3.2.83 \(\int \frac {1}{x (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)
Optimal. Leaf size=194 \[ \frac {1}{3 a^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1}{4 a (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\log (x) (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(a+b x) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1}{2 a^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 194, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used =
{646, 44} \begin {gather*} \frac {1}{2 a^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1}{3 a^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1}{4 a (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\log (x) (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(a+b x) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(x*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
[Out]
1/(a^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + 1/(4*a*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + 1/(3*a^2*(a + b*x)
^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + 1/(2*a^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((a + b*x)*Log[x])/(a^
5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((a + b*x)*Log[a + b*x])/(a^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
Rule 44
Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])
Rule 646
Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p] && NeQ[2*c*d - b*e, 0]
Rubi steps
\begin {align*} \int \frac {1}{x \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{x \left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac {1}{a^5 b^5 x}-\frac {1}{a b^4 (a+b x)^5}-\frac {1}{a^2 b^4 (a+b x)^4}-\frac {1}{a^3 b^4 (a+b x)^3}-\frac {1}{a^4 b^4 (a+b x)^2}-\frac {1}{a^5 b^4 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {1}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1}{4 a (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1}{3 a^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1}{2 a^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) \log (x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(a+b x) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 84, normalized size = 0.43 \begin {gather*} \frac {a \left (25 a^3+52 a^2 b x+42 a b^2 x^2+12 b^3 x^3\right )+12 \log (x) (a+b x)^4-12 (a+b x)^4 \log (a+b x)}{12 a^5 (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/(x*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
[Out]
(a*(25*a^3 + 52*a^2*b*x + 42*a*b^2*x^2 + 12*b^3*x^3) + 12*(a + b*x)^4*Log[x] - 12*(a + b*x)^4*Log[a + b*x])/(1
2*a^5*(a + b*x)^3*Sqrt[(a + b*x)^2])
________________________________________________________________________________________
IntegrateAlgebraic [B] time = 112.50, size = 3884, normalized size = 20.02 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/(x*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
[Out]
(-32*Sqrt[b^2]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(3*a^87 - 3*a^86*b*x + 3*a^85*b^2*x^2 - 3*a^84*b^3*x^3 + 28*a^83*
b^4*x^4 + 4074*a^82*b^5*x^5 + 328392*a^81*b^6*x^6 + 17416344*a^80*b^7*x^7 + 684119040*a^79*b^8*x^8 + 212262124
80*a^78*b^9*x^9 + 541798692864*a^77*b^10*x^10 + 11700119129088*a^76*b^11*x^11 + 218177660073984*a^75*b^12*x^12
