-1708089006242241264480481073293611769771298388785813753364480000/(37*(3 - 2*x)^(37/2)) + (-69674095008990
9200017539783692427216704271188038402697920512000/(3 - 2*x)^(35/2) + (7573666677621473556024460064742611515974
09525795681824661504000000/(3 - 2*x)^(33/2) + (616772664905423340350737254793402194192083509401081655628275875
8400000/(31*(3 - 2*x)^(31/2)) + (980445504127015992472138196645778610361943940861637274650890661068800000/(29*
(3 - 2*x)^(29/2)) + (4496423323436580179825935667807239175646629240803415910250222313472000000/(3 - 2*x)^(27/2
) + (487904184130260773926886832047572655461484781443782543411352841560457216000/(3 - 2*x)^(25/2) + (429268867
21523802306414887155091882259902542088067698170622802545418240000000/(3 - 2*x)^(23/2) + (289369259398036472323
1826294558630623656919099359688069727689450554368000000000/(3 - 2*x)^(21/2) + (1187674764929302643741666332431
40666046068763101817907661320807641190359040000000/(3 - 2*x)^(19/2) + (-23130641371662285970537372414163682847
22516912423159767489332810437803253760000000/(3 - 2*x)^(17/2) + (-99223951965379086042262394895796485235598584
6800936213338418761762097950023680000000/(3 - 2*x)^(15/2) + (-109415183151546322431572415879018096250836012099
731766901467841654602614755123200000000/(3 - 2*x)^(13/2) + (-8073268485314233063840337934095431560069216535225
849300748018943930634745621913600000000/(3 - 2*x)^(11/2) + (-4433798722621123130520736149457228398171520393809
63932483996666511839997547213824000000000/(3 - 2*x)^(9/2) + (-183301908922166977441737067901437000873585615761
36178754174544727578117325359791923200000000/(3 - 2*x)^(7/2) + (-553541210002735957048844214716028245499086746
401723523324780660557661668413725058949120000000/(3 - 2*x)^(5/2) + (-11332385663391839740343974428370683887566
771471384841151672642393999283182139266339840000000000/(3 - 2*x)^(3/2) + (-13272202629081314874038396353552342
7142665518975435293064356777236410088640362467513344000000000/Sqrt[3 - 2*x] + ((Sqrt[(7 - I*Sqrt[7])/2]*(-1858
108368071384082365375489497327979997317265656094102900994881309741240965074545186816000000000 - (3853414006278
103146767987622401496699336335555921865837542016885265897482833115690092544000000000*I)*Sqrt[7])*ArcTanh[(Sqrt
[2]*Sqrt[3 - 2*x])/Sqrt[7 - I*Sqrt[7]]])/(-14 + (2*I)*Sqrt[7]) + (Sqrt[(7 + I*Sqrt[7])/2]*(-185810836807138408
2365375489497327979997317265656094102900994881309741240965074545186816000000000 + (385341400627810314676798762
2401496699336335555921865837542016885265897482833115690092544000000000*I)*Sqrt[7])*ArcTanh[(Sqrt[2]*Sqrt[3 - 2
*x])/Sqrt[7 + I*Sqrt[7]]])/(-14 - (2*I)*Sqrt[7]))/7)/42)/70)/98)/126)/154)/182)/210)/238)/266)/294)/322)/350)/
378)/406)/434)/462)/490)/518)/546)/196)/392)/58...
