3.5.79 \(\int \frac {1}{x (a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\)
Optimal. Leaf size=223 \[ \frac {1}{6 a^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\log (x) \left (a+b x^2\right )}{a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{2 a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 223, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used =
{1112, 266, 44} \begin {gather*} \frac {1}{4 a^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{6 a^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{2 a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\log (x) \left (a+b x^2\right )}{a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]
[Out]
1/(2*a^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(8*a*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(6*a^2*(
a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(4*a^3*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + ((a +
b*x^2)*Log[x])/(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - ((a + b*x^2)*Log[a + b*x^2])/(2*a^5*Sqrt[a^2 + 2*a*b*x^
2 + b^2*x^4])
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 266
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Rule 1112
Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2 + c*x^4)^FracPa
rt[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c,
d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]
Rubi steps
\begin {align*} \int \frac {1}{x \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{x \left (a b+b^2 x^2\right )^5} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {\left (b^4 \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (a b+b^2 x\right )^5} \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {\left (b^4 \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\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,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1}{2 a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{6 a^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{4 a^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a+b x^2\right ) \log (x)}{a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 96, normalized size = 0.43 \begin {gather*} \frac {a \left (25 a^3+52 a^2 b x^2+42 a b^2 x^4+12 b^3 x^6\right )+24 \log (x) \left (a+b x^2\right )^4-12 \left (a+b x^2\right )^4 \log \left (a+b x^2\right )}{24 a^5 \left (a+b x^2\right )^3 \sqrt {\left (a+b x^2\right )^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/(x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]
[Out]
(a*(25*a^3 + 52*a^2*b*x^2 + 42*a*b^2*x^4 + 12*b^3*x^6) + 24*(a + b*x^2)^4*Log[x] - 12*(a + b*x^2)^4*Log[a + b*
x^2])/(24*a^5*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])
________________________________________________________________________________________
IntegrateAlgebraic [B] time = 112.39, size = 3893, normalized size = 17.46 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/(x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]
[Out]
(-16*Sqrt[b^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*(3*a^87 - 3*a^86*b*x^2 + 3*a^85*b^2*x^4 - 3*a^84*b^3*x^6 + 28*a
^83*b^4*x^8 + 4074*a^82*b^5*x^10 + 328392*a^81*b^6*x^12 + 17416344*a^80*b^7*x^14 + 684119040*a^79*b^8*x^16 + 2
1226212480*a^78*b^9*x^18 + 541798692864*a^77*b^10*x^20 + 11700119129088*a^76*b^11*x^22 + 218177660073984*a^75*
b^12*x^24 + 3568299926191104*a^74*b^13*x^26 + 51815594461802496*a^73*b^14*x^28 + 674671446755364864*a^72*b^15*
