3.5.80 \(\int \frac {1}{x^3 (a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\)

Optimal. Leaf size=267 \[ -\frac {b}{8 a^2 \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 b \log (x) \left (a+b x^2\right )}{a^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 b \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {2 b}{a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a+b x^2}{2 a^5 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 b}{4 a^4 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b}{3 a^3 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

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Rubi [A]  time = 0.14, antiderivative size = 267, 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 {3 b}{4 a^4 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b}{3 a^3 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b}{8 a^2 \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {2 b}{a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a+b x^2}{2 a^5 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 b \log (x) \left (a+b x^2\right )}{a^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 b \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^6 \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]

[Out]

(-2*b)/(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - b/(8*a^2*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - b/(3*
a^3*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3*b)/(4*a^4*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])
- (a + b*x^2)/(2*a^5*x^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (5*b*(a + b*x^2)*Log[x])/(a^6*Sqrt[a^2 + 2*a*b*x^
2 + b^2*x^4]) + (5*b*(a + b*x^2)*Log[a + b*x^2])/(2*a^6*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^3 \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^3 \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^2 \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^2}-\frac {5}{a^6 b^4 x}+\frac {1}{a^2 b^3 (a+b x)^5}+\frac {2}{a^3 b^3 (a+b x)^4}+\frac {3}{a^4 b^3 (a+b x)^3}+\frac {4}{a^5 b^3 (a+b x)^2}+\frac {5}{a^6 b^3 (a+b x)}\right ) \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {2 b}{a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b}{8 a^2 \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b}{3 a^3 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 b}{4 a^4 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {a+b x^2}{2 a^5 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 b \left (a+b x^2\right ) \log (x)}{a^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 b \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 119, normalized size = 0.45 \begin {gather*} \frac {-a \left (12 a^4+125 a^3 b x^2+260 a^2 b^2 x^4+210 a b^3 x^6+60 b^4 x^8\right )-120 b x^2 \log (x) \left (a+b x^2\right )^4+60 b x^2 \left (a+b x^2\right )^4 \log \left (a+b x^2\right )}{24 a^6 x^2 \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^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]

[Out]

(-(a*(12*a^4 + 125*a^3*b*x^2 + 260*a^2*b^2*x^4 + 210*a*b^3*x^6 + 60*b^4*x^8)) - 120*b*x^2*(a + b*x^2)^4*Log[x]
+ 60*b*x^2*(a + b*x^2)^4*Log[a + b*x^2])/(24*a^6*x^2*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])

________________________________________________________________________________________

IntegrateAlgebraic [B]  time = 81.20, size = 2844, normalized size = 10.65 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]

[Out]