________________________________________________________________________________________
Maple [A]
time = 1.18, size = 820, normalized size = 0.78
| | |
method |
result |
size |
| | |
trager |
\(\text {Expression too large to display}\) |
\(732\) |
risch |
\(\text {Expression too large to display}\) |
\(761\) |
derivativedivides |
\(\text {Expression too large to display}\) |
\(820\) |
default |
\(\text {Expression too large to display}\) |
\(820\) |
| | |
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(3-2*x)^(41/2)/(2*x^2+x+1)^20,x,method=_RETURNVERBOSE)
[Out]
1/30145677658996078082575630336*(807597736492641378942268937217995835353849465/1048576*(3-2*x)^(1/2)+180866897
1148992206490172102870787954874541181/334114095890432*(3-2*x)^(57/2)-11968977253082880651292892111395530933265
219/25701084299264*(3-2*x)^(59/2)+339556544641293541759958988614814460549873/9826885173248*(3-2*x)^(61/2)-6424
3396719140374998473027009027485263697/29480655519744*(3-2*x)^(63/2)+129886852748727110357425618672922324659/11
33871366144*(3-2*x)^(65/2)-503502693505289734438057515605193725/103079215104*(3-2*x)^(67/2)+133883313322119397
348791732981953297/824633720832*(3-2*x)^(69/2)-3254850748003483429666738850178379/824633720832*(3-2*x)^(71/2)+
360433340020130123942335063779145/5772436045824*(3-2*x)^(73/2)-928342237074576734557978321305/1924145348608*(3
-2*x)^(75/2)-44796329357069082297154473725670903546220392558695/9070970929152*(3-2*x)^(43/2)+28607223317693223
698395672584150593863016075796143/29480655519744*(3-2*x)^(45/2)-5059022664167725408892162874688680417923742003
781/29480655519744*(3-2*x)^(47/2)+73012476452577571533836489036461787385135079265/2680059592704*(3-2*x)^(49/2)
-1939242920901534821454026903132433081580221023737/501171143835648*(3-2*x)^(51/2)+4907385430648794239550771659
87434152441563270473/1002342287671296*(3-2*x)^(53/2)-55011835288361289002011693179378316699033102675/100234228
7671296*(3-2*x)^(55/2)-1006304725834560333245233940167063186576585913370455/10720238370816*(3-2*x)^(39/2)+1380
5722741822612586258592099428566280191230197271405/39307540692992*(3-2*x)^(37/2)-176509423589632626758711731662
29809316744939271143/51904512*(3-2*x)^(11/2)+1186323846453826237212517196312193819452761764018822545/391539956
1216*(3-2*x)^(21/2)+2672239984790337844292019294315182385216573077301785/117922622078976*(3-2*x)^(41/2)-223975
46321209486953062074374795737299957063565/3145728*(3-2*x)^(3/2)+4045315666898833370484992335277819835991876340
17/12582912*(3-2*x)^(5/2)-1188598027552254830082683218064697188605612952419/12582912*(3-2*x)^(7/2)+38315833791
66294091823572953989993625772471445345/18874368*(3-2*x)^(9/2)+997785012616801018716913042477456833097312341255
1261/21592276992*(3-2*x)^(13/2)-1255696718499588580979726331572072320357969297077745/2399141888*(3-2*x)^(15/2)
-755930116404682856570195190192032441294632160945523631/3915399561216*(3-2*x)^(23/2)+8535085022072145119870938
