3.7.40 \(\int \frac {A+B x}{x^5 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx\)
Optimal. Leaf size=256 \[ \frac {(a+b x) (A b-a B)}{3 a^2 x^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x)}{4 a x^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^3 \log (x) (a+b x) (A b-a B)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^3 (a+b x) (A b-a B) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 (a+b x) (A b-a B)}{a^4 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (a+b x) (A b-a B)}{2 a^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.12, antiderivative size = 256, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used =
{770, 77} \begin {gather*} \frac {b^2 (a+b x) (A b-a B)}{a^4 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (a+b x) (A b-a B)}{2 a^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (A b-a B)}{3 a^2 x^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^3 \log (x) (a+b x) (A b-a B)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^3 (a+b x) (A b-a B) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x)}{4 a x^4 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/(x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]
[Out]
-(A*(a + b*x))/(4*a*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*B)*(a + b*x))/(3*a^2*x^3*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) - (b*(A*b - a*B)*(a + b*x))/(2*a^3*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (b^2*(A*b - a*B)*(a + b*x
))/(a^4*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (b^3*(A*b - a*B)*(a + b*x)*Log[x])/(a^5*Sqrt[a^2 + 2*a*b*x + b^2*x^
2]) - (b^3*(A*b - a*B)*(a + b*x)*Log[a + b*x])/(a^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
Rule 77
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Rule 770
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]
Rubi steps
\begin {align*} \int \frac {A+B x}{x^5 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {A+B x}{x^5 \left (a b+b^2 x\right )} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \left (\frac {A}{a b x^5}+\frac {-A b+a B}{a^2 b x^4}+\frac {A b-a B}{a^3 x^3}+\frac {b (-A b+a B)}{a^4 x^2}-\frac {b^2 (-A b+a B)}{a^5 x}+\frac {b^3 (-A b+a B)}{a^5 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {A (a+b x)}{4 a x^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) (a+b x)}{3 a^2 x^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (A b-a B) (a+b x)}{2 a^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 (A b-a B) (a+b x)}{a^4 x \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^3 (A b-a B) (a+b x) \log (x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^3 (A b-a B) (a+b x) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 121, normalized size = 0.47 \begin {gather*} -\frac {(a+b x) \left (a \left (a^3 (3 A+4 B x)-2 a^2 b x (2 A+3 B x)+6 a b^2 x^2 (A+2 B x)-12 A b^3 x^3\right )-12 b^3 x^4 \log (x) (A b-a B)+12 b^3 x^4 (A b-a B) \log (a+b x)\right )}{12 a^5 x^4 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/(x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]