x^30 + 7941063256075862016*a^71*b^16*x^32 + 85067029477553700864*a^70*b^17*x^34 + 834123879538833555456*a^69*b
^18*x^36 + 7523474448566234578944*a^68*b^19*x^38 + 62685534036478928879616*a^67*b^20*x^40 + 484263949539613361
700864*a^66*b^21*x^42 + 3479930411745645615906816*a^65*b^22*x^44 + 23327835134055094829973504*a^64*b^23*x^46 +
146249615315137820985655296*a^63*b^24*x^48 + 859432059122570824614150144*a^62*b^25*x^50 + 4743516432354559922
458853376*a^61*b^26*x^52 + 24634603647181064055495327744*a^60*b^27*x^54 + 120573287346692375306185998336*a^59*
b^28*x^56 + 556993306425737520204547620864*a^58*b^29*x^58 + 2431710423525963544007685439488*a^57*b^30*x^60 + 1
0044990811674606081019087945728*a^56*b^31*x^62 + 39302890124942958048947321438208*a^55*b^32*x^64 + 14579796496
7106670900820916043776*a^54*b^33*x^66 + 513218808138628154416518496518144*a^53*b^34*x^68 + 1715578603609902451
791538956533760*a^52*b^35*x^70 + 5449694039783497785024773904924672*a^51*b^36*x^72 + 1646075331496819230532900
0761262080*a^50*b^37*x^74 + 47301435355834570726904893777379328*a^49*b^38*x^76 + 12937400271657390724845986819
2899072*a^48*b^39*x^78 + 336930636646549696381527413823111168*a^47*b^40*x^80 + 8357962833549846310713359905346
02752*a^46*b^41*x^82 + 1975361005020021175749282788611719168*a^45*b^42*x^84 + 44491101278708956587716353220275
07712*a^44*b^43*x^86 + 9551037404260216847866175737338789888*a^43*b^44*x^88 + 19544508416407422206340829051147
517952*a^42*b^45*x^90 + 38125736076483998462868401157222432768*a^41*b^46*x^92 + 708973476748467328830503817936
65482752*a^40*b^47*x^94 + 125671127881529846697796348885994569728*a^39*b^48*x^96 + 212317935948353209887693339
475650281472*a^38*b^49*x^98 + 341830976243852079520987249876704165888*a^37*b^50*x^100 + 5243402241880474251546
80645061540052992*a^36*b^51*x^102 + 766073281583947528164240486507938316288*a^35*b^52*x^104 + 1065700613371013
117842709614376603615232*a^34*b^53*x^106 + 1411020168689744925287295159943475232768*a^33*b^54*x^108 + 17773025
27925165461341561537146824687616*a^32*b^55*x^110 + 2128564412652177688937195991485965664256*a^31*b^56*x^112 +
2422395406390342874120214563859260768256*a^30*b^57*x^114 + 2617807829735370087962525106611234537472*a^29*b^58*
x^116 + 2684295777372515995379929097395626835968*a^28*b^59*x^118 + 2609450207845959606905876323113652715520*a^
27*b^60*x^120 + 2402576508782798093149416471150968438784*a^26*b^61*x^122 + 20929165076170382274142272025204988
31360*a^25*b^62*x^124 + 1722894322550200785772462871077208457216*a^24*b^63*x^126 + 133852419916295515617184211
7301576925184*a^23*b^64*x^128 + 979986658223088202764220685370425081856*a^22*b^65*x^130 + 67504846378098153749
5401786462871486464*a^21*b^66*x^132 + 436701745764302470805819212429950713856*a^20*b^67*x^134 + 26478490145474
0686099176120664272666624*a^19*b^68*x^136 + 150133777416926053796721610962266750976*a^18*b^69*x^138 + 79403252
661685074394167343394234302464*a^17*b^70*x^140 + 39059779966378067682904413566775853056*a^16*b^71*x^142 + 1781
3285973748092394987873783737483264*a^15*b^72*x^144 + 7503690680834688849004036334253244416*a^14*b^73*x^146 + 2
907226777830056762419313398234742784*a^13*b^74*x^148 + 1030915112359687157926188398124466176*a^12*b^75*x^150 +
332671302995605405712650389373845504*a^11*b^76*x^152 + 97031595508821255286739673941016576*a^10*b^77*x^154 +
25374257060868870577491562299654144*a^9*b^78*x^156 + 5890738842117298555432264066400256*a^8*b^79*x^158 + 11992