66021124087908041634697244059/7830799122432*(3-2*x)^(25/2)-688617380989400554399451644246187148600704200518977
5/125627793408*(3-2*x)^(27/2)+136329987967245395141848253765147208279814148352958009/5527622909952*(3-2*x)^(29
/2)-55066091420817590167865401986871791412011888132876913/5527622909952*(3-2*x)^(31/2)+27374875289284393578691
38774910126923363791747141675/755914244096*(3-2*x)^(33/2)-1166457217021587688420366823074349521448831011337110
5/9826885173248*(3-2*x)^(35/2)+12646629333382722716904430763732665179119615389552413/25098715136*(3-2*x)^(17/2
)-2593673203685044441695042001860835122939346700333136537/6199382638592*(3-2*x)^(19/2))/((3-2*x)^2-7+14*x)^19-
115/3248261265098830736532127368829731369648128*(62541562556792464940960784209*(7+2*14^(1/2))^(1/2)*14^(1/2)-2
34044028404883307655877091262*(7+2*14^(1/2))^(1/2))*ln(3-2*x+14^(1/2)-(3-2*x)^(1/2)*(7+2*14^(1/2))^(1/2))+115/
3248261265098830736532127368829731369648128*(62541562556792464940960784209*(7+2*14^(1/2))^(1/2)*14^(1/2)-23404
4028404883307655877091262*(7+2*14^(1/2))^(1/2))*ln(3-2*x+14^(1/2)+(3-2*x)^(1/2)*(7+2*14^(1/2))^(1/2))-115/8120
65316274707684133031842207432842412032*(30297118912219360725028693061*14^(1/2)+1/2*(62541562556792464940960784
209*(7+2*14^(1/2))^(1/2)*14^(1/2)-234044028404883307655877091262*(7+2*14^(1/2))^(1/2))*(7+2*14^(1/2))^(1/2))/(
-7+2*14^(1/2))^(1/2)*arctan((2*(3-2*x)^(1/2)-(7+2*14^(1/2))^(1/2))/(-7+2*14^(1/2))^(1/2))+683151246370725/3014
5677658996078082575630336/(3-2*x)^(1/2)+355/5266289575642392066/(3-2*x)^(33/2)+52865/277038748585308867472/(3-
2*x)^(31/2)+14333/32395660116830472406/(3-2*x)^(29/2)+1478345/1689042692987850837168/(3-2*x)^(27/2)+475387/312
785683886639043920/(3-2*x)^(25/2)+16575515/7006399319060714583808/(3-2*x)^(23/2)+246866015/7356719285013750312
9984/(3-2*x)^(21/2)+1/3111898385606868039/(3-2*x)^(39/2)+10/2952313853011644037/(3-2*x)^(37/2)+143/78196420971
65976098/(3-2*x)^(35/2)+122484655975/17852305464966700759542784/(3-2*x)^(13/2)+10815878546425/1480368099325700
262983624704/(3-2*x)^(11/2)+8192823353/1863702218870150079292928/(3-2*x)^(19/2)+8972680075/1667523037936450070
946304/(3-2*x)^(17/2)+102495360575/16479051198430800701116416/(3-2*x)^(15/2)+769045155125/10093418859038865429
4338048/(3-2*x)^(9/2)+838467657280275/105509871806486273289014706176/(3-2*x)^(7/2)+9270470094105/1076631344964
145645806272512/(3-2*x)^(5/2)+320421783064625/30145677658996078082575630336/(3-2*x)^(3/2)+115/8120653162747076
84133031842207432842412032*(-30297118912219360725028693061*14^(1/2)-1/2*(62541562556792464940960784209*(7+2*14
^(1/2))^(1/2)*14^(1/2)-234044028404883307655877091262*(7+2*14^(1/2))^(1/2))*(7+2*14^(1/2))^(1/2))/(-7+2*14^(1/
2))^(1/2)*arctan((2*(3-2*x)^(1/2)+(7+2*14^(1/2)...
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(3-2*x)^(41/2)/(2*x^2+x+1)^20,x, algorithm="maxima")
[Out]
integrate(1/((2*x^2 + x + 1)^20*(-2*x + 3)^(41/2)), x)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2763 vs.
\(2 (821) = 1642\).