[Out]
-1/12*((a + b*x)*(a*(-12*A*b^3*x^3 + 6*a*b^2*x^2*(A + 2*B*x) - 2*a^2*b*x*(2*A + 3*B*x) + a^3*(3*A + 4*B*x)) -
12*b^3*(A*b - a*B)*x^4*Log[x] + 12*b^3*(A*b - a*B)*x^4*Log[a + b*x]))/(a^5*x^4*Sqrt[(a + b*x)^2])
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IntegrateAlgebraic [B] time = 144.24, size = 4265, normalized size = 16.66 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(A + B*x)/(x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]
[Out]
(2*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-3*a^57*A*b - 317*a^56*A*b^2*x - 4*a^57*b*B*x - 16435*a^55*A*b^3*x^2 - 4
22*a^56*b^2*B*x^2 - 557229*a^54*A*b^4*x^3 - 21846*a^55*b^3*B*x^3 - 13894402*a^53*A*b^5*x^4 - 739670*a^54*b^4*B
*x^4 - 271672850*a^52*A*b^6*x^5 - 18421342*a^53*b^5*B*x^5 - 4337143836*a^51*A*b^7*x^6 - 359837100*a^52*b^6*B*x
^6 - 58124825800*a^50*A*b^8*x^7 - 5740821008*a^51*b^7*B*x^7 - 667238416880*a^49*A*b^9*x^8 - 76915029568*a^50*b
^8*B*x^8 - 6661915424640*a^48*A*b^10*x^9 - 883125016800*a^49*b^9*B*x^9 - 58546278327040*a^47*A*b^11*x^10 - 882
4603336640*a^48*b^10*B*x^10 - 457219477729280*a^46*A*b^12*x^11 - 77673661112320*a^47*b^11*B*x^11 - 31977687737
85600*a^45*A*b^13*x^12 - 608095358008320*a^46*b^12*B*x^12 - 20158054194181120*a^44*A*b^14*x^13 - 4268185542333
440*a^45*b^13*B*x^13 - 115146312238684160*a^43*A*b^15*x^14 - 27037561545164800*a^44*b^14*B*x^14 - 598690451217
223680*a^42*A*b^16*x^15 - 155445290934312960*a^43*b^15*B*x^15 - 2844164046416896000*a^41*A*b^17*x^16 - 8149951
40379607040*a^42*b^16*B*x^16 - 12385260068851712000*a^40*A*b^18*x^17 - 3912910229957263360*a^41*b^17*B*x^17 -
49572374862619607040*a^39*A*b^19*x^18 - 17265683227604582400*a^40*b^18*B*x^18 - 182793990899023544320*a^38*A*b
^20*x^19 - 70241014807366860800*a^39*b^19*B*x^19 - 622179642382727905280*a^37*A*b^21*x^20 - 264209995758604779
520*a^38*b^20*B*x^20 - 1957960418044046868480*a^36*A*b^22*x^21 - 921204943692974653440*a^37*b^21*B*x^21 - 5704
280886145928396800*a^35*A*b^23*x^22 - 2983982685925258035200*a^36*b^22*B*x^22 - 15401528673924522967040*a^34*A
*b^24*x^23 - 8998181559261268541440*a^35*b^23*B*x^23 - 38569099213499054161920*a^33*A*b^25*x^24 - 253061967128
34465792000*a^34*b^24*B*x^24 - 89633271616377551060992*a^32*A*b^26*x^25 - 66484824642585330974720*a^33*b^25*B*
x^25 - 193370972259236024680448*a^31*A*b^27*x^26 - 163405590100169621241856*a^32*b^26*B*x^26 - 387293735817076
738621440*a^30*A*b^28*x^27 - 376181863176239723839488*a^31*b^27*B*x^27 - 720021611882725251219456*a^29*A*b^29*
x^28 - 812007574252872032845824*a^30*b^28*B*x^28 - 1242004036400206030307328*a^28*A*b^30*x^29 - 16447403386016
17029857280*a^29*b^29*B*x^29 - 1986403135117299107758080*a^27*A*b^31*x^30 - 3127832391630149740658688*a^28*b^3
0*B*x^30 - 2942632751449592010113024*a^26*A*b^32*x^31 - 5586162055607461598986240*a^27*b^31*B*x^31 - 403205999