(-3*a^64*b + 12*a^61*b^4*x^6 + 1493*a^60*b^5*x^8 + 91118*a^59*b^6*x^10 + 3636810*a^58*b^7*x^12 + 106785760*a^5
7*b^8*x^14 + 2460076984*a^56*b^9*x^16 + 46310582976*a^55*b^10*x^18 + 732580660416*a^54*b^11*x^20 + 99387345024
00*a^53*b^12*x^22 + 117445946618880*a^52*b^13*x^24 + 1223637092620800*a^51*b^14*x^26 + 11350399493322240*a^50*
b^15*x^28 + 94488598194831360*a^49*b^16*x^30 + 710617423734220800*a^48*b^17*x^32 + 4855121549399654400*a^47*b^
18*x^34 + 30277909491232112640*a^46*b^19*x^36 + 173049266469218549760*a^45*b^20*x^38 + 909584595569746575360*a
^44*b^21*x^40 + 4410135451820644761600*a^43*b^22*x^42 + 19775527127642947584000*a^42*b^23*x^44 + 8219655679583
0977167360*a^41*b^24*x^46 + 317307653352763348746240*a^40*b^25*x^48 + 1139583932697244587786240*a^39*b^26*x^50
+ 3813174028256128637337600*a^38*b^27*x^52 + 11902704133849368389222400*a^37*b^28*x^54 + 34696391435737807567
454208*a^36*b^29*x^56 + 94533796001415733929050112*a^35*b^30*x^58 + 240916941658790456262131712*a^34*b^31*x^60
+ 574607755442020115378339840*a^33*b^32*x^62 + 1283164402851665258416701440*a^32*b^33*x^64 + 2683621494959139
076750442496*a^31*b^34*x^66 + 5257145964851977442918662144*a^30*b^35*x^68 + 9646533665292061818700693504*a^29*
b^36*x^70 + 16577868267430599772549939200*a^28*b^37*x^72 + 26675109984442157398669393920*a^27*b^38*x^74 + 4017
2301073084658684862136320*a^26*b^39*x^76 + 56591282297530387719562199040*a^25*b^40*x^78 + 74519077727500043969
227653120*a^24*b^41*x^80 + 91643682630143098677205401600*a^23*b^42*x^82 + 105147917610427657204059340800*a^22*
b^43*x^84 + 112415234614316153992359444480*a^21*b^44*x^86 + 111827437082254903764917944320*a^20*b^45*x^88 + 10
3333597240348093867119083520*a^19*b^46*x^90 + 88524263900650858976850739200*a^18*b^47*x^92 + 70151845667324737
433370624000*a^17*b^48*x^94 + 51292238631349321490937937920*a^16*b^49*x^96 + 34499019413506196684176097280*a^1
5*b^50*x^98 + 21271946840740957336664801280*a^14*b^51*x^100 + 11976015405374637268480819200*a^13*b^52*x^102 +
6127537471799578147474636800*a^12*b^53*x^104 + 2833532513821046118461472768*a^11*b^54*x^106 + 1176474844646974
332812132352*a^10*b^55*x^108 + 435119222530177299017367552*a^9*b^56*x^110 + 141970050175924377684541440*a^8*b^
57*x^112 + 40374133580173208809635840*a^7*b^58*x^114 + 9854453508288460730925056*a^6*b^59*x^116 + 202287240508
1186719760384*a^5*b^60*x^118 + 339632949089043788857344*a^4*b^61*x^120 + 44787397574274108620800*a^3*b^62*x^12
2 + 4350116952069709496320*a^2*b^63*x^124 + 276701161105643274240*a*b^64*x^126 + 8646911284551352320*b^65*x^12
8 + Sqrt[b^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*(-3*a^63 + 3*a^62*b*x^2 - 3*a^61*b^2*x^4 - 9*a^60*b^3*x^6 - 1484
*a^59*b^4*x^8 - 89634*a^58*b^5*x^10 - 3547176*a^57*b^6*x^12 - 103238584*a^56*b^7*x^14 - 2356838400*a^55*b^8*x^
16 - 43953744576*a^54*b^9*x^18 - 688626915840*a^53*b^10*x^20 - 9250107586560*a^52*b^11*x^22 - 108195839032320*
a^51*b^12*x^24 - 1115441253588480*a^50*b^13*x^26 - 10234958239733760*a^49*b^14*x^28 - 84253639955097600*a^48*b
^15*x^30 - 626363783779123200*a^47*b^16*x^32 - 4228757765620531200*a^46*b^17*x^34 - 26049151725611581440*a^45*
b^18*x^36 - 147000114743606968320*a^44*b^19*x^38 - 762584480826139607040*a^43*b^20*x^40 - 36475509709945051545
60*a^42*b^21*x^42 - 16127976156648442429440*a^41*b^22*x^44 - 66068580639182534737920*a^40*b^23*x^46 - 25123907
2713580814008320*a^39*b^24*x^48 - 888344859983663773777920*a^38*b^25*x^50 - 2924829168272464863559680*a^37*b^2
