3.2.85 \(\int \frac {1}{x^3 (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)
Optimal. Leaf size=278 \[ \frac {15 b^2 \log (x) (a+b x)}{a^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 b^2 (a+b x) \log (a+b x)}{a^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {10 b^2}{a^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b (a+b x)}{a^6 x \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b^2}{a^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a+b x}{2 a^5 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2}{a^4 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2}{4 a^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.11, antiderivative size = 278, 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^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2}{a^4 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2}{4 a^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {10 b^2}{a^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b (a+b x)}{a^6 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a+b x}{2 a^5 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 b^2 \log (x) (a+b x)}{a^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 b^2 (a+b x) \log (a+b x)}{a^7 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(x^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
[Out]
(10*b^2)/(a^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + b^2/(4*a^3*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + b^2/(a^
4*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*b^2)/(a^5*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (a + b*
x)/(2*a^5*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*b*(a + b*x))/(a^6*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (15*b^2
*(a + b*x)*Log[x])/(a^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (15*b^2*(a + b*x)*Log[a + b*x])/(a^7*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^3 \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^3 \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^3}-\frac {5}{a^6 b^4 x^2}+\frac {15}{a^7 b^3 x}-\frac {1}{a^3 b^2 (a+b x)^5}-\frac {3}{a^4 b^2 (a+b x)^4}-\frac {6}{a^5 b^2 (a+b x)^3}-\frac {10}{a^6 b^2 (a+b x)^2}-\frac {15}{a^7 b^2 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {10 b^2}{a^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2}{4 a^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2}{a^4 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b^2}{a^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a+b x}{2 a^5 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b (a+b x)}{a^6 x \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 b^2 (a+b x) \log (x)}{a^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 b^2 (a+b x) \log (a+b x)}{a^7 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 121, normalized size = 0.44 \begin {gather*} \frac {a \left (-2 a^5+12 a^4 b x+125 a^3 b^2 x^2+260 a^2 b^3 x^3+210 a b^4 x^4+60 b^5 x^5\right )+60 b^2 x^2 \log (x) (a+b x)^4-60 b^2 x^2 (a+b x)^4 \log (a+b x)}{4 a^7 x^2 (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/(x^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
[Out]
(a*(-2*a^5 + 12*a^4*b*x + 125*a^3*b^2*x^2 + 260*a^2*b^3*x^3 + 210*a*b^4*x^4 + 60*b^5*x^5) + 60*b^2*x^2*(a + b*