+ 3568299926191104*a^74*b^13*x^13 + 51815594461802496*a^73*b^14*x^14 + 674671446755364864*a^72*b^15*x^15 + 79
41063256075862016*a^71*b^16*x^16 + 85067029477553700864*a^70*b^17*x^17 + 834123879538833555456*a^69*b^18*x^18
+ 7523474448566234578944*a^68*b^19*x^19 + 62685534036478928879616*a^67*b^20*x^20 + 484263949539613361700864*a^
66*b^21*x^21 + 3479930411745645615906816*a^65*b^22*x^22 + 23327835134055094829973504*a^64*b^23*x^23 + 14624961
5315137820985655296*a^63*b^24*x^24 + 859432059122570824614150144*a^62*b^25*x^25 + 4743516432354559922458853376
*a^61*b^26*x^26 + 24634603647181064055495327744*a^60*b^27*x^27 + 120573287346692375306185998336*a^59*b^28*x^28
+ 556993306425737520204547620864*a^58*b^29*x^29 + 2431710423525963544007685439488*a^57*b^30*x^30 + 1004499081
1674606081019087945728*a^56*b^31*x^31 + 39302890124942958048947321438208*a^55*b^32*x^32 + 14579796496710667090
0820916043776*a^54*b^33*x^33 + 513218808138628154416518496518144*a^53*b^34*x^34 + 1715578603609902451791538956
533760*a^52*b^35*x^35 + 5449694039783497785024773904924672*a^51*b^36*x^36 + 1646075331496819230532900076126208
0*a^50*b^37*x^37 + 47301435355834570726904893777379328*a^49*b^38*x^38 + 129374002716573907248459868192899072*a
^48*b^39*x^39 + 336930636646549696381527413823111168*a^47*b^40*x^40 + 835796283354984631071335990534602752*a^4
6*b^41*x^41 + 1975361005020021175749282788611719168*a^45*b^42*x^42 + 4449110127870895658771635322027507712*a^4
4*b^43*x^43 + 9551037404260216847866175737338789888*a^43*b^44*x^44 + 19544508416407422206340829051147517952*a^
42*b^45*x^45 + 38125736076483998462868401157222432768*a^41*b^46*x^46 + 70897347674846732883050381793665482752*
a^40*b^47*x^47 + 125671127881529846697796348885994569728*a^39*b^48*x^48 + 212317935948353209887693339475650281
472*a^38*b^49*x^49 + 341830976243852079520987249876704165888*a^37*b^50*x^50 + 52434022418804742515468064506154
0052992*a^36*b^51*x^51 + 766073281583947528164240486507938316288*a^35*b^52*x^52 + 1065700613371013117842709614
376603615232*a^34*b^53*x^53 + 1411020168689744925287295159943475232768*a^33*b^54*x^54 + 1777302527925165461341
561537146824687616*a^32*b^55*x^55 + 2128564412652177688937195991485965664256*a^31*b^56*x^56 + 2422395406390342
874120214563859260768256*a^30*b^57*x^57 + 2617807829735370087962525106611234537472*a^29*b^58*x^58 + 2684295777
372515995379929097395626835968*a^28*b^59*x^59 + 2609450207845959606905876323113652715520*a^27*b^60*x^60 + 2402
576508782798093149416471150968438784*a^26*b^61*x^61 + 2092916507617038227414227202520498831360*a^25*b^62*x^62
+ 1722894322550200785772462871077208457216*a^24*b^63*x^63 + 1338524199162955156171842117301576925184*a^23*b^64
*x^64 + 979986658223088202764220685370425081856*a^22*b^65*x^65 + 675048463780981537495401786462871486464*a^21*
b^66*x^66 + 436701745764302470805819212429950713856*a^20*b^67*x^67 + 264784901454740686099176120664272666624*a
^19*b^68*x^68 + 150133777416926053796721610962266750976*a^18*b^69*x^69 + 7940325266168507439416734339423430246
4*a^17*b^70*x^70 + 39059779966378067682904413566775853056*a^16*b^71*x^71 + 17813285973748092394987873783737483
264*a^15*b^72*x^72 + 7503690680834688849004036334253244416*a^14*b^73*x^73 + 2907226777830056762419313398234742
784*a^13*b^74*x^74 + 1030915112359687157926188398124466176*a^12*b^75*x^75 + 3326713029956054057126503893738455