6*x^52 - 8977874965576903525662720*a^36*b^27*x^54 - 25718516470160904041791488*a^35*b^28*x^56 - 68815279531254
829887258624*a^34*b^29*x^58 - 172101662127535626374873088*a^33*b^30*x^60 - 402506093314484489003466752*a^32*b^
31*x^62 - 880658309537180769413234688*a^31*b^32*x^64 - 1802963185421958307337207808*a^30*b^33*x^66 - 345418277
9430019135581454336*a^29*b^34*x^68 - 6192350885862042683119239168*a^28*b^35*x^70 - 103855173815685570894307000
32*a^27*b^36*x^72 - 16289592602873600309238693888*a^26*b^37*x^74 - 23882708470211058375623442432*a^25*b^38*x^7
6 - 32708573827319329343938756608*a^24*b^39*x^78 - 41810503900180714625288896512*a^23*b^40*x^80 - 498331787299
62384051916505088*a^22*b^41*x^82 - 55314738880465273152142835712*a^21*b^42*x^84 - 5710049573385088084021660876
8*a^20*b^43*x^86 - 54726941348404022924701335552*a^19*b^44*x^88 - 48606655891944070942417747968*a^18*b^45*x^90
- 39917608008706788034432991232*a^17*b^46*x^92 - 30234237658617949398937632768*a^16*b^47*x^94 - 2105800097273
1372092000305152*a^15*b^48*x^96 - 13441018440774824592175792128*a^14*b^49*x^98 - 7830928399966132744489009152*
a^13*b^50*x^100 - 4145087005408504523991810048*a^12*b^51*x^102 - 1982450466391073623482826752*a^11*b^52*x^104
- 851082047429972494978646016*a^10*b^53*x^106 - 325392797217001837833486336*a^9*b^54*x^108 - 10972642531317546
1183881216*a^8*b^55*x^110 - 32243624862748916500660224*a^7*b^56*x^112 - 8130508717424292308975616*a^6*b^57*x^1
14 - 1723944790864168421949440*a^5*b^58*x^116 - 298927614217018297810944*a^4*b^59*x^118 - 40705334872025491046
400*a^3*b^60*x^120 - 4082062702248617574400*a^2*b^61*x^122 - 268054249821091921920*a*b^62*x^124 - 864691128455
1352320*b^63*x^126))/(3*a^5*x^8*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*(-8*a^60*b^4 - 936*a^59*b^5*x^2 - 53832*a^58*b
^6*x^4 - 2028600*a^57*b^7*x^6 - 56333328*a^56*b^8*x^8 - 1229251968*a^55*b^9*x^10 - 21948838912*a^54*b^10*x^12
- 329736109824*a^53*b^11*x^14 - 4253142643200*a^52*b^12*x^16 - 47832699273216*a^51*b^13*x^18 - 474726976948224
*a^50*b^14*x^20 - 4198198392760320*a^49*b^15*x^22 - 33343446890557440*a^48*b^16*x^24 - 239403280048128000*a^47
*b^17*x^26 - 1562459053231964160*a^46*b^18*x^28 - 9312555821128089600*a^45*b^19*x^30 - 50890640751160197120*a^
44*b^20*x^32 - 255857896219504803840*a^43*b^21*x^34 - 1186944307588300800000*a^42*b^22*x^36 - 5093751076470545
448960*a^41*b^23*x^38 - 20266419321282499706880*a^40*b^24*x^40 - 74898737599223200481280*a^39*b^25*x^42 - 2575
38426184575127388160*a^38*b^26*x^44 - 825067881224032223232000*a^37*b^27*x^46 - 2465674112032552349859840*a^36
*b^28*x^48 - 6880414330081729610514432*a^35*b^29*x^50 - 17942536074222169724289024*a^34*b^30*x^52 - 4375510777
7792062666047488*a^33*b^31*x^54 - 99831297608843635615334400*a^32*b^32*x^56 - 213181721876762624526385152*a^31
*b^33*x^58 - 426155156775629047172431872*a^30*b^34*x^60 - 797530838364556999815856128*a^29*b^35*x^62 - 1397189
047727722213310201856*a^28*b^36*x^64 - 2290841170528074923297996800*a^27*b^37*x^66 - 3514050815948758604845154
304*a^26*b^38*x^68 - 5040462441364212140879118336*a^25*b^39*x^70 - 6756005325714055365069373440*a^24*b^40*x^72
- 8454788378368570414312980480*a^23*b^41*x^74 - 9868868050118054737084416000*a^22*b^42*x^76 - 107314581937258
56494093598720*a^21*b^43*x^78 - 10855754855129860210792857600*a^20*b^44*x^80 - 10198877550514546149257379840*a