3337903028633600*a^25*A*b^33*x^32 - 9369065544421249522860032*a^26*b^32*B*x^32 - 5100887353195578638991360*a^2
4*A*b^34*x^33 - 14751931790742721671462912*a^25*b^33*B*x^33 - 5943688032831047550894080*a^23*A*b^35*x^34 - 217
91870716543451634073600*a^24*b^34*B*x^34 - 6359209726942002628526080*a^22*A*b^36*x^35 - 3017327834030136526635
0080*a^23*b^35*B*x^35 - 6221440965249143364648960*a^21*A*b^37*x^36 - 39110279855341317012848640*a^22*b^36*B*x^
36 - 5534422942607081681715200*a^20*A*b^38*x^37 - 47383825587164854186147840*a^21*b^37*B*x^37 - 44408531543169
81360394240*a^19*A*b^39*x^38 - 53560665008408226777006080*a^20*b^38*B*x^38 - 3175344362316856893112320*a^18*A*
b^40*x^39 - 56366537702201135569305600*a^19*b^39*B*x^39 - 1982480817847829434204160*a^17*A*b^41*x^40 - 5509530
7178765004486737920*a^18*b^40*B*x^40 - 1038716472230052836147200*a^16*A*b^42*x^41 - 49883074555155751349780480
*a^17*b^41*B*x^41 - 412810542648699912192000*a^15*A*b^43*x^42 - 41708256158480643045457920*a^16*b^42*B*x^42 -
75432821484570521108480*a^14*A*b^44*x^43 - 32095525534669895788134400*a^15*b^43*B*x^43 + 580526649106111240601
60*a^13*A*b^45*x^44 - 22644117749068360948121600*a^14*b^44*B*x^44 + 79865553475357366026240*a^12*A*b^46*x^45 -
14583555296017011431178240*a^13*b^45*B*x^45 + 58971164466871942512640*a^11*A*b^47*x^46 - 85310034619829484047
56480*a^12*b^46*B*x^46 + 32893073859235178086400*a^10*A*b^48*x^47 - 4506625206293205352448000*a^11*b^47*B*x^47
+ 14894684151092682424320*a^9*A*b^49*x^48 - 2135325976384711509934080*a^10*b^48*B*x^48 + 55883331151817447833
60*a^8*A*b^50*x^49 - 900153430695189938176000*a^9*b^49*B*x^49 + 1738723848437343715328*a^7*A*b^51*x^50 - 33429
8147803734726410240*a^8*b^50*B*x^50 + 443727567860816412672*a^6*A*b^52*x^51 - 108048410512384428343296*a^7*b^5
1*B*x^51 + 90862937231966863360*a^5*A*b^53*x^52 - 29924898996052754956288*a^6*b^52*B*x^52 + 143929414591226839
04*a^4*A*b^54*x^53 - 6958707127883990564864*a^5*b^53*B*x^53 + 1658450562779185152*a^3*A*b^55*x^54 - 1321237911
180285050880*a^4*b^54*B*x^54 + 123848989752688640*a^2*A*b^56*x^55 - 196685706526151671808*a^3*b^55*B*x^55 + 45
03599627370496*a*A*b^57*x^56 - 21527206218830970880*a^2*b^56*B*x^56 - 1540231072560709632*a*b^57*B*x^57 - 5404
3195528445952*b^58*B*x^58) + 2*b^3*Sqrt[b^2]*(3*a^58*A + 320*a^57*A*b*x + 4*a^58*B*x + 16752*a^56*A*b^2*x^2 +
426*a^57*b*B*x^2 + 573664*a^55*A*b^3*x^3 + 22268*a^56*b^2*B*x^3 + 14451631*a^54*A*b^4*x^4 + 761516*a^55*b^3*B*
x^4 + 285567252*a^53*A*b^5*x^5 + 19161012*a^54*b^4*B*x^5 + 4608816686*a^52*A*b^6*x^6 + 378258442*a^53*b^5*B*x^
6 + 62461969636*a^51*A*b^7*x^7 + 6100658108*a^52*b^6*B*x^7 + 725363242680*a^50*A*b^8*x^8 + 82655850576*a^51*b^
7*B*x^8 + 7329153841520*a^49*A*b^9*x^9 + 960040046368*a^50*b^8*B*x^9 + 65208193751680*a^48*A*b^10*x^10 + 97077
28353440*a^49*b^9*B*x^10 + 515765756056320*a^47*A*b^11*x^11 + 86498264448960*a^48*b^10*B*x^11 + 36549882515148