x)^4*Log[x] - 60*b^2*x^2*(a + b*x)^4*Log[a + b*x])/(4*a^7*x^2*(a + b*x)^3*Sqrt[(a + b*x)^2])
________________________________________________________________________________________
IntegrateAlgebraic [B] time = 47.49, size = 2538, normalized size = 9.13 \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 + b^2*x^2)^(5/2)),x]
[Out]
(-2*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(a^56*b - a^55*b^2*x + 3*a^54*b^3*x^2 + 185*a^53*b^4*x^3 + 8290*a^52*b^5*x
^4 + 233750*a^51*b^6*x^5 + 4615540*a^50*b^7*x^6 + 65872440*a^49*b^8*x^7 + 652130320*a^48*b^9*x^8 + 3247908480*
a^47*b^10*x^9 - 27325377280*a^46*b^11*x^10 - 933432529920*a^45*b^12*x^11 - 14144235089920*a^44*b^13*x^12 - 157
908917642240*a^43*b^14*x^13 - 1443595273615360*a^42*b^15*x^14 - 11284484800778240*a^41*b^16*x^15 - 77201054974
402560*a^40*b^17*x^16 - 469035705020088320*a^39*b^18*x^17 - 2556214212604723200*a^38*b^19*x^18 - 1258928070478
8889600*a^37*b^20*x^19 - 56346483304179302400*a^36*b^21*x^20 - 230208579043783475200*a^35*b^22*x^21 - 86160395
9510360391680*a^34*b^23*x^22 - 2962604246799564144640*a^33*b^24*x^23 - 9380655672259389685760*a^32*b^25*x^24 -
27403582052278248407040*a^31*b^26*x^25 - 73969691817495033282560*a^30*b^27*x^26 - 184709189424036458790912*a^
29*b^28*x^27 - 427071855502166962733056*a^28*b^29*x^28 - 914882886121433275039744*a^27*b^30*x^29 - 18165572629
22479405367296*a^26*b^31*x^30 - 3343615123830145908998144*a^25*b^32*x^31 - 5704688861302094420770816*a^24*b^33
*x^32 - 9019074305464746024894464*a^23*b^34*x^33 - 13206010959933360293543936*a^22*b^35*x^34 - 178946218395177
44485105664*a^21*b^36*x^35 - 22416441511501712888692736*a^20*b^37*x^36 - 25926200384824170139090944*a^19*b^38*
x^37 - 27640503210365958672613376*a^18*b^39*x^38 - 27111453098263308165709824*a^17*b^40*x^39 - 244098105791471
02277861376*a^16*b^41*x^40 - 20118828048302273382580224*a^15*b^42*x^41 - 15131309701402629123342336*a^14*b^43*
x^42 - 10345299869765624506875904*a^13*b^44*x^43 - 6401132778395795976093696*a^12*b^45*x^44 - 3565346517008726
364258304*a^11*b^46*x^45 - 1776230315962117333712896*a^10*b^47*x^46 - 785381587239128283480064*a^9*b^48*x^47 -
305284675583873306853376*a^8*b^49*x^48 - 103083698718904024039424*a^7*b^50*x^49 - 29778235896487714226176*a^6
*b^51*x^50 - 7212241491031463297024*a^5*b^52*x^51 - 1424471110688731824128*a^4*b^53*x^52 - 2203414265188686233
60*a^3*b^54*x^53 - 25034384428645744640*a^2*b^55*x^54 - 1857734846290329600*a*b^56*x^55 - 67553994410557440*b^
57*x^56) - 2*b*Sqrt[b^2]*(a^57 - 2*a^55*b^2*x^2 - 188*a^54*b^3*x^3 - 8475*a^53*b^4*x^4 - 242040*a^52*b^5*x^5 -
4849290*a^51*b^6*x^6 - 70487980*a^50*b^7*x^7 - 718002760*a^49*b^8*x^8 - 3900038800*a^48*b^9*x^9 + 24077468800
*a^47*b^10*x^10 + 960757907200*a^46*b^11*x^11 + 15077667619840*a^45*b^12*x^12 + 172053152732160*a^44*b^13*x^13
+ 1601504191257600*a^43*b^14*x^14 + 12728080074393600*a^42*b^15*x^15 + 88485539775180800*a^41*b^16*x^16 + 546
236759994490880*a^40*b^17*x^17 + 3025249917624811520*a^39*b^18*x^18 + 15145494917393612800*a^38*b^19*x^19 + 68
935764008968192000*a^37*b^20*x^20 + 286555062347962777600*a^36*b^21*x^21 + 1091812538554143866880*a^35*b^22*x^
22 + 3824208206309924536320*a^34*b^23*x^23 + 12343259919058953830400*a^33*b^24*x^24 + 36784237724537638092800*