89224076687944555612256337920*a^7*b^80*x^160 + 210809734722622322011205101682688*a^6*b^81*x^162 + 313463645139
98779407019781652480*a^5*b^82*x^164 + 3833971628290179974483895386112*a^4*b^83*x^166 + 37035200698730241841288
7605248*a^3*b^84*x^168 + 26492400411034983734511140864*a^2*b^85*x^170 + 1247611445842297308296773632*a*b^86*x^
172 + 29014219670751100192948224*b^87*x^174) - 16*(3*a^88*b - 25*a^84*b^5*x^8 - 4102*a^83*b^6*x^10 - 332466*a^
82*b^7*x^12 - 17744736*a^81*b^8*x^14 - 701535384*a^80*b^9*x^16 - 21910331520*a^79*b^10*x^18 - 563024905344*a^7
8*b^11*x^20 - 12241917821952*a^77*b^12*x^22 - 229877779203072*a^76*b^13*x^24 - 3786477586265088*a^75*b^14*x^26
- 55383894387993600*a^74*b^15*x^28 - 726487041217167360*a^73*b^16*x^30 - 8615734702831226880*a^72*b^17*x^32 -
93008092733629562880*a^71*b^18*x^34 - 919190909016387256320*a^70*b^19*x^36 - 8357598328105068134400*a^69*b^20
*x^38 - 70209008485045163458560*a^68*b^21*x^40 - 546949483576092290580480*a^67*b^22*x^42 - 3964194361285258977
607680*a^66*b^23*x^44 - 26807765545800740445880320*a^65*b^24*x^46 - 169577450449192915815628800*a^64*b^25*x^48
- 1005681674437708645599805440*a^63*b^26*x^50 - 5602948491477130747073003520*a^62*b^27*x^52 - 293781200795356
23977954181120*a^61*b^28*x^54 - 145207890993873439361681326080*a^60*b^29*x^56 - 677566593772429895510733619200
*a^59*b^30*x^58 - 2988703729951701064212233060352*a^58*b^31*x^60 - 12476701235200569625026773385216*a^57*b^32*
x^62 - 49347880936617564129966409383936*a^56*b^33*x^64 - 185100855092049628949768237481984*a^55*b^34*x^66 - 65
9016773105734825317339412561920*a^54*b^35*x^68 - 2228797411748530606208057453051904*a^53*b^36*x^70 - 716527264
3393400236816312861458432*a^52*b^37*x^72 - 21910447354751690090353774666186752*a^51*b^38*x^74 - 63762188670802
763032233894538641408*a^50*b^39*x^76 - 176675438072408477975364761970278400*a^49*b^40*x^78 - 46630463936312360
3629987282016010240*a^48*b^41*x^80 - 1172726920001534327452863404357713920*a^47*b^42*x^82 - 281115728837500580
6820618779146321920*a^46*b^43*x^84 - 6424471132890916834520918110639226880*a^45*b^44*x^86 - 140001475321311125
06637811059366297600*a^44*b^45*x^88 - 29095545820667639054207004788486307840*a^43*b^46*x^90 - 5767024449289142
0669209230208369950720*a^42*b^47*x^92 - 109023083751330731345918782950887915520*a^41*b^48*x^94 - 1965684755563
76579580846730679660052480*a^40*b^49*x^96 - 337989063829883056585489688361644851200*a^39*b^50*x^98 - 554148912
192205289408680589352354447360*a^38*b^51*x^100 - 866171200431899504675667894938244218880*a^37*b^52*x^102 - 129
0413505771994953318921131569478369280*a^36*b^53*x^104 - 1831773894954960646006950100884541931520*a^35*b^54*x^1
06 - 2476720782060758043130004774320078848000*a^34*b^55*x^108 - 3188322696614910386628856697090299920384*a^33*
b^56*x^110 - 3905866940577343150278757528632790351872*a^32*b^57*x^112 - 45509598190425205630574105553452264325
12*a^31*b^58*x^114 - 5040203236125712962082739670470495305728*a^30*b^59*x^116 - 530210360710788608334245420400
6861373440*a^29*b^60*x^118 - 5293745985218475602285805420509279551488*a^28*b^61*x^120 - 5012026716628757700055
292794264621154304*a^27*b^62*x^122 - 4495493016399836320563643673671467270144*a^26*b^63*x^124 - 38158108301672
39013186690073597707288576*a^25*b^64*x^126 - 3061418521713155941944304988378785382400*a^24*b^65*x^128 - 231851
0857386043358936062802672002007040*a^23*b^66*x^130 - 1655035122004069740259622471833296568320*a^22*b^67*x^132