time = 39.48, size = 2763, normalized size = 2.61 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(3-2*x)^(41/2)/(2*x^2+x+1)^20,x, algorithm="fricas")
[Out]
1/392164866433134591452265700785383646021637848990793338705942972429050741081771910613067676554567307313820760
631947790705447918522595322425206164852572160*(616525316537858546962128448983043227187951381815778781478549978
900*579590499192185855665304541951571706717845859384545414208024478076585205782332794174344701326^(1/4)*sqrt(1
2868460882463048970358421717432178503450051394)*sqrt(14)*sqrt(7)*(549755813888*x^58 - 11269994184704*x^57 + 10
7064944754688*x^56 - 630638638006272*x^55 + 2618521301286912*x^54 - 8342252417974272*x^53 + 21849572376576000*
x^52 - 49684091485814784*x^51 + 101394501297242112*x^50 - 188583312363618304*x^49 + 323261995581177856*x^48 -
517079841212727296*x^47 + 778117896260812800*x^46 - 1105641165387988992*x^45 + 1491287028233404416*x^44 - 1919
929663119949824*x^43 + 2363050939901804544*x^42 - 2786274020645928960*x^41 + 3161145685194047488*x^40 - 345375
3931369283584*x^39 + 3634098467102523392*x^38 - 3697893960325791744*x^37 + 3640651752731836416*x^36 - 34617982
12247617536*x^35 + 3194540251789393920*x^34 - 2861544579495297024*x^33 + 2477632938217930752*x^32 - 2088430257
127768064*x^31 + 1712761005459316736*x^30 - 1355447485390974976*x^29 + 1048940886155151360*x^28 - 790511024135
089152*x^27 + 571750925528393856*x^26 - 408374103192240192*x^25 + 282845069599813728*x^24 - 186113897194906128
*x^23 + 123982890381352520*x^22 - 78116367732251996*x^21 + 46488580159296898*x^20 - 29591055660829971*x^19 + 1
6200795673453545*x^18 - 8941894120163277*x^17 + 5578893209169441*x^16 - 2296849711499532*x^15 + 14482898824007
88*x^14 - 756896247319212*x^13 + 182213447974992*x^12 - 240797810407770*x^11 + 25549234281774*x^10 - 265002817
27302*x^9 + 25520701332582*x^8 + 9965507230260*x^7 + 10389354811164*x^6 + 3755740313808*x^5 + 1820618017974*x^
4 + 463742325333*x^3 + 139858796529*x^2 + 19758444939*x + 3486784401)*sqrt(37814840288016788880034681293391537
27662345024772741260943*sqrt(14) + 14149022371848728385570789036684124101210161640127797919744)*arctan(1/34885
55476273159707600878934940824497561724963674913242575009589894914045286581081812447079130476773106112671051669
99787145808229165832263016823558232093156487983192678515257488180949060050957316309922278384344605468898548205
7622250395943920813921700*579590499192185855665304541951571706717845859384545414208024478076585205782332794174
344701326^(3/4)*sqrt(1634857335323112850812492677092639503349451327418417311)*sqrt(643423044123152448517921085
87160892517250256970)*sqrt(12868460882463048970358421717432178503450051394)*sqrt(57959049919218585566530454195
1571706717845859384545414208024478076585205782332794174344701326^(1/4)*sqrt(1286846088246304897035842171743217
8503450051394)*sqrt(-2*x + 3)*sqrt(3781484028801678888003468129339153727662345024772741260943*sqrt(14) + 14149
022371848728385570789036684124101210161640127797919744)*(8061110911143276053983022787*sqrt(14) - 3029711891221
9360725028693061) - 210380976680132535569563443287236823905478719259451204168457324874865216162080856741370745