80*a^46*A*b^12*x^12 + 685769019120640*a^47*b^11*B*x^12 + 23355822967966720*a^45*A*b^13*x^13 + 4876280900341760
*a^46*b^12*B*x^13 + 135304366432865280*a^44*A*b^14*x^14 + 31305747087498240*a^45*b^13*B*x^14 + 713836763455907
840*a^43*A*b^15*x^15 + 182482852479477760*a^44*b^14*B*x^15 + 3442854497634119680*a^42*A*b^16*x^16 + 9704404313
13920000*a^43*b^15*B*x^16 + 15229424115268608000*a^41*A*b^17*x^17 + 4727905370336870400*a^42*b^16*B*x^17 + 619
57634931471319040*a^40*A*b^18*x^18 + 21178593457561845760*a^41*b^17*B*x^18 + 232366365761643151360*a^39*A*b^19
*x^19 + 87506698034971443200*a^40*b^18*B*x^19 + 804973633281751449600*a^38*A*b^20*x^20 + 334451010565971640320
*a^39*b^19*B*x^20 + 2580140060426774773760*a^37*A*b^21*x^21 + 1185414939451579432960*a^38*b^20*B*x^21 + 766224
1304189975265280*a^36*A*b^22*x^22 + 3905187629618232688640*a^37*b^21*B*x^22 + 21105809560070451363840*a^35*A*b
^23*x^23 + 11982164245186526576640*a^36*b^22*B*x^23 + 53970627887423577128960*a^34*A*b^24*x^24 + 3430437827209
5734333440*a^35*b^23*B*x^24 + 128202370829876605222912*a^33*A*b^25*x^25 + 91791021355419796766720*a^34*b^24*B*
x^25 + 283004243875613575741440*a^32*A*b^26*x^26 + 229890414742754952216576*a^33*b^25*B*x^26 + 580664708076312
763301888*a^31*A*b^27*x^27 + 539587453276409345081344*a^32*b^26*B*x^27 + 1107315347699801989840896*a^30*A*b^28
*x^28 + 1188189437429111756685312*a^31*b^27*B*x^28 + 1962025648282931281526784*a^29*A*b^29*x^29 + 245674791285
4489062703104*a^30*b^28*B*x^29 + 3228407171517505138065408*a^28*A*b^30*x^30 + 4772572730231766770515968*a^29*b
^29*B*x^30 + 4929035886566891117871104*a^27*A*b^31*x^31 + 8713994447237611339644928*a^28*b^30*B*x^31 + 6974692
744787495038746624*a^26*A*b^32*x^32 + 14955227600028711121846272*a^27*b^31*B*x^32 + 9132947346533481667624960*
a^25*A*b^33*x^33 + 24120997335163971194322944*a^26*b^32*B*x^33 + 11044575386026626189885440*a^24*A*b^34*x^34 +
36543802507286173305536512*a^25*b^33*B*x^34 + 12302897759773050179420160*a^23*A*b^35*x^35 + 51965149056844816
900423680*a^24*b^34*B*x^35 + 12580650692191145993175040*a^22*A*b^36*x^36 + 69283558195642682279198720*a^23*b^3
5*B*x^36 + 11755863907856225046364160*a^21*A*b^37*x^37 + 86494105442506171198996480*a^22*b^36*B*x^37 + 9975276
096924063042109440*a^20*A*b^38*x^38 + 100944490595573080963153920*a^21*b^37*B*x^38 + 7616197516633838253506560
*a^19*A*b^39*x^39 + 109927202710609362346311680*a^20*b^38*B*x^39 + 5157825180164686327316480*a^18*A*b^40*x^40
+ 111461844880966140056043520*a^19*b^39*B*x^40 + 3021197290077882270351360*a^17*A*b^41*x^41 + 1049783817339207
55836518400*a^18*b^40*B*x^41 + 1451527014878752748339200*a^16*A*b^42*x^42 + 91591330713636394395238400*a^17*b^
41*B*x^42 + 488243364133270433300480*a^15*A*b^43*x^43 + 73803781693150538833592320*a^16*b^42*B*x^43 + 17380156
573959397048320*a^14*A*b^44*x^44 + 54739643283738256736256000*a^15*b^43*B*x^44 - 137918218385968490086400*a^13