- 1111750209545284008301220998892822200320*a^21*b^68*x^134 - 701486647219043156904995333094223380480*a^20*b^69
*x^136 - 414918678871666739895897731626539417600*a^19*b^70*x^138 - 229537030078611128190888954356501053440*a^1
8*b^71*x^140 - 118463032628063142077071756961010155520*a^17*b^72*x^142 - 5687306594012616007789228735051333632
0*a^16*b^73*x^144 - 25316976654582781243991910117990727680*a^15*b^74*x^146 - 104109174586647456114233497324879
87200*a^14*b^75*x^148 - 3938141890189743920345501796359208960*a^13*b^76*x^150 - 136358641535529256363883878749
8311680*a^12*b^77*x^152 - 429702898504426660999390063314862080*a^11*b^78*x^154 - 12240585256969012586423123624
0670720*a^10*b^79*x^156 - 31264995902986169132923826366054400*a^9*b^80*x^158 - 7090028066193986499987876322738
176*a^8*b^81*x^160 - 1410098958799310266566817358020608*a^7*b^82*x^162 - 242156099236621101418224883335168*a^6
*b^83*x^164 - 35180336142288959381503677038592*a^5*b^84*x^166 - 4204323635277482392896782991360*a^4*b^85*x^168
- 396844407398337402147398746112*a^3*b^86*x^170 - 27740011856877281042807914496*a^2*b^87*x^172 - 127662566551
3048408489721856*a*b^88*x^174 - 29014219670751100192948224*b^89*x^176))/(3*a^4*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^
4]*(128*a^84*b^4*x^8 + 21120*a^83*b^5*x^10 + 1721472*a^82*b^6*x^12 + 92406656*a^81*b^7*x^14 + 3674439936*a^80*
b^8*x^16 + 115431837696*a^79*b^9*x^18 + 2983781007360*a^78*b^10*x^20 + 65264862633984*a^77*b^11*x^22 + 1232951
257055232*a^76*b^12*x^24 + 20432990676713472*a^75*b^13*x^26 + 300716130445885440*a^74*b^14*x^28 + 396923874651
8323200*a^73*b^15*x^30 + 47370458487200808960*a^72*b^16*x^32 + 514638307833890734080*a^71*b^17*x^34 + 51189837
72087653498880*a^70*b^18*x^36 + 46847576245279590973440*a^69*b^19*x^38 + 396147929845363900416000*a^68*b^20*x^
40 + 3106721959632715288412160*a^67*b^21*x^42 + 22669023436505501851975680*a^66*b^22*x^44 + 154345386528573397
590343680*a^65*b^23*x^46 + 983080641935232659210895360*a^64*b^24*x^48 + 5870874650091141471613747200*a^63*b^25
*x^50 + 32939251658045736840163491840*a^62*b^26*x^52 + 173944179132957033591283384320*a^61*b^27*x^54 + 8659628
50098397774705978245120*a^60*b^28*x^56 + 4070229396534870045831701987328*a^59*b^29*x^58 + 18086007219228693734
214252625920*a^58*b^30*x^60 + 76065544328296704454991881961472*a^57*b^31*x^62 + 303124377178837558835291835334
656*a^56*b^32*x^64 + 1145674020859022985294059076059136*a^55*b^33*x^66 + 4110416795701119981416309783003136*a^
54*b^34*x^68 + 14009851676518597644792572002959360*a^53*b^35*x^70 + 45394822957658124638063332524294144*a^52*b
^36*x^72 + 139917789751193392435587350802726912*a^51*b^37*x^74 + 410459873858094378326686815750193152*a^50*b^3
8*x^76 + 1146588188790904910371747976404008960*a^49*b^39*x^78 + 3051146630923992568193710004423884800*a^48*b^4
0*x^80 + 7737324336869628583877642484858224640*a^47*b^41*x^82 + 18703305019991137633578654068606238720*a^46*b^
42*x^84 + 43107103684238284774282301309630545920*a^45*b^43*x^86 + 94746125967065558286369665929351004160*a^44*
b^44*x^88 + 198615000002424286648098587431403520000*a^43*b^45*x^90 + 397130692504847361152873125569091338240*a
^42*b^46*x^92 + 757418302509245017132523963169657323520*a^41*b^47*x^94 + 1377864478653181794400277391407075819
520*a^40*b^48*x^96 + 2390618413362513124890580419918786723840*a^39*b^49*x^98 + 3955385015068042115189991389350