^19*b^45*x^82 - 8881831640526360274670714880*a^18*b^46*x^84 - 7153973538484081655808000000*a^17*b^47*x^86 - 53
15869468116817954964766720*a^16*b^48*x^88 - 3633267961926811266066677760*a^15*b^49*x^90 - 22762826014142515602
10268160*a^14*b^50*x^92 - 1302044770810745855693291520*a^13*b^51*x^94 - 676809132086273815609344000*a^12*b^52*
x^96 - 317945254586734678093332480*a^11*b^53*x^98 - 134101230649370624636485632*a^10*b^54*x^100 - 503814703819
39849118613504*a^9*b^55*x^102 - 16697916908414960085762048*a^8*b^56*x^104 - 4823541649938374354534400*a^7*b^57
*x^106 - 1195887430319030342254592*a^6*b^58*x^108 - 249357249723288646582272*a^5*b^59*x^110 - 4252680673411623
3412608*a^4*b^60*x^112 - 5696585154262430908416*a^3*b^61*x^114 - 562049233495837900800*a^2*b^62*x^116 - 363170
27395115679744*a*b^63*x^118 - 1152921504606846976*b^64*x^120) + 3*a^5*Sqrt[b^2]*x^8*(8*a^61*b^3 + 944*a^60*b^4
*x^2 + 54768*a^59*b^5*x^4 + 2082432*a^58*b^6*x^6 + 58361928*a^57*b^7*x^8 + 1285585296*a^56*b^8*x^10 + 23178090
880*a^55*b^9*x^12 + 351684948736*a^54*b^10*x^14 + 4582878753024*a^53*b^11*x^16 + 52085841916416*a^52*b^12*x^18
+ 522559676221440*a^51*b^13*x^20 + 4672925369708544*a^50*b^14*x^22 + 37541645283317760*a^49*b^15*x^24 + 27274
6726938685440*a^48*b^16*x^26 + 1801862333280092160*a^47*b^17*x^28 + 10875014874360053760*a^46*b^18*x^30 + 6020
3196572288286720*a^45*b^19*x^32 + 306748536970665000960*a^44*b^20*x^34 + 1442802203807805603840*a^43*b^21*x^36
+ 6280695384058846248960*a^42*b^22*x^38 + 25360170397753045155840*a^41*b^23*x^40 + 95165156920505700188160*a^
40*b^24*x^42 + 332437163783798327869440*a^39*b^25*x^44 + 1082606307408607350620160*a^38*b^26*x^46 + 3290741993
256584573091840*a^37*b^27*x^48 + 9346088442114281960374272*a^36*b^28*x^50 + 24822950404303899334803456*a^35*b^
29*x^52 + 61697643852014232390336512*a^34*b^30*x^54 + 143586405386635698281381888*a^33*b^31*x^56 + 31301301948
5606260141719552*a^32*b^32*x^58 + 639336878652391671698817024*a^31*b^33*x^60 + 1223685995140186046988288000*a^
30*b^34*x^62 + 2194719886092279213126057984*a^29*b^35*x^64 + 3688030218255797136608198656*a^28*b^36*x^66 + 580
4891986476833528143151104*a^27*b^37*x^68 + 8554513257312970745724272640*a^26*b^38*x^70 + 117964677670782675059
48491776*a^25*b^39*x^72 + 15210793704082625779382353920*a^24*b^40*x^74 + 18323656428486625151397396480*a^23*b^
41*x^76 + 20600326243843911231178014720*a^22*b^42*x^78 + 21587213048855716704886456320*a^21*b^43*x^80 + 210546
32405644406360050237440*a^20*b^44*x^82 + 19080709191040906423928094720*a^19*b^45*x^84 + 1603580517901044193047
8714880*a^18*b^46*x^86 + 12469843006600899610772766720*a^17*b^47*x^88 + 8949137430043629221031444480*a^16*b^48
*x^90 + 5909550563341062826276945920*a^15*b^49*x^92 + 3578327372224997415903559680*a^14*b^50*x^94 + 1978853902
897019671302635520*a^13*b^51*x^96 + 994754386673008493702676480*a^12*b^52*x^98 + 452046485236105302729818112*a
^11*b^53*x^100 + 184482701031310473755099136*a^10*b^54*x^102 + 67079387290354809204375552*a^9*b^55*x^104 + 215
21458558353334440296448*a^8*b^56*x^106 + 6019429080257404696788992*a^7*b^57*x^108 + 1445244680042318988836864*
a^6*b^58*x^110 + 291884056457404879994880*a^5*b^59*x^112 + 48223391888378664321024*a^4*b^60*x^114 + 6258634387
758268809216*a^3*b^61*x^116 + 598366260890953580544*a^2*b^62*x^118 + 37469948899722526720*a*b^63*x^120 + 11529
21504606846976*b^64*x^122)) - (5*b*ArcTanh[(Sqrt[b^2]*x^2)/a - Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/a])/a^6