650892815340*x + 963732050599621794425456308219340060829468062999882820661390*sqrt(166789371965963959581098742
817586289130679764812156476721038706576007991289033281726) + 3155714650201988033543451649308552358582180788891
76806252685987312297824243121285112056118476339223010)*(30297118912219360725028693061*sqrt(14)*sqrt(7) - 11285
5552756005864755762319018*sqrt(7))*sqrt(3781484028801678888003468129339153727662345024772741260943*sqrt(14) +
14149022371848728385570789036684124101210161640127797919744) - 1/331641722680775415760424069447358035430711841
28057805445740643992848947205475131833297639875732592434272266883677954804521721584006729715127306903510*57959
0499192185855665304541951571706717845859384545414208024478076585205782332794174344701326^(3/4)*sqrt(1286846088
2463048970358421717432178503450051394)*(30297118912219360725028693061*sqrt(14)*sqrt(7) - 112855552756005864755
762319018*sqrt(7))*sqrt(-2*x + 3)*sqrt(3781484028801678888003468129339153727662345024772741260943*sqrt(14) + 1
4149022371848728385570789036684124101210161640127797919744) + 2/7*sqrt(14)*sqrt(7) + sqrt(7)) + 61652531653785
8546962128448983043227187951381815778781478549978900*579590499192185855665304541951571706717845859384545414208
024478076585205782332794174344701326^(1/4)*sqrt(12868460882463048970358421717432178503450051394)*sqrt(14)*sqrt
(7)*(549755813888*x^58 - 11269994184704*x^57 + 107064944754688*x^56 - 630638638006272*x^55 + 2618521301286912*
x^54 - 8342252417974272*x^53 + 21849572376576000*x^52 - 49684091485814784*x^51 + 101394501297242112*x^50 - 188
583312363618304*x^49 + 323261995581177856*x^48 - 517079841212727296*x^47 + 778117896260812800*x^46 - 110564116
5387988992*x^45 + 1491287028233404416*x^44 - 1919929663119949824*x^43 + 2363050939901804544*x^42 - 27862740206
45928960*x^41 + 3161145685194047488*x^40 - 3453753931369283584*x^39 + 3634098467102523392*x^38 - 3697893960325
791744*x^37 + 3640651752731836416*x^36 - 3461798212247617536*x^35 + 3194540251789393920*x^34 - 286154457949529
7024*x^33 + 2477632938217930752*x^32 - 20884302...
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(3-2*x)**(41/2)/(2*x**2+x+1)**20,x)
[Out]
Exception raised: SystemError >> excessive stack use: stack is 7262 deep
________________________________________________________________________________________
Giac [A]
time = 2.53, size = 1410, normalized size = 1.33 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(3-2*x)^(41/2)/(2*x^2+x+1)^20,x, algorithm="giac")
[Out]
-115/363805261691069042491598265308929913400590336*sqrt(7)*(24183332733429828161949068361*14^(3/4)*sqrt(7)*(sq
rt(14) + 4)*sqrt(-2*sqrt(14) + 8) + 8061110911143276053983022787*14^(3/4)*sqrt(7)*(sqrt(14) - 4)*sqrt(-2*sqrt(
14) + 8) - 56427776378002932377881159509*14^(3/4)*sqrt(2*sqrt(14) + 8)*(sqrt(14) + 4) - 1692833291340087971336
43478527*14^(3/4)*sqrt(2*sqrt(14) + 8)*(sqrt(14) - 4) + 242376951297754885800229544488*14^(1/4)*sqrt(7)*sqrt(-
2*sqrt(14) + 8) - 1696638659084284200601606811416*14^(1/4)*sqrt(2*sqrt(14) + 8))*arctan(1/28*14^(3/4)*(14^(1/4
)*sqrt(1/2)*sqrt(sqrt(14) + 4) + 2*sqrt(-2*x + 3))/sqrt(-1/8*sqrt(14) + 1/2)) - 115/36380526169106904249159826
5308929913400590336*sqrt(7)*(24183332733429828161949068361*14^(3/4)*sqrt(7)*(sqrt(14) + 4)*sqrt(-2*sqrt(14) +