644940800*a^38*b^50*x^100 + 6239651737215173635140320005662399528960*a^37*b^51*x^102 + 93825188091319868332876
46369303054254080*a^36*b^52*x^104 + 13444227011136623928559678656620034785280*a^35*b^53*x^106 + 18350748809159
245654215695088179610648576*a^34*b^54*x^108 + 23850139665038281564949273182080325386240*a^33*b^55*x^110 + 2950
1011442370924678193433541945719783424*a^32*b^56*x^112 + 34709779639129526516799866639524772708352*a^31*b^57*x^
114 + 38820803891602154416762573083734716710912*a^30*b^58*x^116 + 41244886350924311041670535470940728328192*a^
29*b^59*x^118 + 41593718455288022589388471161144339333120*a^28*b^60*x^120 + 3977933777171484162541623936545840
7456768*a^27*b^61*x^122 + 36044348394238624400918223255712630308864*a^26*b^62*x^124 + 309099170190585394604103
88414124412370944*a^25*b^63*x^126 + 25056451871195099922047049138792730460160*a^24*b^64*x^128 + 19174574965327
876626062519403677535436800*a^23*b^65*x^130 + 13831731297442951886921500151546055229440*a^22*b^66*x^132 + 9389
906208869529129738482371217451909120*a^21*b^67*x^134 + 5988100438665955998606600464981368504320*a^20*b^68*x^13
6 + 3579959214285221715179046877486708162560*a^19*b^69*x^138 + 2001897592698205886195342610222022656000*a^18*b
^70*x^140 + 1044415366922352700482947391151763619840*a^17*b^71*x^142 + 506902354822510510083032763037195960320
*a^16*b^72*x^144 + 228129429525827274249013529575497400320*a^15*b^73*x^146 + 948491330605084400599782882145625
70240*a^14*b^74*x^148 + 36277068986647020067202576304360652800*a^13*b^75*x^150 + 12701061310210824422967616769
422786560*a^12*b^76*x^152 + 4047255710028362687341349707846778880*a^11*b^77*x^154 + 11658603122098405545241880
84357038080*a^10*b^78*x^156 + 301141170995507566848412970190372864*a^9*b^79*x^158 + 69062315585083645783933993
101557760*a^8*b^80*x^160 + 13891106442322941803683011662708736*a^7*b^81*x^162 + 241261569446184837820339897289
9328*a^6*b^82*x^164 + 354493762970299331082628280352768*a^5*b^83*x^166 + 42847967496007858756144393617408*a^4*
b^84*x^168 + 4090618117313628445869793607680*a^3*b^85*x^170 + 289213741678046966723307896832*a^2*b^86*x^172 +
13462597927228510489527975936*a*b^87*x^174 + 309485009821345068724781056*b^88*x^176) + 3*a^4*b*Sqrt[b^2]*(-128
*a^85*b^3*x^8 - 21248*a^84*b^4*x^10 - 1742592*a^83*b^5*x^12 - 94128128*a^82*b^6*x^14 - 3766846592*a^81*b^7*x^1
6 - 119106277632*a^80*b^8*x^18 - 3099212845056*a^79*b^9*x^20 - 68248643641344*a^78*b^10*x^22 - 129821611968921
6*a^77*b^11*x^24 - 21665941933768704*a^76*b^12*x^26 - 321149121122598912*a^75*b^13*x^28 - 4269954876964208640*
a^74*b^14*x^30 - 51339697233719132160*a^73*b^15*x^32 - 562008766321091543040*a^72*b^16*x^34 - 5633622079921544
232960*a^71*b^17*x^36 - 51966560017367244472320*a^70*b^18*x^38 - 442995506090643491389440*a^69*b^19*x^40 - 350
2869889478079188828160*a^68*b^20*x^42 - 25775745396138217140387840*a^67*b^21*x^44 - 17701440996507889944231936
0*a^66*b^22*x^46 - 1137426028463806056801239040*a^65*b^23*x^48 - 6853955292026374130824642560*a^64*b^24*x^50 -
38810126308136878311777239040*a^63*b^25*x^52 - 206883430791002770431446876160*a^62*b^26*x^54 - 10399070292313
54808297261629440*a^61*b^27*x^56 - 4936192246633267820537680232448*a^60*b^28*x^58 - 22156236615763563780045954
613248*a^59*b^29*x^60 - 94151551547525398189206134587392*a^58*b^30*x^62 - 379189921507134263290283717296128*a^
57*b^31*x^64 - 1448798398037860544129350911393792*a^56*b^32*x^66 - 5256090816560142966710368859062272*a^55*b^3