*A*b^45*x^45 + 37227673045085372379299840*a^14*b^44*B*x^45 - 138836717942229308538880*a^12*A*b^46*x^46 + 23114
558757999959835934720*a^13*b^45*B*x^46 - 91864238326107120599040*a^11*A*b^47*x^47 + 13037628668276153757204480
*a^12*b^46*B*x^47 - 47787758010327860510720*a^10*A*b^48*x^48 + 6641951182677916862382080*a^11*b^47*B*x^48 - 20
483017266274427207680*a^9*A*b^49*x^49 + 3035479407079901448110080*a^10*b^48*B*x^49 - 7327056963619088498688*a^
8*A*b^50*x^50 + 1234451578498924664586240*a^9*b^49*B*x^50 - 2182451416298160128000*a^7*A*b^51*x^51 + 442346558
316119154753536*a^8*b^50*B*x^51 - 534590505092783276032*a^6*A*b^52*x^52 + 137973309508437183299584*a^7*b^51*B*
x^52 - 105255878691089547264*a^5*A*b^53*x^53 + 36883606123936745521152*a^6*b^52*B*x^53 - 16051392021901869056*
a^4*A*b^54*x^54 + 8279945039064275615744*a^5*b^53*B*x^54 - 1782299552531873792*a^3*A*b^55*x^55 + 1517923617706
436722688*a^4*b^54*B*x^55 - 128352589380059136*a^2*A*b^56*x^56 + 218212912744982642688*a^3*b^55*B*x^56 - 45035
99627370496*a*A*b^57*x^57 + 23067437291391680512*a^2*b^56*B*x^57 + 1594274268089155584*a*b^57*B*x^58 + 5404319
5528445952*b^58*B*x^59))/(3*a^3*Sqrt[b^2]*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-8*a^55*b^3 - 856*a^54*b^4*x - 44
952*a^53*b^5*x^2 - 1544200*a^52*b^6*x^3 - 39024128*a^51*b^7*x^4 - 773577792*a^50*b^8*x^5 - 12524936320*a^49*b^
9*x^6 - 170296730240*a^48*b^10*x^7 - 1984120320000*a^47*b^11*x^8 - 20114496962560*a^46*b^12*x^9 - 179566706708
480*a^45*b^13*x^10 - 1425197285191680*a^44*b^14*x^11 - 10135466892328960*a^43*b^15*x^12 - 65002731573248000*a^
42*b^16*x^13 - 377988278551511040*a^41*b^17*x^14 - 2001964503866736640*a^40*b^18*x^15 - 9694802380060098560*a^
39*b^19*x^16 - 43067676513699102720*a^38*b^20*x^17 - 175999137245102080000*a^37*b^21*x^18 - 663215667843765370
880*a^36*b^22*x^19 - 2309235977416291123200*a^35*b^23*x^20 - 7442172334443656642560*a^34*b^24*x^21 - 222319136
87939148677120*a^33*b^25*x^22 - 61633746954246684672000*a^32*b^26*x^23 - 158724528652723242926080*a^31*b^27*x^
24 - 379996732316684052856832*a^30*b^28*x^25 - 846185238869335798185984*a^29*b^29*x^26 - 175330280985969560440
0128*a^28*b^30*x^27 - 3380879175384770071756800*a^27*b^31*x^28 - 6067075116758367775752192*a^26*b^32*x^29 - 10
130299260198164646330368*a^25*b^33*x^30 - 15732354628864642687959040*a^24*b^34*x^31 - 227114977441879134358732
80*a^23*b^35*x^32 - 30453924162774845358080000*a^22*b^36*x^33 - 37892949692201258262200320*a^21*b^37*x^34 - 43
698211760659985344757760*a^20*b^38*x^35 - 46636119738715911602831360*a^19*b^39*x^36 - 459808743429103649239859
20*a^18*b^40*x^37 - 41796027702620083716096000*a^17*b^41*x^38 - 34942032882729039775662080*a^16*b^42*x^39 - 26
791115326544811214766080*a^15*b^43*x^40 - 18777086658591383586078720*a^14*b^44*x^41 - 119833350775924333110886
40*a^13*b^45*x^42 - 6931978138751221104640000*a^12*b^46*x^43 - 3615072567383554586050560*a^11*b^47*x^44 - 1688
681358787565374668800*a^10*b^48*x^45 - 701054748617095008747520*a^9*b^49*x^46 - 256189999332369408983040*a^8*b
