3.7.39 \(\int \frac {A+B x}{x^4 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=211 \[ \frac {(a+b x) (A b-a B)}{2 a^2 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x)}{3 a x^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2 \log (x) (a+b x) (A b-a B)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 (a+b x) (A b-a B) \log (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (a+b x) (A b-a B)}{a^3 x \sqrt {a^2+2 a b x+b^2 x^2}} \]

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Rubi [A]  time = 0.10, antiderivative size = 211, 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 (a+b x) (A b-a B)}{a^3 x \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (A b-a B)}{2 a^2 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2 \log (x) (a+b x) (A b-a B)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 (a+b x) (A b-a B) \log (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x)}{3 a x^3 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

-(A*(a + b*x))/(3*a*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*B)*(a + b*x))/(2*a^2*x^2*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) - (b*(A*b - a*B)*(a + b*x))/(a^3*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (b^2*(A*b - a*B)*(a + b*x)*Lo
g[x])/(a^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (b^2*(A*b - a*B)*(a + b*x)*Log[a + b*x])/(a^4*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^4 \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^4 \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^4}+\frac {-A b+a B}{a^2 b x^3}+\frac {A b-a B}{a^3 x^2}+\frac {b (-A b+a B)}{a^4 x}-\frac {b^2 (-A b+a B)}{a^4 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {A (a+b x)}{3 a x^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) (a+b x)}{2 a^2 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (A b-a B) (a+b x)}{a^3 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2 (A b-a B) (a+b x) \log (x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 (A b-a B) (a+b x) \log (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 102, normalized size = 0.48 \begin {gather*} -\frac {(a+b x) \left (a \left (a^2 (2 A+3 B x)-3 a b x (A+2 B x)+6 A b^2 x^2\right )+6 b^2 x^3 \log (x) (A b-a B)+6 b^2 x^3 (a B-A b) \log (a+b x)\right )}{6 a^4 x^3 \sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

-1/6*((a + b*x)*(a*(6*A*b^2*x^2 - 3*a*b*x*(A + 2*B*x) + a^2*(2*A + 3*B*x)) + 6*b^2*(A*b - a*B)*x^3*Log[x] + 6*
b^2*(-(A*b) + a*B)*x^3*Log[a + b*x]))/(a^4*x^3*Sqrt[(a + b*x)^2])

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IntegrateAlgebraic [B]  time = 27.26, size = 3036, normalized size = 14.39 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)/(x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