3*x^68 - 18120268472219717626208881785962496*a^54*b^34*x^70 - 59404674634176722282855904527253504*a^53*b^35*x^
72 - 185312612708851517073650683327021056*a^52*b^36*x^74 - 550377663609287770762274166552920064*a^51*b^37*x^76
- 1557048062648999288698434792154202112*a^50*b^38*x^78 - 4197734819714897478565457980827893760*a^49*b^39*x^80
- 10788470967793621152071352489282109440*a^48*b^40*x^82 - 26440629356860766217456296553464463360*a^47*b^41*x^
84 - 61810408704229422407860955378236784640*a^46*b^42*x^86 - 137853229651303843060651967238981550080*a^45*b^43
*x^88 - 293361125969489844934468253360754524160*a^44*b^44*x^90 - 595745692507271647800971713000494858240*a^43*
b^45*x^92 - 1154548995014092378285397088738748661760*a^42*b^46*x^94 - 2135282781162426811532801354576733143040
*a^41*b^47*x^96 - 3768482892015694919290857811325862543360*a^40*b^48*x^98 - 6346003428430555240080571809269431
664640*a^39*b^49*x^100 - 10195036752283215750330311395013044469760*a^38*b^50*x^102 - 1562217054634716046842796
6374965453783040*a^37*b^51*x^104 - 22826745820268610761847325025923089039360*a^36*b^52*x^106 - 317949758202958
69582775373744799645433856*a^35*b^53*x^108 - 42200888474197527219164968270259936034816*a^34*b^54*x^110 - 53351
151107409206243142706724026045169664*a^33*b^55*x^112 - 64210791081500451194993300181470492491776*a^32*b^56*x^1
14 - 73530583530731680933562439723259489419264*a^31*b^57*x^116 - 80065690242526465458433108554675445039104*a^3
0*b^58*x^118 - 82838604806212333631059006632085067661312*a^29*b^59*x^120 - 81373056227002864214804710526602746
789888*a^28*b^60*x^122 - 75823686165953466026334462621171037765632*a^27*b^61*x^124 - 6695426541329716386132861
1669837042679808*a^26*b^62*x^126 - 55966368890253639382457437552917142831104*a^25*b^63*x^128 - 442310268365229
76548109568542470265896960*a^24*b^64*x^130 - 33006306262770828512984019555223590666240*a^23*b^65*x^132 - 23221
637506312481016659982522763507138560*a^22*b^66*x^134 - 15378006647535485128345082836198820413440*a^21*b^67*x^1
36 - 9568059652951177713785647342468076666880*a^20*b^68*x^138 - 5581856806983427601374389487708730818560*a^19*
b^69*x^140 - 3046312959620558586678290001373786275840*a^18*b^70*x^142 - 15513177217448632105659801541889595801
60*a^17*b^71*x^144 - 735031784348337784332046292612693360640*a^16*b^72*x^146 - 3229785625863357143089918177900
59970560*a^15*b^73*x^148 - 131126202047155460127180864518923223040*a^14*b^74*x^150 - 4897813029685784449017019
3073783439360*a^13*b^75*x^152 - 16748317020239187110308966477269565440*a^12*b^76*x^154 - 521311602223820324186
5537792203816960*a^11*b^77*x^156 - 1467001483205348121372601054547410944*a^10*b^78*x^158 - 3702034865805912126
32346963291930624*a^9*b^79*x^160 - 82953422027406587587617004764266496*a^8*b^80*x^162 - 1630372213678479018188
6410635608064*a^7*b^81*x^164 - 2767109457432147709286027253252096*a^6*b^82*x^166 - 397341730466307189838772673
970176*a^5*b^83*x^168 - 46938585613321487202014187225088*a^4*b^84*x^170 - 4379831858991675412593101504512*a^3*
b^85*x^172 - 302676339605275477212835872768*a^2*b^86*x^174 - 13772082937049855558252756992*a*b^87*x^176 - 3094
85009821345068724781056*b^88*x^178)) + ArcTanh[(Sqrt[b^2]*x^2)/a - Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/a]/a^5