04*a^11*b^76*x^76 + 97031595508821255286739673941016576*a^10*b^77*x^77 + 25374257060868870577491562299654144*a
^9*b^78*x^78 + 5890738842117298555432264066400256*a^8*b^79*x^79 + 1199289224076687944555612256337920*a^7*b^80*
x^80 + 210809734722622322011205101682688*a^6*b^81*x^81 + 31346364513998779407019781652480*a^5*b^82*x^82 + 3833
971628290179974483895386112*a^4*b^83*x^83 + 370352006987302418412887605248*a^3*b^84*x^84 + 2649240041103498373
4511140864*a^2*b^85*x^85 + 1247611445842297308296773632*a*b^86*x^86 + 29014219670751100192948224*b^87*x^87) -
32*(3*a^88*b - 25*a^84*b^5*x^4 - 4102*a^83*b^6*x^5 - 332466*a^82*b^7*x^6 - 17744736*a^81*b^8*x^7 - 701535384*a
^80*b^9*x^8 - 21910331520*a^79*b^10*x^9 - 563024905344*a^78*b^11*x^10 - 12241917821952*a^77*b^12*x^11 - 229877
779203072*a^76*b^13*x^12 - 3786477586265088*a^75*b^14*x^13 - 55383894387993600*a^74*b^15*x^14 - 72648704121716
7360*a^73*b^16*x^15 - 8615734702831226880*a^72*b^17*x^16 - 93008092733629562880*a^71*b^18*x^17 - 9191909090163
87256320*a^70*b^19*x^18 - 8357598328105068134400*a^69*b^20*x^19 - 70209008485045163458560*a^68*b^21*x^20 - 546
949483576092290580480*a^67*b^22*x^21 - 3964194361285258977607680*a^66*b^23*x^22 - 26807765545800740445880320*a
^65*b^24*x^23 - 169577450449192915815628800*a^64*b^25*x^24 - 1005681674437708645599805440*a^63*b^26*x^25 - 560
2948491477130747073003520*a^62*b^27*x^26 - 29378120079535623977954181120*a^61*b^28*x^27 - 14520789099387343936
1681326080*a^60*b^29*x^28 - 677566593772429895510733619200*a^59*b^30*x^29 - 2988703729951701064212233060352*a^
58*b^31*x^30 - 12476701235200569625026773385216*a^57*b^32*x^31 - 49347880936617564129966409383936*a^56*b^33*x^
32 - 185100855092049628949768237481984*a^55*b^34*x^33 - 659016773105734825317339412561920*a^54*b^35*x^34 - 222
8797411748530606208057453051904*a^53*b^36*x^35 - 7165272643393400236816312861458432*a^52*b^37*x^36 - 219104473
54751690090353774666186752*a^51*b^38*x^37 - 63762188670802763032233894538641408*a^50*b^39*x^38 - 1766754380724
08477975364761970278400*a^49*b^40*x^39 - 466304639363123603629987282016010240*a^48*b^41*x^40 - 117272692000153
4327452863404357713920*a^47*b^42*x^41 - 2811157288375005806820618779146321920*a^46*b^43*x^42 - 642447113289091
6834520918110639226880*a^45*b^44*x^43 - 14000147532131112506637811059366297600*a^44*b^45*x^44 - 29095545820667
639054207004788486307840*a^43*b^46*x^45 - 57670244492891420669209230208369950720*a^42*b^47*x^46 - 109023083751
330731345918782950887915520*a^41*b^48*x^47 - 196568475556376579580846730679660052480*a^40*b^49*x^48 - 33798906
3829883056585489688361644851200*a^39*b^50*x^49 - 554148912192205289408680589352354447360*a^38*b^51*x^50 - 8661
71200431899504675667894938244218880*a^37*b^52*x^51 - 1290413505771994953318921131569478369280*a^36*b^53*x^52 -
1831773894954960646006950100884541931520*a^35*b^54*x^53 - 2476720782060758043130004774320078848000*a^34*b^55*
x^54 - 3188322696614910386628856697090299920384*a^33*b^56*x^55 - 3905866940577343150278757528632790351872*a^32
*b^57*x^56 - 4550959819042520563057410555345226432512*a^31*b^58*x^57 - 504020323612571296208273967047049530572
8*a^30*b^59*x^58 - 5302103607107886083342454204006861373440*a^29*b^60*x^59 - 529374598521847560228580542050927
9551488*a^28*b^61*x^60 - 5012026716628757700055292794264621154304*a^27*b^62*x^61 - 449549301639983632056364367