(-16*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-2*a^39*A*b^3 - 145*a^38*A*b^4*x - 3*a^39*b^3*B*x - 5114*a^37*A*b^5*x^2 -
216*a^38*b^4*B*x^2 - 116901*a^36*A*b^6*x^3 - 7551*a^37*b^5*B*x^3 - 1946910*a^35*A*b^7*x^4 - 170706*a^36*b^6*B*
x^4 - 25177068*a^34*A*b^8*x^5 - 2804328*a^35*b^7*B*x^5 - 263109000*a^33*A*b^9*x^6 - 35662032*a^34*b^8*B*x^6 -
2283278976*a^32*A*b^10*x^7 - 365151024*a^33*b^9*B*x^7 - 16779949824*a^31*A*b^11*x^8 - 3091280544*a^32*b^10*B*x
^8 - 105974725120*a^30*A*b^12*x^9 - 22045630464*a^31*b^11*B*x^9 - 581687118848*a^29*A*b^13*x^10 - 134239598592
*a^30*b^12*B*x^10 - 2799513041920*a^28*A*b^14*x^11 - 704731180032*a^29*b^13*B*x^11 - 11896314587136*a^27*A*b^1
5*x^12 - 3211055622144*a^28*b^14*B*x^12 - 44883791253504*a^26*A*b^16*x^13 - 12748821356544*a^27*b^15*B*x^13 -
151017362841600*a^25*A*b^17*x^14 - 44154197950464*a^26*b^16*B*x^14 - 454702828584960*a^24*A*b^18*x^15 - 133071
057764352*a^25*b^17*B*x^15 - 1228421033164800*a^23*A*b^19*x^16 - 346215966474240*a^24*b^18*B*x^16 - 2983578273
054720*a^22*A*b^20*x^17 - 763419389460480*a^23*b^19*B*x^17 - 6523447219322880*a^21*A*b^21*x^18 - 1365440743342
080*a^22*b^20*B*x^18 - 12849688500633600*a^20*A*b^22*x^19 - 1731924258324480*a^21*b^21*B*x^19 - 22807023243755
520*a^19*A*b^23*x^20 - 507100983459840*a^20*b^22*B*x^20 - 36462975477350400*a^18*A*b^24*x^21 + 518369894203392
0*a^19*b^23*B*x^21 - 52462389575024640*a^17*A*b^25*x^22 + 19811818316759040*a^18*b^24*B*x^22 - 678277710191001
60*a^16*A*b^26*x^23 + 48242645941616640*a^17*b^25*B*x^23 - 78633368970854400*a^15*A*b^27*x^24 + 92914873364643
840*a^16*b^26*B*x^24 - 81511015665106944*a^14*A*b^28*x^25 + 150366847311544320*a^15*b^27*B*x^25 - 752757298849
38240*a^13*A*b^29*x^26 + 209516678961168384*a^14*b^28*B*x^26 - 61649279210487808*a^12*A*b^30*x^27 + 2541282795
17134848*a^13*b^29*B*x^27 - 44518268702031872*a^11*A*b^31*x^28 + 269528035416342528*a^12*b^30*B*x^28 - 2814243
6320542720*a^10*A*b^32*x^29 + 250112958548410368*a^11*b^31*B*x^29 - 15433601516568576*a^9*A*b^33*x^30 + 202625
541507907584*a^10*b^32*B*x^30 - 7258548417331200*a^8*A*b^34*x^31 + 142646439183384576*a^9*b^33*B*x^31 - 288414
3553708032*a^7*A*b^35*x^32 + 86643575705567232*a^8*b^34*B*x^32 - 949114757971968*a^6*A*b^36*x^33 + 44953605916
065792*a^7*b^35*B*x^33 - 251642133872640*a^5*A*b^37*x^34 + 19650815408996352*a^6*b^36*B*x^34 - 51629801865216*
a^4*A*b^38*x^35 + 7101990416941056*a^5*b^37*B*x^35 - 7687991459840*a^3*A*b^39*x^36 + 2066248636563456*a^4*b^38
*B*x^36 - 738734374912*a^2*A*b^40*x^37 + 465196497764352*a^3*b^39*B*x^37 - 34359738368*a*A*b^41*x^38 + 7607246
0746752*a^2*b^40*B*x^38 + 8040178778112*a*b^41*B*x^39 + 412316860416*b^42*B*x^40) - 16*Sqrt[b^2]*(2*a^40*A*b^2
+ 147*a^39*A*b^3*x + 3*a^40*b^2*B*x + 5259*a^38*A*b^4*x^2 + 219*a^39*b^3*B*x^2 + 122015*a^37*A*b^5*x^3 + 7767
*a^38*b^4*B*x^3 + 2063811*a^36*A*b^6*x^4 + 178257*a^37*b^5*B*x^4 + 27123978*a^35*A*b^7*x^5 + 2975034*a^36*b^6*
B*x^5 + 288286068*a^34*A*b^8*x^6 + 38466360*a^35*b^7*B*x^6 + 2546387976*a^33*A*b^9*x^7 + 400813056*a^34*b^8*B*
x^7 + 19063228800*a^32*A*b^10*x^8 + 3456431568*a^33*b^9*B*x^8 + 122754674944*a^31*A*b^11*x^9 + 25136911008*a^3
2*b^10*B*x^9 + 687661843968*a^30*A*b^12*x^10 + 156285229056*a^31*b^11*B*x^10 + 3381200160768*a^29*A*b^13*x^11
+ 838970778624*a^30*b^12*B*x^11 + 14695827629056*a^28*A*b^14*x^12 + 3915786802176*a^29*b^13*B*x^12 + 567801058
40640*a^27*A*b^15*x^13 + 15959876978688*a^28*b^14*B*x^13 + 195901154095104*a^26*A*b^16*x^14 + 56903019307008*a
^27*b^15*B*x^14 + 605720191426560*a^25*A*b^17*x^15 + 177225255714816*a^26*b^16*B*x^15 + 1683123861749760*a^24*
A*b^18*x^16 + 479287024238592*a^25*b^17*B*x^16 + 4211999306219520*a^23*A*b^19*x^17 + 1109635355934720*a^24*b^1
8*B*x^17 + 9507025492377600*a^22*A*b^20*x^18 + 2128860132802560*a^23*b^19*B*x^18 + 19373135719956480*a^21*A*b^
21*x^19 + 3097365001666560*a^22*b^20*B*x^19 + 35656711744389120*a^20*A*b^22*x^20 + 2239025241784320*a^21*b^21*
B*x^20 + 59269998721105920*a^19*A*b^23*x^21 - 4676597958574080*a^20*b^22*B*x^21 + 88925365052375040*a^18*A*b^2
4*x^22 - 24995517258792960*a^19*b^23*B*x^22 + 120290160594124800*a^17*A*b^25*x^23 - 68054464258375680*a^18*b^2
