3.2.84 \(\int \frac {1}{x^2 (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)
Optimal. Leaf size=235 \[ -\frac {b}{4 a^2 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 b \log (x) (a+b x)}{a^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b (a+b x) \log (a+b x)}{a^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {4 b}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a+b x}{a^5 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b}{2 a^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b}{3 a^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 235, 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 {3 b}{2 a^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b}{3 a^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b}{4 a^2 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {4 b}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a+b x}{a^5 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 b \log (x) (a+b x)}{a^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b (a+b x) \log (a+b x)}{a^6 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(x^2*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
[Out]
(-4*b)/(a^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - b/(4*a^2*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*b)/(3*a^
3*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3*b)/(2*a^4*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (a + b*
x)/(a^5*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*b*(a + b*x)*Log[x])/(a^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*b*(
a + b*x)*Log[a + b*x])/(a^6*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^2 \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^2 \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^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}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {4 b}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b}{4 a^2 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b}{3 a^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b}{2 a^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a+b x}{a^5 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 b (a+b x) \log (x)}{a^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b (a+b x) \log (a+b x)}{a^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 103, normalized size = 0.44 \begin {gather*} \frac {-a \left (12 a^4+125 a^3 b x+260 a^2 b^2 x^2+210 a b^3 x^3+60 b^4 x^4\right )-60 b x \log (x) (a+b x)^4+60 b x (a+b x)^4 \log (a+b x)}{12 a^6 x (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/(x^2*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
[Out]
(-(a*(12*a^4 + 125*a^3*b*x + 260*a^2*b^2*x^2 + 210*a*b^3*x^3 + 60*b^4*x^4)) - 60*b*x*(a + b*x)^4*Log[x] + 60*b
*x*(a + b*x)^4*Log[a + b*x])/(12*a^6*x*(a + b*x)^3*Sqrt[(a + b*x)^2])
________________________________________________________________________________________
IntegrateAlgebraic [B] time = 68.64, size = 2834, normalized size = 12.06 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/(x^2*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
[Out]
(-2*Sqrt[b^2]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(3*a^63 - 3*a^62*b*x + 3*a^61*b^2*x^2 + 9*a^60*b^3*x^3 + 1484*a^59
*b^4*x^4 + 89634*a^58*b^5*x^5 + 3547176*a^57*b^6*x^6 + 103238584*a^56*b^7*x^7 + 2356838400*a^55*b^8*x^8 + 4395
3744576*a^54*b^9*x^9 + 688626915840*a^53*b^10*x^10 + 9250107586560*a^52*b^11*x^11 + 108195839032320*a^51*b^12*