3671467270144*a^26*b^63*x^62 - 3815810830167239013186690073597707288576*a^25*b^64*x^63 - 306141852171315594194
4304988378785382400*a^24*b^65*x^64 - 2318510857386043358936062802672002007040*a^23*b^66*x^65 - 165503512200406
9740259622471833296568320*a^22*b^67*x^66 - 1111750209545284008301220998892822200320*a^21*b^68*x^67 - 701486647
219043156904995333094223380480*a^20*b^69*x^68 - 414918678871666739895897731626539417600*a^19*b^70*x^69 - 22953
7030078611128190888954356501053440*a^18*b^71*x^70 - 118463032628063142077071756961010155520*a^17*b^72*x^71 - 5
6873065940126160077892287350513336320*a^16*b^73*x^72 - 25316976654582781243991910117990727680*a^15*b^74*x^73 -
10410917458664745611423349732487987200*a^14*b^75*x^74 - 3938141890189743920345501796359208960*a^13*b^76*x^75
- 1363586415355292563638838787498311680*a^12*b^77*x^76 - 429702898504426660999390063314862080*a^11*b^78*x^77 -
122405852569690125864231236240670720*a^10*b^79*x^78 - 31264995902986169132923826366054400*a^9*b^80*x^79 - 709
0028066193986499987876322738176*a^8*b^81*x^80 - 1410098958799310266566817358020608*a^7*b^82*x^81 - 24215609923
6621101418224883335168*a^6*b^83*x^82 - 35180336142288959381503677038592*a^5*b^84*x^83 - 4204323635277482392896
782991360*a^4*b^85*x^84 - 396844407398337402147398746112*a^3*b^86*x^85 - 27740011856877281042807914496*a^2*b^8
7*x^86 - 1276625665513048408489721856*a*b^88*x^87 - 29014219670751100192948224*b^89*x^88))/(3*a^4*b*Sqrt[a^2 +
2*a*b*x + b^2*x^2]*(128*a^84*b^4*x^4 + 21120*a^83*b^5*x^5 + 1721472*a^82*b^6*x^6 + 92406656*a^81*b^7*x^7 + 36
74439936*a^80*b^8*x^8 + 115431837696*a^79*b^9*x^9 + 2983781007360*a^78*b^10*x^10 + 65264862633984*a^77*b^11*x^
11 + 1232951257055232*a^76*b^12*x^12 + 20432990676713472*a^75*b^13*x^13 + 300716130445885440*a^74*b^14*x^14 +
3969238746518323200*a^73*b^15*x^15 + 47370458487200808960*a^72*b^16*x^16 + 514638307833890734080*a^71*b^17*x^1
7 + 5118983772087653498880*a^70*b^18*x^18 + 46847576245279590973440*a^69*b^19*x^19 + 396147929845363900416000*
a^68*b^20*x^20 + 3106721959632715288412160*a^67*b^21*x^21 + 22669023436505501851975680*a^66*b^22*x^22 + 154345
386528573397590343680*a^65*b^23*x^23 + 983080641935232659210895360*a^64*b^24*x^24 + 58708746500911414716137472
00*a^63*b^25*x^25 + 32939251658045736840163491840*a^62*b^26*x^26 + 173944179132957033591283384320*a^61*b^27*x^
27 + 865962850098397774705978245120*a^60*b^28*x^28 + 4070229396534870045831701987328*a^59*b^29*x^29 + 18086007
219228693734214252625920*a^58*b^30*x^30 + 76065544328296704454991881961472*a^57*b^31*x^31 + 303124377178837558
835291835334656*a^56*b^32*x^32 + 1145674020859022985294059076059136*a^55*b^33*x^33 + 4110416795701119981416309
783003136*a^54*b^34*x^34 + 14009851676518597644792572002959360*a^53*b^35*x^35 + 453948229576581246380633325242
94144*a^52*b^36*x^36 + 139917789751193392435587350802726912*a^51*b^37*x^37 + 410459873858094378326686815750193
152*a^50*b^38*x^38 + 1146588188790904910371747976404008960*a^49*b^39*x^39 + 3051146630923992568193710004423884
800*a^48*b^40*x^40 + 7737324336869628583877642484858224640*a^47*b^41*x^41 + 1870330501999113763357865406860623
8720*a^46*b^42*x^42 + 43107103684238284774282301309630545920*a^45*b^43*x^43 + 94746125967065558286369665929351