a^32*b^25*x^25 + 101373273869773281689600*a^31*b^26*x^26 + 258678881241531492073472*a^30*b^27*x^27 + 611781044
926203421523968*a^29*b^28*x^28 + 1341954741623600237772800*a^28*b^29*x^29 + 2731440149043912680407040*a^27*b^3
0*x^30 + 5160172386752625314365440*a^26*b^31*x^31 + 9048303985132240329768960*a^25*b^32*x^32 + 147237631667668
40445665280*a^24*b^33*x^33 + 22225085265398106318438400*a^23*b^34*x^34 + 31100632799451104778649600*a^22*b^35*
x^35 + 40311063351019457373798400*a^21*b^36*x^36 + 48342641896325883027783680*a^20*b^37*x^37 + 535667035951901
28811704320*a^19*b^38*x^38 + 54751956308629266838323200*a^18*b^39*x^39 + 51521263677410410443571200*a^17*b^40*
x^40 + 44528638627449375660441600*a^16*b^41*x^41 + 35250137749704902505922560*a^15*b^42*x^42 + 254766095711682
53630218240*a^14*b^43*x^43 + 16746432648161420482969600*a^13*b^44*x^44 + 9966479295404522340352000*a^12*b^45*x
^45 + 5341576832970843697971200*a^11*b^46*x^46 + 2561611903201245617192960*a^10*b^47*x^47 + 109066626282300159
0333440*a^9*b^48*x^48 + 408368374302777330892800*a^8*b^49*x^49 + 132861934615391738265600*a^7*b^50*x^50 + 3699
0477387519177523200*a^6*b^51*x^51 + 8636712601720195121152*a^5*b^52*x^52 + 1644812537207600447488*a^4*b^53*x^5
3 + 245375810947514368000*a^3*b^54*x^54 + 26892119274936074240*a^2*b^55*x^55 + 1925288840700887040*a*b^56*x^56
+ 67553994410557440*b^57*x^57))/(a^6*Sqrt[b^2]*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-8*a^53*b^3 - 824*a^52*b^4*
x - 41624*a^51*b^5*x^2 - 1374408*a^50*b^6*x^3 - 33360000*a^49*b^7*x^4 - 634640160*a^48*b^8*x^5 - 9852935680*a^
47*b^9*x^6 - 128346426880*a^46*b^10*x^7 - 1431322869760*a^45*b^11*x^8 - 13875819776000*a^44*b^12*x^9 - 1183381
36125440*a^43*b^13*x^10 - 896341461585920*a^42*b^14*x^11 - 6076748501483520*a^41*b^15*x^12 - 37110371720970240
*a^40*b^16*x^13 - 205239797661696000*a^39*b^17*x^14 - 1032563826336071680*a^38*b^18*x^15 - 4743587884069027840
*a^37*b^19*x^16 - 19963069672078704640*a^36*b^20*x^17 - 77172507020511150080*a^35*b^21*x^18 - 2746733610734059
52000*a^34*b^22*x^19 - 901852505040622714880*a^33*b^23*x^20 - 2736068869987541975040*a^32*b^24*x^21 - 76802281
87826489917440*a^31*b^25*x^22 - 19968558722990557102080*a^30*b^26*x^23 - 48129381009455054848000*a^29*b^27*x^2
4 - 107604973386901605056512*a^28*b^28*x^25 - 223247821283909158567936*a^27*b^29*x^26 - 4298916311764525499023
36*a^26*b^30*x^27 - 768321365543323237875712*a^25*b^31*x^28 - 1274223129879264624640000*a^24*b^32*x^29 - 19601
21278507813196267520*a^23*b^33*x^30 - 2794976995316331404328960*a^22*b^34*x^31 - 3691104648891335033487360*a^2
1*b^35*x^32 - 4509597585944179606814720*a^20*b^36*x^33 - 5090140752859199700992000*a^19*b^37*x^34 - 5299258405
446468113530880*a^18*b^38*x^35 - 5078523105493240344739840*a^17*b^39*x^36 - 4469748299151531090903040*a^16*b^4
0*x^37 - 3602942084040997973524480*a^15*b^41*x^38 - 2651271349958923517952000*a^14*b^42*x^39 - 177426159054512
5248860160*a^13*b^43*x^40 - 1074954896575188518830080*a^12*b^44*x^41 - 586469129111178240327680*a^11*b^45*x^42
- 286282036005754068008960*a^10*b^46*x^43 - 124067906915825352704000*a^9*b^47*x^44 - 47281587101247691816960*
a^8*b^48*x^45 - 15656772548802830663680*a^7*b^49*x^46 - 4436560732167319060480*a^6*b^50*x^47 - 105428141376836