4*B*x^23 + 146461139989954560*a^16*A*b^26*x^24 - 141157519306260480*a^17*b^25*B*x^24 + 160144384635961344*a^15
*A*b^27*x^25 - 243281720676188160*a^16*b^26*B*x^25 + 156786745550045184*a^14*A*b^28*x^26 - 359883526272712704*
a^15*b^27*B*x^26 + 136925009095426048*a^13*A*b^29*x^27 - 463644958478303232*a^14*b^28*B*x^27 + 106167547912519
680*a^12*A*b^30*x^28 - 523656314933477376*a^13*b^29*B*x^28 + 72660705022574592*a^11*A*b^31*x^29 - 519640993964
752896*a^12*b^30*B*x^29 + 43576037837111296*a^10*A*b^32*x^30 - 452738500056317952*a^11*b^31*B*x^30 + 226921499
33899776*a^9*A*b^33*x^31 - 345271980691292160*a^10*b^32*B*x^31 + 10142691971039232*a^8*A*b^34*x^32 - 229290014
888951808*a^9*b^33*B*x^32 + 3833258311680000*a^7*A*b^35*x^33 - 131597181621633024*a^8*b^34*B*x^33 + 1200756891
844608*a^6*A*b^36*x^34 - 64604421325062144*a^7*b^35*B*x^34 + 303271935737856*a^5*A*b^37*x^35 - 267528058259374
08*a^6*b^36*B*x^35 + 59317793325056*a^4*A*b^38*x^36 - 9168239053504512*a^5*b^37*B*x^36 + 8426725834752*a^3*A*b
^39*x^37 - 2531445134327808*a^4*b^38*B*x^37 + 773094113280*a^2*A*b^40*x^38 - 541268958511104*a^3*b^39*B*x^38 +
34359738368*a*A*b^41*x^39 - 84112639524864*a^2*b^40*B*x^39 - 8452495638528*a*b^41*B*x^40 - 412316860416*b^42*
B*x^41))/(3*a^2*Sqrt[b^2]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(32*a^38*b^2*x^3 + 2368*a^37*b^3*x^4 + 85280*a^36*b^4*
x^5 + 1991424*a^35*b^5*x^6 + 33895680*a^34*b^6*x^7 + 448186368*a^33*b^7*x^8 + 4791316992*a^32*b^8*x^9 + 425562
93120*a^31*b^9*x^10 + 320265977856*a^30*b^10*x^11 + 2072421007360*a^29*b^11*x^12 + 11661974601728*a^28*b^12*x^
13 + 57575209172992*a^27*b^13*x^14 + 251137846149120*a^26*b^14*x^15 + 973253803769856*a^25*b^15*x^16 + 3365932
223692800*a^24*b^16*x^17 + 10424834756444160*a^23*b^17*x^18 + 28992809667133440*a^22*b^18*x^19 + 7255032059658
2400*a^21*b^19*x^20 + 163574499948625920*a^20*b^20*x^21 + 332558077054156800*a^19*b^21*x^22 + 6098233653932851
20*a^18*b^22*x^23 + 1008320668741140480*a^17*b^23*x^24 + 1502053114105036800*a^16*b^24*x^25 + 2013014245653872
640*a^15*b^25*x^26 + 2422115453317939200*a^14*b^26*x^27 + 2609386331050082304*a^13*b^27*x^28 + 250807101391739
2896*a^12*b^28*x^29 + 2141176316727132160*a^11*b^29*x^30 + 1614481075604553728*a^10*b^30*x^31 + 10676230417914
26560*a^9*b^31*x^32 + 613684065626750976*a^8*b^32*x^33 + 303169990394118144*a^7*b^33*x^34 + 126834851016867840
*a^6*b^34*x^35 + 44061004337774592*a^5*b^35*x^36 + 12367444228177920*a^4*b^36*x^37 + 2694902999678976*a^3*b^37
*x^38 + 427710023204864*a^2*b^38*x^39 + 43980465111040*a*b^39*x^40 + 2199023255552*b^40*x^41) + 3*a^2*(-32*a^3
9*b^3*x^3 - 2400*a^38*b^4*x^4 - 87648*a^37*b^5*x^5 - 2076704*a^36*b^6*x^6 - 35887104*a^35*b^7*x^7 - 482082048*
a^34*b^8*x^8 - 5239503360*a^33*b^9*x^9 - 47347610112*a^32*b^10*x^10 - 362822270976*a^31*b^11*x^11 - 2392686985
216*a^30*b^12*x^12 - 13734395609088*a^29*b^13*x^13 - 69237183774720*a^28*b^14*x^14 - 308713055322112*a^27*b^15
*x^15 - 1224391649918976*a^26*b^16*x^16 - 4339186027462656*a^25*b^17*x^17 - 13790766980136960*a^24*b^18*x^18 -
39417644423577600*a^23*b^19*x^19 - 101543130263715840*a^22*b^20*x^20 - 236124820545208320*a^21*b^21*x^21 - 49
6132577002782720*a^20*b^22*x^22 - 942381442447441920*a^19*b^23*x^23 - 1618144034134425600*a^18*b^24*x^24 - 251
0373782846177280*a^17*b^25*x^25 - 3515067359758909440*a^16*b^26*x^26 - 4435129698971811840*a^15*b^27*x^27 - 50
31501784368021504*a^14*b^28*x^28 - 5117457344967475200*a^13*b^29*x^29 - 4649247330644525056*a^12*b^30*x^30 - 3
755657392331685888*a^11*b^31*x^31 - 2682104117395980288*a^10*b^32*x^32 - 1681307107418177536*a^9*b^33*x^33 - 9
16854056020869120*a^8*b^34*x^34 - 430004841410985984*a^7*b^35*x^35 - 170895855354642432*a^6*b^36*x^36 - 564284
48565952512*a^5*b^37*x^37 - 15062347227856896*a^4*b^38*x^38 - 3122613022883840*a^3*b^39*x^39 - 471690488315904
*a^2*b^40*x^40 - 46179488366592*a*b^41*x^41 - 2199023255552*b^42*x^42)) + ((-2*A*b^3)/a^3 + (2*(b^2)^(3/2)*B*x
*ArcTanh[(-(Sqrt[b^2]*x) + Sqrt[a^2 + 2*a*b*x + b^2*x^2])/a])/a^3 - (2*b^2*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Arc
Tanh[(-(Sqrt[b^2]*x) + Sqrt[a^2 + 2*a*b*x + b^2*x^2])/a])/a^3)/(-(Sqrt[b^2]*x) + Sqrt[a^2 + 2*a*b*x + b^2*x^2]
) - (2*A*b^3*ArcTanh[(Sqrt[b^2]*x)/a - Sqrt[a^2 + 2*a*b*x + b^2*x^2]/a])/a^4