8) + 8061110911143276053983022787*14^(3/4)*sqrt(7)*(sqrt(14) - 4)*sqrt(-2*sqrt(14) + 8) - 56427776378002932377
881159509*14^(3/4)*sqrt(2*sqrt(14) + 8)*(sqrt(14) + 4) - 169283329134008797133643478527*14^(3/4)*sqrt(2*sqrt(1
4) + 8)*(sqrt(14) - 4) + 242376951297754885800229544488*14^(1/4)*sqrt(7)*sqrt(-2*sqrt(14) + 8) - 1696638659084
284200601606811416*14^(1/4)*sqrt(2*sqrt(14) + 8))*arctan(-1/28*14^(3/4)*(14^(1/4)*sqrt(1/2)*sqrt(sqrt(14) + 4)
- 2*sqrt(-2*x + 3))/sqrt(-1/8*sqrt(14) + 1/2)) - 115/727610523382138084983196530617859826801180672*sqrt(7)*(8
061110911143276053983022787*14^(3/4)*sqrt(7)*sqrt(2*sqrt(14) + 8)*(sqrt(14) + 4) + 241833327334298281619490683
61*14^(3/4)*sqrt(7)*sqrt(2*sqrt(14) + 8)*(sqrt(14) - 4) + 169283329134008797133643478527*14^(3/4)*(sqrt(14) +
4)*sqrt(-2*sqrt(14) + 8) + 56427776378002932377881159509*14^(3/4)*(sqrt(14) - 4)*sqrt(-2*sqrt(14) + 8) + 24237
6951297754885800229544488*14^(1/4)*sqrt(7)*sqrt(2*sqrt(14) + 8) + 1696638659084284200601606811416*14^(1/4)*sqr
t(-2*sqrt(14) + 8))*log(14^(1/4)*sqrt(1/2)*sqrt(-2*x + 3)*sqrt(sqrt(14) + 4) - 2*x + sqrt(14) + 3) + 115/72761
0523382138084983196530617859826801180672*sqrt(7)*(8061110911143276053983022787*14^(3/4)*sqrt(7)*sqrt(2*sqrt(14
) + 8)*(sqrt(14) + 4) + 24183332733429828161949068361*14^(3/4)*sqrt(7)*sqrt(2*sqrt(14) + 8)*(sqrt(14) - 4) + 1
69283329134008797133643478527*14^(3/4)*(sqrt(14) + 4)*sqrt(-2*sqrt(14) + 8) + 56427776378002932377881159509*14
^(3/4)*(sqrt(14) - 4)*sqrt(-2*sqrt(14) + 8) + 242376951297754885800229544488*14^(1/4)*sqrt(7)*sqrt(2*sqrt(14)
+ 8) + 1696638659084284200601606811416*14^(1/4)*sqrt(-2*sqrt(14) + 8))*log(-14^(1/4)*sqrt(1/2)*sqrt(-2*x + 3)*
sqrt(sqrt(14) + 4) - 2*x + sqrt(14) + 3) + 1/24112597431479447071556104988390860001680293888*(3859127962941386
23132486146144809805*(2*x - 3)^37*sqrt(-2*x + 3) + 49944166626569370884317542782684785215*(2*x - 3)^36*sqrt(-2
*x + 3) + 3157104325190190818790417015768672100251*(2*x - 3)^35*sqrt(-2*x + 3) + 12986266353974282972701016844
8772257537793*(2*x - 3)^34*sqrt(-2*x + 3) + 3907056032933059027385185682832433217956200*(2*x - 3)^33*sqrt(-2*x
+ 3) + 91626342308240062913659469031676941328847688*(2*x - 3)^32*sqrt(-2*x + 3) + 174305183978371665445857016
8808933730174627004*(2*x - 3)^31*sqrt(-2*x + 3) + 27638544507622729125093621837291437830917462708*(2*x - 3)^30
*sqrt(-2*x + 3) + 372498510070445411629537388290851713705080145718*(2*x - 3)^29*sqrt(-2*x + 3) + 4329953516930
687342337472014272666363969651587314*(2*x - 3)^28*sqrt(-2*x + 3) + 4389944456011230862360533115714389672582841
5934650*(2*x - 3)^27*sqrt(-2*x + 3) + 391609357365773780316151578457972453648367489837454*(2*x - 3)^26*sqrt(-2
*x + 3) + 3095031701758849575040626937399363198202032753884252*(2*x - 3)^25*sqrt(-2*x + 3) + 21790719622224681
379416567825910093368668334676797780*(2*x - 3)^24*sqrt(-2*x + 3) + 1372614029241987257940621631160532770991069
68046586092*(2*x - 3)^23*sqrt(-2*x + 3) + 776171183055652545384871388553173912691352168500951876*(2*x - 3)^22*
sqrt(-2*x + 3) + 3950095526376994607880784338655934603802167995433166405*(2*x - 3)^21*sqrt(-2*x + 3) + 1812580
3816832861597832766873339882118924015183338007655*(2*x - 3)^20*sqrt(-2*x + 3) + 750834145086940501444266399776