4687360*a^5*b^51*x^48 - 204350833091936256000*a^4*b^52*x^49 - 31027549632769032192*a^3*b^53*x^50 - 34610163136
34226176*a^2*b^54*x^51 - 252201579132747776*a*b^55*x^52 - 9007199254740992*b^56*x^53) + a^6*x^4*(8*a^54*b^4 +
832*a^53*b^5*x + 42448*a^52*b^6*x^2 + 1416032*a^51*b^7*x^3 + 34734408*a^50*b^8*x^4 + 668000160*a^49*b^9*x^5 +
10487575840*a^48*b^10*x^6 + 138199362560*a^47*b^11*x^7 + 1559669296640*a^46*b^12*x^8 + 15307142645760*a^45*b^1
3*x^9 + 132213955901440*a^44*b^14*x^10 + 1014679597711360*a^43*b^15*x^11 + 6973089963069440*a^42*b^16*x^12 + 4
3187120222453760*a^41*b^17*x^13 + 242350169382666240*a^40*b^18*x^14 + 1237803623997767680*a^39*b^19*x^15 + 577
6151710405099520*a^38*b^20*x^16 + 24706657556147732480*a^37*b^21*x^17 + 97135576692589854720*a^36*b^22*x^18 +
351845868093917102080*a^35*b^23*x^19 + 1176525866114028666880*a^34*b^24*x^20 + 3637921375028164689920*a^33*b^2
5*x^21 + 10416297057814031892480*a^32*b^26*x^22 + 27648786910817047019520*a^31*b^27*x^23 + 6809793973244561195
0080*a^30*b^28*x^24 + 155734354396356659904512*a^29*b^29*x^25 + 330852794670810763624448*a^28*b^30*x^26 + 6531
39452460361708470272*a^27*b^31*x^27 + 1198212996719775787778048*a^26*b^32*x^28 + 2042544495422587862515712*a^2
5*b^33*x^29 + 3234344408387077820907520*a^24*b^34*x^30 + 4755098273824144600596480*a^23*b^35*x^31 + 6486081644
207666437816320*a^22*b^36*x^32 + 8200702234835514640302080*a^21*b^37*x^33 + 9599738338803379307806720*a^20*b^3
8*x^34 + 10389399158305667814522880*a^19*b^39*x^35 + 10377781510939708458270720*a^18*b^40*x^36 + 9548271404644
771435642880*a^17*b^41*x^37 + 8072690383192529064427520*a^16*b^42*x^38 + 6254213433999921491476480*a^15*b^43*x
^39 + 4425532940504048766812160*a^14*b^44*x^40 + 2849216487120313767690240*a^13*b^45*x^41 + 166142402568636675
9157760*a^12*b^46*x^42 + 872751165116932308336640*a^11*b^47*x^43 + 410349942921579420712960*a^10*b^48*x^44 + 1
71349494017073044520960*a^9*b^49*x^45 + 62938359650050522480640*a^8*b^50*x^46 + 20093333280970149724160*a^7*b^
51*x^47 + 5490842145935683747840*a^6*b^52*x^48 + 1258632246860300943360*a^5*b^53*x^49 + 235378382724705288192*
a^4*b^54*x^50 + 34488565946403258368*a^3*b^55*x^51 + 3713217892766973952*a^2*b^56*x^52 + 261208778387488768*a*
b^57*x^53 + 9007199254740992*b^58*x^54)) + (30*b^2*ArcTanh[(Sqrt[b^2]*x)/a - Sqrt[a^2 + 2*a*b*x + b^2*x^2]/a])
/a^7
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fricas [A] time = 0.41, size = 218, normalized size = 0.78 \begin {gather*} \frac {60 \, a b^{5} x^{5} + 210 \, a^{2} b^{4} x^{4} + 260 \, a^{3} b^{3} x^{3} + 125 \, a^{4} b^{2} x^{2} + 12 \, a^{5} b x - 2 \, a^{6} - 60 \, {\left (b^{6} x^{6} + 4 \, a b^{5} x^{5} + 6 \, a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{3} + a^{4} b^{2} x^{2}\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{6} x^{6} + 4 \, a b^{5} x^{5} + 6 \, a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{3} + a^{4} b^{2} x^{2}\right )} \log \relax (x)}{4 \, {\left (a^{7} b^{4} x^{6} + 4 \, a^{8} b^{3} x^{5} + 6 \, a^{9} b^{2} x^{4} + 4 \, a^{10} b x^{3} + a^{11} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")
[Out]
1/4*(60*a*b^5*x^5 + 210*a^2*b^4*x^4 + 260*a^3*b^3*x^3 + 125*a^4*b^2*x^2 + 12*a^5*b*x - 2*a^6 - 60*(b^6*x^6 + 4