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fricas [A]  time = 0.43, size = 94, normalized size = 0.45 \begin {gather*} -\frac {6 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} \log \left (b x + a\right ) - 6 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} \log \relax (x) + 2 \, A a^{3} - 6 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} + 3 \, {\left (B a^{3} - A a^{2} b\right )} x}{6 \, a^{4} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(6*(B*a*b^2 - A*b^3)*x^3*log(b*x + a) - 6*(B*a*b^2 - A*b^3)*x^3*log(x) + 2*A*a^3 - 6*(B*a^2*b - A*a*b^2)*
x^2 + 3*(B*a^3 - A*a^2*b)*x)/(a^4*x^3)

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giac [A]  time = 0.16, size = 153, normalized size = 0.73 \begin {gather*} \frac {{\left (B a b^{2} \mathrm {sgn}\left (b x + a\right ) - A b^{3} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | x \right |}\right )}{a^{4}} - \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 | b x + a \right |}\right )}{a^{4} b} - \frac {2 \, A a^{3} \mathrm {sgn}\left (b x + a\right ) - 6 \, {\left (B a^{2} b \mathrm {sgn}\left (b x + a\right ) - A a b^{2} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 3 \, {\left (B a^{3} \mathrm {sgn}\left (b x + a\right ) - A a^{2} b \mathrm {sgn}\left (b x + a\right )\right )} x}{6 \, a^{4} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*a*b^2*sgn(b*x + a) - A*b^3*sgn(b*x + a))*log(abs(x))/a^4 - (B*a*b^3*sgn(b*x + a) - A*b^4*sgn(b*x + a))*log(
abs(b*x + a))/(a^4*b) - 1/6*(2*A*a^3*sgn(b*x + a) - 6*(B*a^2*b*sgn(b*x + a) - A*a*b^2*sgn(b*x + a))*x^2 + 3*(B
*a^3*sgn(b*x + a) - A*a^2*b*sgn(b*x + a))*x)/(a^4*x^3)