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fricas [A]  time = 0.67, size = 207, normalized size = 0.78 \begin {gather*} -\frac {60 \, a b^{4} x^{8} + 210 \, a^{2} b^{3} x^{6} + 260 \, a^{3} b^{2} x^{4} + 125 \, a^{4} b x^{2} + 12 \, a^{5} - 60 \, {\left (b^{5} x^{10} + 4 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} + 4 \, a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )} \log \left (b x^{2} + a\right ) + 120 \, {\left (b^{5} x^{10} + 4 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} + 4 \, a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )} \log \relax (x)}{24 \, {\left (a^{6} b^{4} x^{10} + 4 \, a^{7} b^{3} x^{8} + 6 \, a^{8} b^{2} x^{6} + 4 \, a^{9} b x^{4} + a^{10} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/24*(60*a*b^4*x^8 + 210*a^2*b^3*x^6 + 260*a^3*b^2*x^4 + 125*a^4*b*x^2 + 12*a^5 - 60*(b^5*x^10 + 4*a*b^4*x^8
+ 6*a^2*b^3*x^6 + 4*a^3*b^2*x^4 + a^4*b*x^2)*log(b*x^2 + a) + 120*(b^5*x^10 + 4*a*b^4*x^8 + 6*a^2*b^3*x^6 + 4*
a^3*b^2*x^4 + a^4*b*x^2)*log(x))/(a^6*b^4*x^10 + 4*a^7*b^3*x^8 + 6*a^8*b^2*x^6 + 4*a^9*b*x^4 + a^10*x^2)