x^12 + 1115441253588480*a^50*b^13*x^13 + 10234958239733760*a^49*b^14*x^14 + 84253639955097600*a^48*b^15*x^15 +
626363783779123200*a^47*b^16*x^16 + 4228757765620531200*a^46*b^17*x^17 + 26049151725611581440*a^45*b^18*x^18
+ 147000114743606968320*a^44*b^19*x^19 + 762584480826139607040*a^43*b^20*x^20 + 3647550970994505154560*a^42*b^
21*x^21 + 16127976156648442429440*a^41*b^22*x^22 + 66068580639182534737920*a^40*b^23*x^23 + 251239072713580814
008320*a^39*b^24*x^24 + 888344859983663773777920*a^38*b^25*x^25 + 2924829168272464863559680*a^37*b^26*x^26 + 8
977874965576903525662720*a^36*b^27*x^27 + 25718516470160904041791488*a^35*b^28*x^28 + 688152795312548298872586
24*a^34*b^29*x^29 + 172101662127535626374873088*a^33*b^30*x^30 + 402506093314484489003466752*a^32*b^31*x^31 +
880658309537180769413234688*a^31*b^32*x^32 + 1802963185421958307337207808*a^30*b^33*x^33 + 3454182779430019135
581454336*a^29*b^34*x^34 + 6192350885862042683119239168*a^28*b^35*x^35 + 10385517381568557089430700032*a^27*b^
36*x^36 + 16289592602873600309238693888*a^26*b^37*x^37 + 23882708470211058375623442432*a^25*b^38*x^38 + 327085
73827319329343938756608*a^24*b^39*x^39 + 41810503900180714625288896512*a^23*b^40*x^40 + 4983317872996238405191
6505088*a^22*b^41*x^41 + 55314738880465273152142835712*a^21*b^42*x^42 + 57100495733850880840216608768*a^20*b^4
3*x^43 + 54726941348404022924701335552*a^19*b^44*x^44 + 48606655891944070942417747968*a^18*b^45*x^45 + 3991760
8008706788034432991232*a^17*b^46*x^46 + 30234237658617949398937632768*a^16*b^47*x^47 + 21058000972731372092000
305152*a^15*b^48*x^48 + 13441018440774824592175792128*a^14*b^49*x^49 + 7830928399966132744489009152*a^13*b^50*
x^50 + 4145087005408504523991810048*a^12*b^51*x^51 + 1982450466391073623482826752*a^11*b^52*x^52 + 85108204742
9972494978646016*a^10*b^53*x^53 + 325392797217001837833486336*a^9*b^54*x^54 + 109726425313175461183881216*a^8*
b^55*x^55 + 32243624862748916500660224*a^7*b^56*x^56 + 8130508717424292308975616*a^6*b^57*x^57 + 1723944790864
168421949440*a^5*b^58*x^58 + 298927614217018297810944*a^4*b^59*x^59 + 40705334872025491046400*a^3*b^60*x^60 +
4082062702248617574400*a^2*b^61*x^61 + 268054249821091921920*a*b^62*x^62 + 8646911284551352320*b^63*x^63) - 2*
(3*a^64*b - 12*a^61*b^4*x^3 - 1493*a^60*b^5*x^4 - 91118*a^59*b^6*x^5 - 3636810*a^58*b^7*x^6 - 106785760*a^57*b
^8*x^7 - 2460076984*a^56*b^9*x^8 - 46310582976*a^55*b^10*x^9 - 732580660416*a^54*b^11*x^10 - 9938734502400*a^5
3*b^12*x^11 - 117445946618880*a^52*b^13*x^12 - 1223637092620800*a^51*b^14*x^13 - 11350399493322240*a^50*b^15*x
^14 - 94488598194831360*a^49*b^16*x^15 - 710617423734220800*a^48*b^17*x^16 - 4855121549399654400*a^47*b^18*x^1
7 - 30277909491232112640*a^46*b^19*x^18 - 173049266469218549760*a^45*b^20*x^19 - 909584595569746575360*a^44*b^
21*x^20 - 4410135451820644761600*a^43*b^22*x^21 - 19775527127642947584000*a^42*b^23*x^22 - 8219655679583097716
7360*a^41*b^24*x^23 - 317307653352763348746240*a^40*b^25*x^24 - 1139583932697244587786240*a^39*b^26*x^25 - 381
3174028256128637337600*a^38*b^27*x^26 - 11902704133849368389222400*a^37*b^28*x^27 - 34696391435737807567454208
*a^36*b^29*x^28 - 94533796001415733929050112*a^35*b^30*x^29 - 240916941658790456262131712*a^34*b^31*x^30 - 574
607755442020115378339840*a^33*b^32*x^31 - 1283164402851665258416701440*a^32*b^33*x^32 - 2683621494959139076750
442496*a^31*b^34*x^33 - 5257145964851977442918662144*a^30*b^35*x^34 - 9646533665292061818700693504*a^29*b^36*x