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maple [A]  time = 0.07, size = 119, normalized size = 0.56 \begin {gather*} \frac {\left (b x +a \right ) \left (-6 A \,b^{3} x^{3} \ln \relax (x )+6 A \,b^{3} x^{3} \ln \left (b x +a \right )+6 B a \,b^{2} x^{3} \ln \relax (x )-6 B a \,b^{2} x^{3} \ln \left (b x +a \right )-6 A a \,b^{2} x^{2}+6 B \,a^{2} b \,x^{2}+3 A \,a^{2} b x -3 B \,a^{3} x -2 A \,a^{3}\right )}{6 \sqrt {\left (b x +a \right )^{2}}\, a^{4} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^4/((b*x+a)^2)^(1/2),x)

[Out]

1/6*(b*x+a)*(6*A*ln(b*x+a)*x^3*b^3-6*A*b^3*x^3*ln(x)-6*B*ln(b*x+a)*x^3*a*b^2+6*B*a*b^2*x^3*ln(x)-6*A*a*b^2*x^2
+6*B*a^2*b*x^2+3*A*a^2*b*x-3*B*a^3*x-2*A*a^3)/((b*x+a)^2)^(1/2)/a^4/x^3

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maxima [A]  time = 0.67, size = 224, normalized size = 1.06 \begin {gather*} -\frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} B b^{2} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{3}} + \frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} A b^{3} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{4}} + \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b}{2 \, a^{3} x} - \frac {11 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{2}}{6 \, a^{4} x} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B}{2 \, a^{2} x^{2}} + \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b}{6 \, a^{3} x^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A}{3 \, a^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-1)^(2*a*b*x + 2*a^2)*B*b^2*log(2*a*b*x/abs(x) + 2*a^2/abs(x))/a^3 + (-1)^(2*a*b*x + 2*a^2)*A*b^3*log(2*a*b*
x/abs(x) + 2*a^2/abs(x))/a^4 + 3/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*b/(a^3*x) - 11/6*sqrt(b^2*x^2 + 2*a*b*x + a
^2)*A*b^2/(a^4*x) - 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B/(a^2*x^2) + 5/6*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A*b/(a^3
*x^2) - 1/3*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A/(a^2*x^3)

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

[Out]

int((A + B*x)/(x^4*((a + b*x)^2)^(1/2)), x)

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sympy [A]  time = 0.53, size = 165, normalized size = 0.78 \begin {gather*} \frac {- 2 A a^{2} + x^{2} \left (- 6 A b^{2} + 6 B a b\right ) + x \left (3 A a b - 3 B a^{2}\right )}{6 a^{3} x^{3}} + \frac {b^{2} \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{3} + B a^{2} b^{2} - a b^{2} \left (- A b + B a\right )}{- 2 A b^{4} + 2 B a b^{3}} \right )}}{a^{4}} - \frac {b^{2} \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{3} + B a^{2} b^{2} + a b^{2} \left (- A b + B a\right )}{- 2 A b^{4} + 2 B a b^{3}} \right )}}{a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**4/((b*x+a)**2)**(1/2),x)

[Out]

(-2*A*a**2 + x**2*(-6*A*b**2 + 6*B*a*b) + x*(3*A*a*b - 3*B*a**2))/(6*a**3*x**3) + b**2*(-A*b + B*a)*log(x + (-
A*a*b**3 + B*a**2*b**2 - a*b**2*(-A*b + B*a))/(-2*A*b**4 + 2*B*a*b**3))/a**4 - b**2*(-A*b + B*a)*log(x + (-A*a
*b**3 + B*a**2*b**2 + a*b**2*(-A*b + B*a))/(-2*A*b**4 + 2*B*a*b**3))/a**4

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