^50*x^47 - 81427614537648963584000*a^7*b^51*x^48 - 22167719416834671247360*a^6*b^52*x^49 - 5065556537073972805
632*a^5*b^53*x^50 - 944974547212455378944*a^4*b^54*x^51 - 138206465364745781248*a^3*b^55*x^52 - 14861878770322
636800*a^2*b^56*x^53 - 1044835113549955072*a*b^57*x^54 - 36028797018963968*b^58*x^55) + 3*a^3*x^4*(8*a^56*b^4
+ 864*a^55*b^5*x + 45808*a^54*b^6*x^2 + 1589152*a^53*b^7*x^3 + 40568328*a^52*b^8*x^4 + 812601920*a^51*b^9*x^5
+ 13298514112*a^50*b^10*x^6 + 182821666560*a^49*b^11*x^7 + 2154417050240*a^48*b^12*x^8 + 22098617282560*a^47*b
^13*x^9 + 199681203671040*a^46*b^14*x^10 + 1604763991900160*a^45*b^15*x^11 + 11560664177520640*a^44*b^16*x^12
+ 75138198465576960*a^43*b^17*x^13 + 442991010124759040*a^42*b^18*x^14 + 2379952782418247680*a^41*b^19*x^15 +
11696766883926835200*a^40*b^20*x^16 + 52762478893759201280*a^39*b^21*x^17 + 219066813758801182720*a^38*b^22*x^
18 + 839214805088867450880*a^37*b^23*x^19 + 2972451645260056494080*a^36*b^24*x^20 + 9751408311859947765760*a^3
5*b^25*x^21 + 29674086022382805319680*a^34*b^26*x^22 + 83865660642185833349120*a^33*b^27*x^23 + 22035827560696
9927598080*a^32*b^28*x^24 + 538721260969407295782912*a^31*b^29*x^25 + 1226181971186019851042816*a^30*b^30*x^26
+ 2599488048729031402586112*a^29*b^31*x^27 + 5134181985244465676156928*a^28*b^32*x^28 + 944795429214313784750
8992*a^27*b^33*x^29 + 16197374376956532422082560*a^26*b^34*x^30 + 25862653889062807334289408*a^25*b^35*x^31 +
38443852373052556123832320*a^24*b^36*x^32 + 53165421906962758793953280*a^23*b^37*x^33 + 6834687385497610362028
0320*a^22*b^38*x^34 + 81591161452861243606958080*a^21*b^39*x^35 + 90334331499375896947589120*a^20*b^40*x^36 +
92616994081626276526817280*a^19*b^41*x^37 + 87776902045530448640081920*a^18*b^42*x^38 + 7673806058534912349175
8080*a^17*b^43*x^39 + 61733148209273850990428160*a^16*b^44*x^40 + 45568201985136194800844800*a^15*b^45*x^41 +
30760421736183816897167360*a^14*b^46*x^42 + 18915313216343654415728640*a^13*b^47*x^43 + 1054705070613477569069
0560*a^12*b^48*x^44 + 5303753926171119960719360*a^11*b^49*x^45 + 2389736107404660383416320*a^10*b^50*x^46 + 95
7244747949464417730560*a^9*b^51*x^47 + 337617613870018372567040*a^8*b^52*x^48 + 103595333954483634831360*a^7*b
^53*x^49 + 27233275953908644052992*a^6*b^54*x^50 + 6010531084286428184576*a^5*b^55*x^51 + 10831810125772011601
92*a^4*b^56*x^52 + 153068344135068418048*a^3*b^57*x^53 + 15906713883872591872*a^2*b^58*x^54 + 1080863910568919
040*a*b^59*x^55 + 36028797018963968*b^60*x^56)) + ((2*A*b^4)/a^4 - (2*b^3*Sqrt[b^2]*B*x*ArcTanh[(-(Sqrt[b^2]*x
) + Sqrt[a^2 + 2*a*b*x + b^2*x^2])/a])/a^4 + (2*b^3*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(-(Sqrt[b^2]*x) +
Sqrt[a^2 + 2*a*b*x + b^2*x^2])/a])/a^4)/(-(Sqrt[b^2]*x) + Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*A*b^4*ArcTanh[(S
qrt[b^2]*x)/a - Sqrt[a^2 + 2*a*b*x + b^2*x^2]/a])/a^5