^35 - 16577868267430599772549939200*a^28*b^37*x^36 - 26675109984442157398669393920*a^27*b^38*x^37 - 4017230107
3084658684862136320*a^26*b^39*x^38 - 56591282297530387719562199040*a^25*b^40*x^39 - 74519077727500043969227653
120*a^24*b^41*x^40 - 91643682630143098677205401600*a^23*b^42*x^41 - 105147917610427657204059340800*a^22*b^43*x
^42 - 112415234614316153992359444480*a^21*b^44*x^43 - 111827437082254903764917944320*a^20*b^45*x^44 - 10333359
7240348093867119083520*a^19*b^46*x^45 - 88524263900650858976850739200*a^18*b^47*x^46 - 70151845667324737433370
624000*a^17*b^48*x^47 - 51292238631349321490937937920*a^16*b^49*x^48 - 34499019413506196684176097280*a^15*b^50
*x^49 - 21271946840740957336664801280*a^14*b^51*x^50 - 11976015405374637268480819200*a^13*b^52*x^51 - 61275374
71799578147474636800*a^12*b^53*x^52 - 2833532513821046118461472768*a^11*b^54*x^53 - 11764748446469743328121323
52*a^10*b^55*x^54 - 435119222530177299017367552*a^9*b^56*x^55 - 141970050175924377684541440*a^8*b^57*x^56 - 40
374133580173208809635840*a^7*b^58*x^57 - 9854453508288460730925056*a^6*b^59*x^58 - 2022872405081186719760384*a
^5*b^60*x^59 - 339632949089043788857344*a^4*b^61*x^60 - 44787397574274108620800*a^3*b^62*x^61 - 43501169520697
09496320*a^2*b^63*x^62 - 276701161105643274240*a*b^64*x^63 - 8646911284551352320*b^65*x^64))/(3*a^5*x^4*Sqrt[a
^2 + 2*a*b*x + b^2*x^2]*(-8*a^60*b^4 - 936*a^59*b^5*x - 53832*a^58*b^6*x^2 - 2028600*a^57*b^7*x^3 - 56333328*a
^56*b^8*x^4 - 1229251968*a^55*b^9*x^5 - 21948838912*a^54*b^10*x^6 - 329736109824*a^53*b^11*x^7 - 4253142643200
*a^52*b^12*x^8 - 47832699273216*a^51*b^13*x^9 - 474726976948224*a^50*b^14*x^10 - 4198198392760320*a^49*b^15*x^
11 - 33343446890557440*a^48*b^16*x^12 - 239403280048128000*a^47*b^17*x^13 - 1562459053231964160*a^46*b^18*x^14
- 9312555821128089600*a^45*b^19*x^15 - 50890640751160197120*a^44*b^20*x^16 - 255857896219504803840*a^43*b^21*
x^17 - 1186944307588300800000*a^42*b^22*x^18 - 5093751076470545448960*a^41*b^23*x^19 - 20266419321282499706880
*a^40*b^24*x^20 - 74898737599223200481280*a^39*b^25*x^21 - 257538426184575127388160*a^38*b^26*x^22 - 825067881
224032223232000*a^37*b^27*x^23 - 2465674112032552349859840*a^36*b^28*x^24 - 6880414330081729610514432*a^35*b^2
9*x^25 - 17942536074222169724289024*a^34*b^30*x^26 - 43755107777792062666047488*a^33*b^31*x^27 - 9983129760884
3635615334400*a^32*b^32*x^28 - 213181721876762624526385152*a^31*b^33*x^29 - 426155156775629047172431872*a^30*b
^34*x^30 - 797530838364556999815856128*a^29*b^35*x^31 - 1397189047727722213310201856*a^28*b^36*x^32 - 22908411
70528074923297996800*a^27*b^37*x^33 - 3514050815948758604845154304*a^26*b^38*x^34 - 50404624413642121408791183
36*a^25*b^39*x^35 - 6756005325714055365069373440*a^24*b^40*x^36 - 8454788378368570414312980480*a^23*b^41*x^37
- 9868868050118054737084416000*a^22*b^42*x^38 - 10731458193725856494093598720*a^21*b^43*x^39 - 108557548551298
60210792857600*a^20*b^44*x^40 - 10198877550514546149257379840*a^19*b^45*x^41 - 8881831640526360274670714880*a^
18*b^46*x^42 - 7153973538484081655808000000*a^17*b^47*x^43 - 5315869468116817954964766720*a^16*b^48*x^44 - 363
3267961926811266066677760*a^15*b^49*x^45 - 2276282601414251560210268160*a^14*b^50*x^46 - 130204477081074585569
3291520*a^13*b^51*x^47 - 676809132086273815609344000*a^12*b^52*x^48 - 317945254586734678093332480*a^11*b^53*x^
49 - 134101230649370624636485632*a^10*b^54*x^50 - 50381470381939849118613504*a^9*b^55*x^51 - 16697916908414960