004160*a^44*b^44*x^44 + 198615000002424286648098587431403520000*a^43*b^45*x^45 + 39713069250484736115287312556
9091338240*a^42*b^46*x^46 + 757418302509245017132523963169657323520*a^41*b^47*x^47 + 1377864478653181794400277
391407075819520*a^40*b^48*x^48 + 2390618413362513124890580419918786723840*a^39*b^49*x^49 + 3955385015068042115
189991389350644940800*a^38*b^50*x^50 + 6239651737215173635140320005662399528960*a^37*b^51*x^51 + 9382518809131
986833287646369303054254080*a^36*b^52*x^52 + 13444227011136623928559678656620034785280*a^35*b^53*x^53 + 183507
48809159245654215695088179610648576*a^34*b^54*x^54 + 23850139665038281564949273182080325386240*a^33*b^55*x^55
+ 29501011442370924678193433541945719783424*a^32*b^56*x^56 + 34709779639129526516799866639524772708352*a^31*b^
57*x^57 + 38820803891602154416762573083734716710912*a^30*b^58*x^58 + 41244886350924311041670535470940728328192
*a^29*b^59*x^59 + 41593718455288022589388471161144339333120*a^28*b^60*x^60 + 397793377717148416254162393654584
07456768*a^27*b^61*x^61 + 36044348394238624400918223255712630308864*a^26*b^62*x^62 + 3090991701905853946041038
8414124412370944*a^25*b^63*x^63 + 25056451871195099922047049138792730460160*a^24*b^64*x^64 + 19174574965327876
626062519403677535436800*a^23*b^65*x^65 + 13831731297442951886921500151546055229440*a^22*b^66*x^66 + 938990620
8869529129738482371217451909120*a^21*b^67*x^67 + 5988100438665955998606600464981368504320*a^20*b^68*x^68 + 357
9959214285221715179046877486708162560*a^19*b^69*x^69 + 2001897592698205886195342610222022656000*a^18*b^70*x^70
+ 1044415366922352700482947391151763619840*a^17*b^71*x^71 + 506902354822510510083032763037195960320*a^16*b^72
*x^72 + 228129429525827274249013529575497400320*a^15*b^73*x^73 + 94849133060508440059978288214562570240*a^14*b
^74*x^74 + 36277068986647020067202576304360652800*a^13*b^75*x^75 + 12701061310210824422967616769422786560*a^12
*b^76*x^76 + 4047255710028362687341349707846778880*a^11*b^77*x^77 + 1165860312209840554524188084357038080*a^10
*b^78*x^78 + 301141170995507566848412970190372864*a^9*b^79*x^79 + 69062315585083645783933993101557760*a^8*b^80
*x^80 + 13891106442322941803683011662708736*a^7*b^81*x^81 + 2412615694461848378203398972899328*a^6*b^82*x^82 +
354493762970299331082628280352768*a^5*b^83*x^83 + 42847967496007858756144393617408*a^4*b^84*x^84 + 4090618117
313628445869793607680*a^3*b^85*x^85 + 289213741678046966723307896832*a^2*b^86*x^86 + 1346259792722851048952797
5936*a*b^87*x^87 + 309485009821345068724781056*b^88*x^88) + 3*a^4*b*Sqrt[b^2]*(-128*a^85*b^3*x^4 - 21248*a^84*
b^4*x^5 - 1742592*a^83*b^5*x^6 - 94128128*a^82*b^6*x^7 - 3766846592*a^81*b^7*x^8 - 119106277632*a^80*b^8*x^9 -
3099212845056*a^79*b^9*x^10 - 68248643641344*a^78*b^10*x^11 - 1298216119689216*a^77*b^11*x^12 - 2166594193376
8704*a^76*b^12*x^13 - 321149121122598912*a^75*b^13*x^14 - 4269954876964208640*a^74*b^14*x^15 - 513396972337191
32160*a^73*b^15*x^16 - 562008766321091543040*a^72*b^16*x^17 - 5633622079921544232960*a^71*b^17*x^18 - 51966560
017367244472320*a^70*b^18*x^19 - 442995506090643491389440*a^69*b^19*x^20 - 3502869889478079188828160*a^68*b^20
*x^21 - 25775745396138217140387840*a^67*b^21*x^22 - 177014409965078899442319360*a^66*b^22*x^23 - 1137426028463
806056801239040*a^65*b^23*x^24 - 6853955292026374130824642560*a^64*b^24*x^25 - 38810126308136878311777239040*a