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fricas [A] time = 0.42, size = 117, normalized size = 0.46 \begin {gather*} \frac {12 \, {\left (B a b^{3} - A b^{4}\right )} x^{4} \log \left (b x + a\right ) - 12 \, {\left (B a b^{3} - A b^{4}\right )} x^{4} \log \relax (x) - 3 \, A a^{4} - 12 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} + 6 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} - 4 \, {\left (B a^{4} - A a^{3} b\right )} x}{12 \, a^{5} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/x^5/((b*x+a)^2)^(1/2),x, algorithm="fricas")
[Out]
1/12*(12*(B*a*b^3 - A*b^4)*x^4*log(b*x + a) - 12*(B*a*b^3 - A*b^4)*x^4*log(x) - 3*A*a^4 - 12*(B*a^2*b^2 - A*a*
b^3)*x^3 + 6*(B*a^3*b - A*a^2*b^2)*x^2 - 4*(B*a^4 - A*a^3*b)*x)/(a^5*x^4)
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giac [A] time = 0.23, size = 188, normalized size = 0.73 \begin {gather*} -\frac {{\left (B a b^{3} \mathrm {sgn}\left (b x + a\right ) - A b^{4} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac {{\left (B a b^{4} \mathrm {sgn}\left (b x + a\right ) - A b^{5} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{a^{5} b} - \frac {3 \, A a^{4} \mathrm {sgn}\left (b x + a\right ) + 12 \, {\left (B a^{2} b^{2} \mathrm {sgn}\left (b x + a\right ) - A a b^{3} \mathrm {sgn}\left (b x + a\right )\right )} x^{3} - 6 \, {\left (B a^{3} b \mathrm {sgn}\left (b x + a\right ) - A a^{2} b^{2} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 4 \, {\left (B a^{4} \mathrm {sgn}\left (b x + a\right ) - A a^{3} b \mathrm {sgn}\left (b x + a\right )\right )} x}{12 \, a^{5} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/x^5/((b*x+a)^2)^(1/2),x, algorithm="giac")
[Out]
-(B*a*b^3*sgn(b*x + a) - A*b^4*sgn(b*x + a))*log(abs(x))/a^5 + (B*a*b^4*sgn(b*x + a) - A*b^5*sgn(b*x + a))*log
(abs(b*x + a))/(a^5*b) - 1/12*(3*A*a^4*sgn(b*x + a) + 12*(B*a^2*b^2*sgn(b*x + a) - A*a*b^3*sgn(b*x + a))*x^3 -
6*(B*a^3*b*sgn(b*x + a) - A*a^2*b^2*sgn(b*x + a))*x^2 + 4*(B*a^4*sgn(b*x + a) - A*a^3*b*sgn(b*x + a))*x)/(a^5
*x^4)
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maple [A] time = 0.07, size = 143, normalized size = 0.56 \begin {gather*} -\frac {\left (b x +a \right ) \left (-12 A \,b^{4} x^{4} \ln \relax (x )+12 A \,b^{4} x^{4} \ln \left (b x +a \right )+12 B a \,b^{3} x^{4} \ln \relax (x )-12 B a \,b^{3} x^{4} \ln \left (b x +a \right )-12 A a \,b^{3} x^{3}+12 B \,a^{2} b^{2} x^{3}+6 A \,a^{2} b^{2} x^{2}-6 B \,a^{3} b \,x^{2}-4 A \,a^{3} b x +4 B \,a^{4} x +3 A \,a^{4}\right )}{12 \sqrt {\left (b x +a \right )^{2}}\, a^{5} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)/x^5/((b*x+a)^2)^(1/2),x)
[Out]
-1/12*(b*x+a)*(12*A*ln(b*x+a)*x^4*b^4-12*A*ln(x)*x^4*b^4-12*B*ln(b*x+a)*x^4*a*b^3+12*B*ln(x)*x^4*a*b^3-12*A*a*
b^3*x^3+12*B*x^3*a^2*b^2+6*A*a^2*b^2*x^2-6*B*x^2*a^3*b-4*A*a^3*b*x+4*B*a^4*x+3*A*a^4)/((b*x+a)^2)^(1/2)/x^4/a^
5