085762048*a^8*b^56*x^52 - 4823541649938374354534400*a^7*b^57*x^53 - 1195887430319030342254592*a^6*b^58*x^54 -
249357249723288646582272*a^5*b^59*x^55 - 42526806734116233412608*a^4*b^60*x^56 - 5696585154262430908416*a^3*b^
61*x^57 - 562049233495837900800*a^2*b^62*x^58 - 36317027395115679744*a*b^63*x^59 - 1152921504606846976*b^64*x^
60) + 3*a^5*Sqrt[b^2]*x^4*(8*a^61*b^3 + 944*a^60*b^4*x + 54768*a^59*b^5*x^2 + 2082432*a^58*b^6*x^3 + 58361928*
a^57*b^7*x^4 + 1285585296*a^56*b^8*x^5 + 23178090880*a^55*b^9*x^6 + 351684948736*a^54*b^10*x^7 + 4582878753024
*a^53*b^11*x^8 + 52085841916416*a^52*b^12*x^9 + 522559676221440*a^51*b^13*x^10 + 4672925369708544*a^50*b^14*x^
11 + 37541645283317760*a^49*b^15*x^12 + 272746726938685440*a^48*b^16*x^13 + 1801862333280092160*a^47*b^17*x^14
+ 10875014874360053760*a^46*b^18*x^15 + 60203196572288286720*a^45*b^19*x^16 + 306748536970665000960*a^44*b^20
*x^17 + 1442802203807805603840*a^43*b^21*x^18 + 6280695384058846248960*a^42*b^22*x^19 + 2536017039775304515584
0*a^41*b^23*x^20 + 95165156920505700188160*a^40*b^24*x^21 + 332437163783798327869440*a^39*b^25*x^22 + 10826063
07408607350620160*a^38*b^26*x^23 + 3290741993256584573091840*a^37*b^27*x^24 + 9346088442114281960374272*a^36*b
^28*x^25 + 24822950404303899334803456*a^35*b^29*x^26 + 61697643852014232390336512*a^34*b^30*x^27 + 14358640538
6635698281381888*a^33*b^31*x^28 + 313013019485606260141719552*a^32*b^32*x^29 + 639336878652391671698817024*a^3
1*b^33*x^30 + 1223685995140186046988288000*a^30*b^34*x^31 + 2194719886092279213126057984*a^29*b^35*x^32 + 3688
030218255797136608198656*a^28*b^36*x^33 + 5804891986476833528143151104*a^27*b^37*x^34 + 8554513257312970745724
272640*a^26*b^38*x^35 + 11796467767078267505948491776*a^25*b^39*x^36 + 15210793704082625779382353920*a^24*b^40
*x^37 + 18323656428486625151397396480*a^23*b^41*x^38 + 20600326243843911231178014720*a^22*b^42*x^39 + 21587213
048855716704886456320*a^21*b^43*x^40 + 21054632405644406360050237440*a^20*b^44*x^41 + 190807091910409064239280
94720*a^19*b^45*x^42 + 16035805179010441930478714880*a^18*b^46*x^43 + 12469843006600899610772766720*a^17*b^47*
x^44 + 8949137430043629221031444480*a^16*b^48*x^45 + 5909550563341062826276945920*a^15*b^49*x^46 + 35783273722
24997415903559680*a^14*b^50*x^47 + 1978853902897019671302635520*a^13*b^51*x^48 + 994754386673008493702676480*a
^12*b^52*x^49 + 452046485236105302729818112*a^11*b^53*x^50 + 184482701031310473755099136*a^10*b^54*x^51 + 6707
9387290354809204375552*a^9*b^55*x^52 + 21521458558353334440296448*a^8*b^56*x^53 + 6019429080257404696788992*a^
7*b^57*x^54 + 1445244680042318988836864*a^6*b^58*x^55 + 291884056457404879994880*a^5*b^59*x^56 + 4822339188837
8664321024*a^4*b^60*x^57 + 6258634387758268809216*a^3*b^61*x^58 + 598366260890953580544*a^2*b^62*x^59 + 374699
48899722526720*a*b^63*x^60 + 1152921504606846976*b^64*x^61)) - (10*b*ArcTanh[(Sqrt[b^2]*x)/a - Sqrt[a^2 + 2*a*
b*x + b^2*x^2]/a])/a^6
________________________________________________________________________________________
fricas [A] time = 0.43, size = 197, normalized size = 0.84 \begin {gather*} -\frac {60 \, a b^{4} x^{4} + 210 \, a^{2} b^{3} x^{3} + 260 \, a^{3} b^{2} x^{2} + 125 \, a^{4} b x + 12 \, a^{5} - 60 \, {\left (b^{5} x^{5} + 4 \, a b^{4} x^{4} + 6 \, a^{2} b^{3} x^{3} + 4 \, a^{3} b^{2} x^{2} + a^{4} b x\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{5} x^{5} + 4 \, a b^{4} x^{4} + 6 \, a^{2} b^{3} x^{3} + 4 \, a^{3} b^{2} x^{2} + a^{4} b x\right )} \log \relax (x)}{12 \, {\left (a^{6} b^{4} x^{5} + 4 \, a^{7} b^{3} x^{4} + 6 \, a^{8} b^{2} x^{3} + 4 \, a^{9} b x^{2} + a^{10} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^2/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")