^63*b^25*x^26 - 206883430791002770431446876160*a^62*b^26*x^27 - 1039907029231354808297261629440*a^61*b^27*x^28
- 4936192246633267820537680232448*a^60*b^28*x^29 - 22156236615763563780045954613248*a^59*b^29*x^30 - 94151551
547525398189206134587392*a^58*b^30*x^31 - 379189921507134263290283717296128*a^57*b^31*x^32 - 14487983980378605
44129350911393792*a^56*b^32*x^33 - 5256090816560142966710368859062272*a^55*b^33*x^34 - 18120268472219717626208
881785962496*a^54*b^34*x^35 - 59404674634176722282855904527253504*a^53*b^35*x^36 - 185312612708851517073650683
327021056*a^52*b^36*x^37 - 550377663609287770762274166552920064*a^51*b^37*x^38 - 15570480626489992886984347921
54202112*a^50*b^38*x^39 - 4197734819714897478565457980827893760*a^49*b^39*x^40 - 10788470967793621152071352489
282109440*a^48*b^40*x^41 - 26440629356860766217456296553464463360*a^47*b^41*x^42 - 618104087042294224078609553
78236784640*a^46*b^42*x^43 - 137853229651303843060651967238981550080*a^45*b^43*x^44 - 293361125969489844934468
253360754524160*a^44*b^44*x^45 - 595745692507271647800971713000494858240*a^43*b^45*x^46 - 11545489950140923782
85397088738748661760*a^42*b^46*x^47 - 2135282781162426811532801354576733143040*a^41*b^47*x^48 - 37684828920156
94919290857811325862543360*a^40*b^48*x^49 - 6346003428430555240080571809269431664640*a^39*b^49*x^50 - 10195036
752283215750330311395013044469760*a^38*b^50*x^51 - 15622170546347160468427966374965453783040*a^37*b^51*x^52 -
22826745820268610761847325025923089039360*a^36*b^52*x^53 - 31794975820295869582775373744799645433856*a^35*b^53
*x^54 - 42200888474197527219164968270259936034816*a^34*b^54*x^55 - 53351151107409206243142706724026045169664*a
^33*b^55*x^56 - 64210791081500451194993300181470492491776*a^32*b^56*x^57 - 73530583530731680933562439723259489
419264*a^31*b^57*x^58 - 80065690242526465458433108554675445039104*a^30*b^58*x^59 - 828386048062123336310590066
32085067661312*a^29*b^59*x^60 - 81373056227002864214804710526602746789888*a^28*b^60*x^61 - 7582368616595346602
6334462621171037765632*a^27*b^61*x^62 - 66954265413297163861328611669837042679808*a^26*b^62*x^63 - 55966368890
253639382457437552917142831104*a^25*b^63*x^64 - 44231026836522976548109568542470265896960*a^24*b^64*x^65 - 330
06306262770828512984019555223590666240*a^23*b^65*x^66 - 23221637506312481016659982522763507138560*a^22*b^66*x^
67 - 15378006647535485128345082836198820413440*a^21*b^67*x^68 - 9568059652951177713785647342468076666880*a^20*
b^68*x^69 - 5581856806983427601374389487708730818560*a^19*b^69*x^70 - 3046312959620558586678290001373786275840
*a^18*b^70*x^71 - 1551317721744863210565980154188959580160*a^17*b^71*x^72 - 7350317843483377843320462926126933
60640*a^16*b^72*x^73 - 322978562586335714308991817790059970560*a^15*b^73*x^74 - 131126202047155460127180864518
923223040*a^14*b^74*x^75 - 48978130296857844490170193073783439360*a^13*b^75*x^76 - 167483170202391871103089664
77269565440*a^12*b^76*x^77 - 5213116022238203241865537792203816960*a^11*b^77*x^78 - 14670014832053481213726010
54547410944*a^10*b^78*x^79 - 370203486580591212632346963291930624*a^9*b^79*x^80 - 8295342202740658758761700476
4266496*a^8*b^80*x^81 - 16303722136784790181886410635608064*a^7*b^81*x^82 - 2767109457432147709286027253252096