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maxima [A] time = 0.54, size = 284, normalized size = 1.11 \begin {gather*} \frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} B b^{3} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{4}} - \frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} A b^{4} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{5}} - \frac {11 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b^{2}}{6 \, a^{4} x} + \frac {25 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{3}}{12 \, a^{5} x} + \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b}{6 \, a^{3} x^{2}} - \frac {13 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{2}}{12 \, a^{4} x^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B}{3 \, a^{2} x^{3}} + \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b}{12 \, a^{3} x^{3}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A}{4 \, a^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/x^5/((b*x+a)^2)^(1/2),x, algorithm="maxima")
[Out]
(-1)^(2*a*b*x + 2*a^2)*B*b^3*log(2*a*b*x/abs(x) + 2*a^2/abs(x))/a^4 - (-1)^(2*a*b*x + 2*a^2)*A*b^4*log(2*a*b*x
/abs(x) + 2*a^2/abs(x))/a^5 - 11/6*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*b^2/(a^4*x) + 25/12*sqrt(b^2*x^2 + 2*a*b*x
+ a^2)*A*b^3/(a^5*x) + 5/6*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*b/(a^3*x^2) - 13/12*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A
*b^2/(a^4*x^2) - 1/3*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B/(a^2*x^3) + 7/12*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A*b/(a^3*x
^3) - 1/4*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A/(a^2*x^4)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{x^5\,\sqrt {{\left (a+b\,x\right )}^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*x)/(x^5*((a + b*x)^2)^(1/2)),x)
[Out]
int((A + B*x)/(x^5*((a + b*x)^2)^(1/2)), x)
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sympy [A] time = 0.59, size = 189, normalized size = 0.74 \begin {gather*} \frac {- 3 A a^{3} + x^{3} \left (12 A b^{3} - 12 B a b^{2}\right ) + x^{2} \left (- 6 A a b^{2} + 6 B a^{2} b\right ) + x \left (4 A a^{2} b - 4 B a^{3}\right )}{12 a^{4} x^{4}} - \frac {b^{3} \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{4} + B a^{2} b^{3} - a b^{3} \left (- A b + B a\right )}{- 2 A b^{5} + 2 B a b^{4}} \right )}}{a^{5}} + \frac {b^{3} \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{4} + B a^{2} b^{3} + a b^{3} \left (- A b + B a\right )}{- 2 A b^{5} + 2 B a b^{4}} \right )}}{a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/x**5/((b*x+a)**2)**(1/2),x)
[Out]
(-3*A*a**3 + x**3*(12*A*b**3 - 12*B*a*b**2) + x**2*(-6*A*a*b**2 + 6*B*a**2*b) + x*(4*A*a**2*b - 4*B*a**3))/(12
*a**4*x**4) - b**3*(-A*b + B*a)*log(x + (-A*a*b**4 + B*a**2*b**3 - a*b**3*(-A*b + B*a))/(-2*A*b**5 + 2*B*a*b**
4))/a**5 + b**3*(-A*b + B*a)*log(x + (-A*a*b**4 + B*a**2*b**3 + a*b**3*(-A*b + B*a))/(-2*A*b**5 + 2*B*a*b**4))
/a**5
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