[Out]
-1/12*(60*a*b^4*x^4 + 210*a^2*b^3*x^3 + 260*a^3*b^2*x^2 + 125*a^4*b*x + 12*a^5 - 60*(b^5*x^5 + 4*a*b^4*x^4 + 6
*a^2*b^3*x^3 + 4*a^3*b^2*x^2 + a^4*b*x)*log(b*x + a) + 60*(b^5*x^5 + 4*a*b^4*x^4 + 6*a^2*b^3*x^3 + 4*a^3*b^2*x
^2 + a^4*b*x)*log(x))/(a^6*b^4*x^5 + 4*a^7*b^3*x^4 + 6*a^8*b^2*x^3 + 4*a^9*b*x^2 + a^10*x)
________________________________________________________________________________________
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^2/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
[Out]
sage0*x
________________________________________________________________________________________
maple [A] time = 0.15, size = 199, normalized size = 0.85 \begin {gather*} \frac {\left (-60 b^{5} x^{5} \ln \relax (x )+60 b^{5} x^{5} \ln \left (b x +a \right )-240 a \,b^{4} x^{4} \ln \relax (x )+240 a \,b^{4} x^{4} \ln \left (b x +a \right )-360 a^{2} b^{3} x^{3} \ln \relax (x )+360 a^{2} b^{3} x^{3} \ln \left (b x +a \right )-60 a \,b^{4} x^{4}-240 a^{3} b^{2} x^{2} \ln \relax (x )+240 a^{3} b^{2} x^{2} \ln \left (b x +a \right )-210 a^{2} b^{3} x^{3}-60 a^{4} b x \ln \relax (x )+60 a^{4} b x \ln \left (b x +a \right )-260 a^{3} b^{2} x^{2}-125 a^{4} b x -12 a^{5}\right ) \left (b x +a \right )}{12 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} a^{6} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^2/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
[Out]
1/12*(60*ln(b*x+a)*x^5*b^5-60*b^5*x^5*ln(x)+240*a*b^4*x^4*ln(b*x+a)-240*a*b^4*x^4*ln(x)+360*a^2*b^3*x^3*ln(b*x
+a)-360*a^2*b^3*x^3*ln(x)-60*a*b^4*x^4+240*a^3*b^2*x^2*ln(b*x+a)-240*a^3*b^2*x^2*ln(x)-210*a^2*b^3*x^3+60*a^4*
b*x*ln(b*x+a)-60*a^4*b*x*ln(x)-260*a^3*b^2*x^2-125*a^4*b*x-12*a^5)*(b*x+a)/x/a^6/((b*x+a)^2)^(5/2)
________________________________________________________________________________________
maxima [A] time = 1.40, size = 148, normalized size = 0.63 \begin {gather*} \frac {5 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{6}} - \frac {5 \, b}{3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3}} - \frac {5 \, b}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{5}} - \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} x} - \frac {5}{2 \, a^{4} b {\left (x + \frac {a}{b}\right )}^{2}} - \frac {1}{4 \, a^{2} b^{3} {\left (x + \frac {a}{b}\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^2/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")
[Out]
5*(-1)^(2*a*b*x + 2*a^2)*b*log(2*a*b*x/abs(x) + 2*a^2/abs(x))/a^6 - 5/3*b/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3
) - 5*b/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^5) - 1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*x) - 5/2/(a^4*b*(x + a/b)
^2) - 1/4/(a^2*b^3*(x + a/b)^4)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^2\,{\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^2*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)),x)
[Out]
int(1/(x^2*(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^{2} \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**2/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
Integral(1/(x**2*((a + b*x)**2)**(5/2)), x)
________________________________________________________________________________________