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fricas [A] time = 1.10, size = 178, normalized size = 0.80 \begin {gather*} \frac {12 \, a b^{3} x^{6} + 42 \, a^{2} b^{2} x^{4} + 52 \, a^{3} b x^{2} + 25 \, a^{4} - 12 \, {\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \log \left (b x^{2} + a\right ) + 24 \, {\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \log \relax (x)}{24 \, {\left (a^{5} b^{4} x^{8} + 4 \, a^{6} b^{3} x^{6} + 6 \, a^{7} b^{2} x^{4} + 4 \, a^{8} b x^{2} + a^{9}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")
[Out]
1/24*(12*a*b^3*x^6 + 42*a^2*b^2*x^4 + 52*a^3*b*x^2 + 25*a^4 - 12*(b^4*x^8 + 4*a*b^3*x^6 + 6*a^2*b^2*x^4 + 4*a^
3*b*x^2 + a^4)*log(b*x^2 + a) + 24*(b^4*x^8 + 4*a*b^3*x^6 + 6*a^2*b^2*x^4 + 4*a^3*b*x^2 + a^4)*log(x))/(a^5*b^
4*x^8 + 4*a^6*b^3*x^6 + 6*a^7*b^2*x^4 + 4*a^8*b*x^2 + a^9)
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giac [A] time = 0.27, size = 101, normalized size = 0.45 \begin {gather*} -\frac {\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{5} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {\log \left ({\left | x \right |}\right )}{a^{5} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {12 \, a b^{3} x^{6} + 42 \, a^{2} b^{2} x^{4} + 52 \, a^{3} b x^{2} + 25 \, a^{4}}{24 \, {\left (b x^{2} + a\right )}^{4} a^{5} \mathrm {sgn}\left (b x^{2} + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")
[Out]
-1/2*log(abs(b*x^2 + a))/(a^5*sgn(b*x^2 + a)) + log(abs(x))/(a^5*sgn(b*x^2 + a)) + 1/24*(12*a*b^3*x^6 + 42*a^2
*b^2*x^4 + 52*a^3*b*x^2 + 25*a^4)/((b*x^2 + a)^4*a^5*sgn(b*x^2 + a))
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maple [A] time = 0.02, size = 193, normalized size = 0.87 \begin {gather*} \frac {\left (24 b^{4} x^{8} \ln \relax (x )-12 b^{4} x^{8} \ln \left (b \,x^{2}+a \right )+96 a \,b^{3} x^{6} \ln \relax (x )-48 a \,b^{3} x^{6} \ln \left (b \,x^{2}+a \right )+12 a \,b^{3} x^{6}+144 a^{2} b^{2} x^{4} \ln \relax (x )-72 a^{2} b^{2} x^{4} \ln \left (b \,x^{2}+a \right )+42 a^{2} b^{2} x^{4}+96 a^{3} b \,x^{2} \ln \relax (x )-48 a^{3} b \,x^{2} \ln \left (b \,x^{2}+a \right )+52 a^{3} b \,x^{2}+24 a^{4} \ln \relax (x )-12 a^{4} \ln \left (b \,x^{2}+a \right )+25 a^{4}\right ) \left (b \,x^{2}+a \right )}{24 \left (\left (b \,x^{2}+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^4+2*a*b*x^2+a^2)^(5/2),x)
[Out]
1/24*(24*ln(x)*x^8*b^4-12*b^4*x^8*ln(b*x^2+a)+96*ln(x)*x^6*a*b^3-48*a*b^3*x^6*ln(b*x^2+a)+12*a*b^3*x^6+144*ln(
x)*x^4*a^2*b^2-72*a^2*b^2*x^4*ln(b*x^2+a)+42*a^2*b^2*x^4+96*ln(x)*x^2*a^3*b-48*a^3*b*x^2*ln(b*x^2+a)+52*a^3*b*
x^2+24*a^4*ln(x)-12*a^4*ln(b*x^2+a)+25*a^4)*(b*x^2+a)/a^5/((b*x^2+a)^2)^(5/2)
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maxima [A] time = 1.44, size = 101, normalized size = 0.45 \begin {gather*} \frac {12 \, b^{3} x^{6} + 42 \, a b^{2} x^{4} + 52 \, a^{2} b x^{2} + 25 \, a^{3}}{24 \, {\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )}} - \frac {\log \left (b x^{2} + a\right )}{2 \, a^{5}} + \frac {\log \relax (x)}{a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")
[Out]
1/24*(12*b^3*x^6 + 42*a*b^2*x^4 + 52*a^2*b*x^2 + 25*a^3)/(a^4*b^4*x^8 + 4*a^5*b^3*x^6 + 6*a^6*b^2*x^4 + 4*a^7*
b*x^2 + a^8) - 1/2*log(b*x^2 + a)/a^5 + log(x)/a^5
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\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^4 + 2*a*b*x^2)^(5/2)),x)
[Out]
int(1/(x*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (\left (a + b x^{2}\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**4+2*a*b*x**2+a**2)**(5/2),x)
[Out]
Integral(1/(x*((a + b*x**2)**2)**(5/2)), x)
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