*a*b^5*x^5 + 6*a^2*b^4*x^4 + 4*a^3*b^3*x^3 + a^4*b^2*x^2)*log(b*x + a) + 60*(b^6*x^6 + 4*a*b^5*x^5 + 6*a^2*b^4
*x^4 + 4*a^3*b^3*x^3 + a^4*b^2*x^2)*log(x))/(a^7*b^4*x^6 + 4*a^8*b^3*x^5 + 6*a^9*b^2*x^4 + 4*a^10*b*x^3 + a^11
*x^2)
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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^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
[Out]
sage0*x
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maple [A] time = 0.07, size = 218, normalized size = 0.78 \begin {gather*} -\frac {\left (-60 b^{6} x^{6} \ln \relax (x )+60 b^{6} x^{6} \ln \left (b x +a \right )-240 a \,b^{5} x^{5} \ln \relax (x )+240 a \,b^{5} x^{5} \ln \left (b x +a \right )-360 a^{2} b^{4} x^{4} \ln \relax (x )+360 a^{2} b^{4} x^{4} \ln \left (b x +a \right )-60 a \,b^{5} x^{5}-240 a^{3} b^{3} x^{3} \ln \relax (x )+240 a^{3} b^{3} x^{3} \ln \left (b x +a \right )-210 a^{2} b^{4} x^{4}-60 a^{4} b^{2} x^{2} \ln \relax (x )+60 a^{4} b^{2} x^{2} \ln \left (b x +a \right )-260 a^{3} b^{3} x^{3}-125 a^{4} b^{2} x^{2}-12 a^{5} b x +2 a^{6}\right ) \left (b x +a \right )}{4 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} a^{7} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
[Out]
-1/4*(60*ln(b*x+a)*x^6*b^6-60*ln(x)*x^6*b^6+240*ln(b*x+a)*x^5*a*b^5-240*ln(x)*x^5*a*b^5+360*a^2*b^4*x^4*ln(b*x
+a)-360*ln(x)*x^4*a^2*b^4-60*a*b^5*x^5+240*a^3*b^3*x^3*ln(b*x+a)-240*ln(x)*x^3*a^3*b^3-210*a^2*b^4*x^4+60*a^4*
b^2*x^2*ln(b*x+a)-60*ln(x)*x^2*a^4*b^2-260*a^3*b^3*x^3-125*a^4*b^2*x^2-12*a^5*b*x+2*a^6)*(b*x+a)/x^2/a^7/((b*x
+a)^2)^(5/2)
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maxima [A] time = 1.38, size = 178, normalized size = 0.64 \begin {gather*} -\frac {15 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} b^{2} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{7}} + \frac {5 \, b^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{4}} + \frac {15 \, b^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{6}} + \frac {7 \, b}{2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} x} - \frac {1}{2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} x^{2}} + \frac {15}{2 \, a^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {1}{4 \, a^{3} b^{2} {\left (x + \frac {a}{b}\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")
[Out]
-15*(-1)^(2*a*b*x + 2*a^2)*b^2*log(2*a*b*x/abs(x) + 2*a^2/abs(x))/a^7 + 5*b^2/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)
*a^4) + 15*b^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*a^6) + 7/2*b/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3*x) - 1/2/((b^2
*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*x^2) + 15/2/(a^5*(x + a/b)^2) + 1/4/(a^3*b^2*(x + a/b)^4)
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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+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^3*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)),x)
[Out]
int(1/(x^3*(a^2 + b^2*x^2 + 2*a*b*x)^(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\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**2+2*a*b*x+a**2)**(5/2),x)
[Out]
Integral(1/(x**3*((a + b*x)**2)**(5/2)), x)
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