85085540038804754309758915*(2*x - 3)^19*sqrt(-2*x + 3) + 28093265207334834351777609063127189523561134328427582
0345*(2*x - 3)^18*sqrt(-2*x + 3) + 949449516366891514866641779309597536478490489987954462580*(2*x - 3)^17*sqrt
(-2*x + 3) + 2896666953760570249650513456393600983703549509654469117900*(2*x - 3)^16*sqrt(-2*x + 3) + 79682836
92957988567650795129108295704483768260379820818752*(2*x - 3)^15*sqrt(-2*x + 3) + 19727494578812277658606009712
831861626922226523266435734336*(2*x - 3)^14*sqrt(-2*x + 3) + 4384410337942369584248003032076011666649117227803
5172870400*(2*x - 3)^13*sqrt(-2*x + 3) + 87180772449453719112409715850861698835279004734515297162496*(2*x - 3)
^12*sqrt(-2*x + 3) + 154427451620079851403012035013949923367197814895239131529728*(2*x - 3)^11*sqrt(-2*x + 3)
+ 242351725944359254347670713000225450988365795247877220072960*(2*x - 3)^10*sqrt(-2*x + 3) + 33464609143225917
4045261099248092390902126268663782608549888*(2*x - 3)^9*sqrt(-2*x + 3) + 4030345196682619867089916868903813178
41470126237337802123264*(2*x - 3)^8*sqrt(-2*x +...
________________________________________________________________________________________
Mupad [B]
time = 0.97, size = 2500, normalized size = 2.36 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((3 - 2*x)^(41/2)*(x + 2*x^2 + 1)^20),x)
[Out]
((64356352*(2*x - 3)^2)/38073 - (5767168*x)/1443 - (7517962240*(2*x - 3)^3)/5444439 + (1357449428992*(2*x - 3)
^4)/1181443263 - (34130408095744*(2*x - 3)^5)/34261854627 + (1965832636456960*(2*x - 3)^6)/2158496841501 - (95
52588571922432*(2*x - 3)^7)/10792484207505 + (69571472879183872*(2*x - 3)^8)/75547389452535 - (520483872994611
2*(2*x - 3)^9)/5036492630169 + (325082052781755904*(2*x - 3)^10)/257635969158645 - (461538785202937088*(2*x -
3)^11)/272428464995505 + (17726678744562203264*(2*x - 3)^12)/6992330601551295 - (1432471149647610304*(2*x - 3)
^13)/332968123883395 + (2043463601243388704*(2*x - 3)^14)/241114848329355 - (96972768477343976816*(2*x - 3)^15
)/4840844262612435 + (10833870670122545927656*(2*x - 3)^16)/181389282075536535 - (44340157049832305729324*(2*x
- 3)^17)/181389282075536535 + (691509778132186261807282*(2*x - 3)^18)/423241658176251915 - (13577358331537082
239703407*(2*x - 3)^19)/423241658176251915 + (5094959438589599396407530394650672614981*(2*x - 3)^20)/203594616
979243053623625646080 + (47475340273724148225749886260884632526403*(2*x - 3)^21)/20359461697924305362362564608
0 + (547362406727667345868176230754600752341499*(2*x - 3)^22)/518240843219891409223774371840 + (13632173991688
46741803250531443496167647559*(2*x - 3)^23)/438511482724523500112424468480 + (40035704814224807138997531075224
0020201388159*(2*x - 3)^24)/59856817391897457765345939947520 + (167803532186710618751778512174316524508553291*
(2*x - 3)^25)/14964204347974364441336484986880 + (51108771060698315319124863093548144195799415067*(2*x - 3)^26
)/3351981773946257634859372637061120 + (393987083187206735082003889381221664346090053*(2*x - 3)^27)/2280259710
1675222005846072360960 + (194509919512254900809288150922829785396777195281*(2*x - 3)^28)/116889620835048984189
96786631802880 + (39904941217415859849678112809547525787872838871677*(2*x - 3)^29)/288717363462570990949220629