*a^6*b^82*x^83 - 397341730466307189838772673970176*a^5*b^83*x^84 - 46938585613321487202014187225088*a^4*b^84*x
^85 - 4379831858991675412593101504512*a^3*b^85*x^86 - 302676339605275477212835872768*a^2*b^86*x^87 - 137720829
37049855558252756992*a*b^87*x^88 - 309485009821345068724781056*b^88*x^89)) + (2*ArcTanh[(Sqrt[b^2]*x)/a - Sqrt
[a^2 + 2*a*b*x + b^2*x^2]/a])/a^5
________________________________________________________________________________________
fricas [A] time = 0.42, size = 168, normalized size = 0.87 \begin {gather*} \frac {12 \, a b^{3} x^{3} + 42 \, a^{2} b^{2} x^{2} + 52 \, a^{3} b x + 25 \, a^{4} - 12 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \log \left (b x + a\right ) + 12 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \log \relax (x)}{12 \, {\left (a^{5} b^{4} x^{4} + 4 \, a^{6} b^{3} x^{3} + 6 \, a^{7} b^{2} x^{2} + 4 \, a^{8} b x + a^{9}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")
[Out]
1/12*(12*a*b^3*x^3 + 42*a^2*b^2*x^2 + 52*a^3*b*x + 25*a^4 - 12*(b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*
b*x + a^4)*log(b*x + a) + 12*(b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a^4)*log(x))/(a^5*b^4*x^4 +
4*a^6*b^3*x^3 + 6*a^7*b^2*x^2 + 4*a^8*b*x + a^9)
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
[Out]
sage0*x
________________________________________________________________________________________
maple [A] time = 0.06, size = 173, normalized size = 0.89 \begin {gather*} -\frac {\left (-12 b^{4} x^{4} \ln \relax (x )+12 b^{4} x^{4} \ln \left (b x +a \right )-48 a \,b^{3} x^{3} \ln \relax (x )+48 a \,b^{3} x^{3} \ln \left (b x +a \right )-72 a^{2} b^{2} x^{2} \ln \relax (x )+72 a^{2} b^{2} x^{2} \ln \left (b x +a \right )-12 a \,b^{3} x^{3}-48 a^{3} b x \ln \relax (x )+48 a^{3} b x \ln \left (b x +a \right )-42 a^{2} b^{2} x^{2}-12 a^{4} \ln \relax (x )+12 a^{4} \ln \left (b x +a \right )-52 a^{3} b x -25 a^{4}\right ) \left (b x +a \right )}{12 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
[Out]
-1/12*(12*b^4*x^4*ln(b*x+a)-12*b^4*x^4*ln(x)+48*a*b^3*x^3*ln(b*x+a)-48*a*b^3*x^3*ln(x)+72*a^2*b^2*x^2*ln(b*x+a
)-72*a^2*b^2*x^2*ln(x)-12*a*b^3*x^3+48*a^3*b*x*ln(b*x+a)-48*ln(x)*x*a^3*b-42*a^2*b^2*x^2+12*a^4*ln(b*x+a)-12*l
n(x)*a^4-52*a^3*b*x-25*a^4)*(b*x+a)/a^5/((b*x+a)^2)^(5/2)
________________________________________________________________________________________
maxima [A] time = 1.34, size = 118, normalized size = 0.61 \begin {gather*} -\frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{5}} + \frac {1}{3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2}} + \frac {1}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{4}} + \frac {1}{2 \, a^{3} b^{2} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {1}{4 \, a b^{4} {\left (x + \frac {a}{b}\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")
[Out]
-(-1)^(2*a*b*x + 2*a^2)*log(2*a*b*x/abs(x) + 2*a^2/abs(x))/a^5 + 1/3/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2) + 1
/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^4) + 1/2/(a^3*b^2*(x + a/b)^2) + 1/4/(a*b^4*(x + a/b)^4)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)),x)
[Out]
int(1/(x*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
Integral(1/(x*((a + b*x)**2)**(5/2)), x)
________________________________________________________________________________________