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giac [A]  time = 0.22, size = 118, normalized size = 0.44 \begin {gather*} \frac {5 \, b \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{6} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {5 \, b \log \left ({\left | x \right |}\right )}{a^{6} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {60 \, a b^{4} x^{8} + 210 \, a^{2} b^{3} x^{6} + 260 \, a^{3} b^{2} x^{4} + 125 \, a^{4} b x^{2} + 12 \, a^{5}}{24 \, {\left (b x^{2} + a\right )}^{4} a^{6} x^{2} \mathrm {sgn}\left (b x^{2} + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")

[Out]

5/2*b*log(abs(b*x^2 + a))/(a^6*sgn(b*x^2 + a)) - 5*b*log(abs(x))/(a^6*sgn(b*x^2 + a)) - 1/24*(60*a*b^4*x^8 + 2
10*a^2*b^3*x^6 + 260*a^3*b^2*x^4 + 125*a^4*b*x^2 + 12*a^5)/((b*x^2 + a)^4*a^6*x^2*sgn(b*x^2 + a))

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maple [A]  time = 0.02, size = 219, normalized size = 0.82 \begin {gather*} -\frac {\left (120 b^{5} x^{10} \ln \relax (x )-60 b^{5} x^{10} \ln \left (b \,x^{2}+a \right )+480 a \,b^{4} x^{8} \ln \relax (x )-240 a \,b^{4} x^{8} \ln \left (b \,x^{2}+a \right )+60 a \,b^{4} x^{8}+720 a^{2} b^{3} x^{6} \ln \relax (x )-360 a^{2} b^{3} x^{6} \ln \left (b \,x^{2}+a \right )+210 a^{2} b^{3} x^{6}+480 a^{3} b^{2} x^{4} \ln \relax (x )-240 a^{3} b^{2} x^{4} \ln \left (b \,x^{2}+a \right )+260 a^{3} b^{2} x^{4}+120 a^{4} b \,x^{2} \ln \relax (x )-60 a^{4} b \,x^{2} \ln \left (b \,x^{2}+a \right )+125 a^{4} b \,x^{2}+12 a^{5}\right ) \left (b \,x^{2}+a \right )}{24 \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} a^{6} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

-1/24*(120*b^5*x^10*ln(x)-60*ln(b*x^2+a)*x^10*b^5+480*a*b^4*x^8*ln(x)-240*a*b^4*x^8*ln(b*x^2+a)+60*a*b^4*x^8+7
20*a^2*b^3*x^6*ln(x)-360*a^2*b^3*x^6*ln(b*x^2+a)+210*a^2*b^3*x^6+480*a^3*b^2*x^4*ln(x)-240*a^3*b^2*x^4*ln(b*x^
2+a)+260*a^3*b^2*x^4+120*a^4*b*x^2*ln(x)-60*a^4*b*x^2*ln(b*x^2+a)+125*a^4*b*x^2+12*a^5)*(b*x^2+a)/x^2/a^6/((b*
x^2+a)^2)^(5/2)

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maxima [A]  time = 1.42, size = 119, normalized size = 0.45 \begin {gather*} -\frac {60 \, b^{4} x^{8} + 210 \, a b^{3} x^{6} + 260 \, a^{2} b^{2} x^{4} + 125 \, a^{3} b x^{2} + 12 \, a^{4}}{24 \, {\left (a^{5} b^{4} x^{10} + 4 \, a^{6} b^{3} x^{8} + 6 \, a^{7} b^{2} x^{6} + 4 \, a^{8} b x^{4} + a^{9} x^{2}\right )}} + \frac {5 \, b \log \left (b x^{2} + a\right )}{2 \, a^{6}} - \frac {5 \, b \log \relax (x)}{a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")

[Out]

-1/24*(60*b^4*x^8 + 210*a*b^3*x^6 + 260*a^2*b^2*x^4 + 125*a^3*b*x^2 + 12*a^4)/(a^5*b^4*x^10 + 4*a^6*b^3*x^8 +
6*a^7*b^2*x^6 + 4*a^8*b*x^4 + a^9*x^2) + 5/2*b*log(b*x^2 + a)/a^6 - 5*b*log(x)/a^6

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^3\,{\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^3*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)),x)

[Out]

int(1/(x^3*(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^{3} \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**3/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Integral(1/(x**3*((a + b*x**2)**2)**(5/2)), x)

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