8055311360 + (298202908298252068565416529654031351573999658954519*(2*x - 3)^30)/297834753887704811716038123378
33738240 + (17278370717837126498790206579435572552986029824411*(2*x - 3)^31)/270758867170640737923671021253033
9840 + (136589909140623157483229616961110867609087469195457*(2*x - 3)^32)/379062414038897033093139429754247577
60 + (12124448510282132213121066777925721516746772830847*(2*x - 3)^33)/6689336718333477054584813466251427840 +
(5268103225464003924284598756770514565895682824129*(2*x - 3)^34)/6458669934942667500978440588104826880 + (617
17610092862026266313005902016039510287732711413*(2*x - 3)^35)/187301428113337357528374777055039979520 + (23627
9120368028123256783334911177879141056326577387*(2*x - 3)^36)/1972268029364372858760322438733412433920 + (10069
18289966448819369741773577875830109223667348001*(2*x - 3)^37)/25639484381736847163884191703534361640960 + (834
3152514122341340412513706840068954518337251868859*(2*x - 3)^38)/717905562688631720588757367698962125946880 + (
6690164526112934310361705118130577674249448391954923*(2*x - 3)^39)/2153716688065895161766272103096886377840640
+ (305581065207833944849384015433140408874881230574613*(2*x - 3)^40)/4074599139584125981719974249102217471590
40 + (731867339371195846981841457176808134814103613309*(2*x - 3)^41)/4477581472070468111780191482529909309440
+ (9815653611211549232290414629069353244130713267641*(2*x - 3)^42)/305594935468809448628998068682666310369280
+ (11199801517259481678687287141859390404145132617*(2*x - 3)^43)/1971580228831028700832245604404298776576 + (1
365647472724278381706307194167071805455474221*(2*x - 3)^44)/1514223059760507936375862611533082132480 + (403050
1165904934786218654181916754194500565501*(2*x - 3)^45)/31462190241690553789142923150742928752640 + (1428009628
445556490988667295522054915842433631*(2*x - 3)^46)/88094132676733550609600184822080200507392 + (16008905392663
3694221849846408842457682603621*(2*x - 3)^47)/88094132676733550609600184822080200507392 + (2100199814096720892
415827167854475800682460389*(2*x - 3)^48)/11716519646005562231076824581336666667483136 + (73152102949146076476
2993572365861799703833*(2*x - 3)^49)/47435302210548834943630868750350877196288 + (1452782501142468088174528794
40670605483477*(2*x - 3)^50)/126959191210586587643247325184762641907712 + (30541762468911990334017682046220545
95917*(2*x - 3)^51)/42319730403528862547749108394920880635904 + (43226241215596960235839037876452347793*(2*x -
3)^52)/11393773570180847609009375337094083248128 + (167572141694212657464927107565976575*(2*x - 3)^53)/103579
7597289167964455397757917643931648 + (935756145095208333386444273642906999*(2*x - 3)^54)/174013996344580218028
506823330164180516864 + (3250015519725523200399609528788299*(2*x - 3)^55)/248591423349400311469295461900234543
59552 + (359910711199433658030176367535945*(2*x - 3)^56)/174013996344580218028506823330164180516864 + (9270277
54781476746208047620505*(2*x - 3)^57)/58004665448193406009502274443388060172288 + 79953920/10101)/(59763039589
48914397184*(3 - 2*x)^(39/2) - 56774887610014686773248*(3 - 2*x)^(41/2) + 263597692475068188590080*(3 - 2*x)^(
43/2) - 796876101097706139353088*(3 - 2*x)^(45/...
________________________________________________________________________________________