3.2.67 \(\int \frac {1}{x^4 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx\)
Optimal. Leaf size=184 \[ \frac {b (a+b x)}{2 a^2 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {-a-b x}{3 a x^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^3 \log (x) (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^3 (a+b x) \log (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2 (a+b x)}{a^3 x \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.05, antiderivative size = 181, normalized size of antiderivative = 0.98,
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 {b^2 (a+b x)}{a^3 x \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (a+b x)}{2 a^2 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a+b x}{3 a x^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^3 \log (x) (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^3 (a+b x) \log (a+b x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]
[Out]
-(a + b*x)/(3*a*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (b*(a + b*x))/(2*a^2*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(b^2*(a + b*x))/(a^3*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (b^3*(a + b*x)*Log[x])/(a^4*Sqrt[a^2 + 2*a*b*x + b^2*
x^2]) + (b^3*(a + b*x)*Log[a + b*x])/(a^4*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^4 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {1}{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 {1}{a b x^4}-\frac {1}{a^2 x^3}+\frac {b}{a^3 x^2}-\frac {b^2}{a^4 x}+\frac {b^3}{a^4 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {a+b x}{3 a x^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (a+b x)}{2 a^2 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2 (a+b x)}{a^3 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^3 (a+b x) \log (x)}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^3 (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 = 0.02, size = 72, normalized size = 0.39 \begin {gather*} -\frac {(a+b x) \left (a \left (2 a^2-3 a b x+6 b^2 x^2\right )-6 b^3 x^3 \log (a+b x)+6 b^3 x^3 \log (x)\right )}{6 a^4 x^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/(x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]
[Out]
-1/6*((a + b*x)*(a*(2*a^2 - 3*a*b*x + 6*b^2*x^2) + 6*b^3*x^3*Log[x] - 6*b^3*x^3*Log[a + b*x]))/(a^4*x^3*Sqrt[(
a + b*x)^2])
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IntegrateAlgebraic [B] time = 15.52, size = 1877, normalized size = 10.20
result too large to display
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/(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^40*b^3 - 145*a^39*b^4*x - 5114*a^38*b^5*x^2 - 116913*a^37*b^6*x^3 - 1
947774*a^36*b^7*x^4 - 25207320*a^35*b^8*x^5 - 263795280*a^34*b^9*x^6 - 2294617296*a^33*b^10*x^7 - 16925343072*
a^32*b^11*x^8 - 107480682496*a^31*b^12*x^9 - 594633814016*a^30*b^13*x^10 - 2893719393280*a^29*b^14*x^11 - 1248
5059762176*a^28*b^15*x^12 - 48079541379072*a^27*b^16*x^13 - 166216566030336*a^26*b^17*x^14 - 518481114513408*a
^25*b^18*x^15 - 1465834637721600*a^24*b^19*x^16 - 3770975647825920*a^23*b^20*x^17 - 8857965503447040*a^22*b^21
*x^18 - 19052955557560320*a^21*b^22*x^19 - 37606859353620480*a^20*b^23*x^20 - 68203740738355200*a^19*b^24*x^21
- 113690137948323840*a^18*b^25*x^22 - 174056036294983680*a^17*b^26*x^23 - 244297089197015040*a^16*b^27*x^24 -
313453493002174464*a^15*b^28*x^25 - 366271117331005440*a^14*b^29*x^26 - 387951799312580608*a^13*b^30*x^27 - 3
70433102641627136*a^12*b^31*x^28 - 316839398660374528*a^11*b^32*x^29 - 240980795609579520*a^10*b^33*x^30 - 161
594563583016960*a^9*b^34*x^31 - 94570753894121472*a^8*b^35*x^32 - 47707418687176704*a^7*b^36*x^33 - 2042378067
3257472*a^6*b^37*x^34 - 7270421854420992*a^5*b^38*x^35 - 2092980513013760*a^4*b^39*x^36 - 467945276833792*a^3*
b^40*x^37 - 76209899700224*a^2*b^41*x^38 - 8040178778112*a*b^42*x^39 - 412316860416*b^43*x^40) - 16*Sqrt[b^2]*
(2*a^41*b^2 + 147*a^40*b^3*x + 5259*a^39*b^4*x^2 + 122027*a^38*b^5*x^3 + 2064687*a^37*b^6*x^4 + 27155094*a^36*
b^7*x^5 + 289002600*a^35*b^8*x^6 + 2558412576*a^34*b^9*x^7 + 19219960368*a^33*b^10*x^8 + 124406025568*a^32*b^1
1*x^9 + 702114496512*a^31*b^12*x^10 + 3488353207296*a^30*b^13*x^11 + 15378779155456*a^29*b^14*x^12 + 605646011
41248*a^28*b^15*x^13 + 214296107409408*a^27*b^16*x^14 + 684697680543744*a^26*b^17*x^15 + 1984315752235008*a^25
*b^18*x^16 + 5236810285547520*a^24*b^19*x^17 + 12628941151272960*a^23*b^20*x^18 + 27910921061007360*a^22*b^21*
x^19 + 56659814911180800*a^21*b^22*x^20 + 105810600091975680*a^20*b^23*x^21 + 181893878686679040*a^19*b^24*x^2
2 + 287746174243307520*a^18*b^25*x^23 + 418353125491998720*a^17*b^26*x^24 + 557750582199189504*a^16*b^27*x^25
+ 679724610333179904*a^15*b^28*x^26 + 754222916643586048*a^14*b^29*x^27 + 758384901954207744*a^13*b^30*x^28 +
687272501302001664*a^12*b^31*x^29 + 557820194269954048*a^11*b^32*x^30 + 402575359192596480*a^10*b^33*x^31 + 25
6165317477138432*a^9*b^34*x^32 + 142278172581298176*a^8*b^35*x^33 + 68131199360434176*a^7*b^36*x^34 + 27694202
527678464*a^6*b^37*x^35 + 9363402367434752*a^5*b^38*x^36 + 2560925789847552*a^4*b^39*x^37 + 544155176534016*a^
3*b^40*x^38 + 84250078478336*a^2*b^41*x^39 + 8452495638528*a*b^42*x^40 + 412316860416*b^43*x^41))/(3*a^3*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 + 42556293120*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 + 3365932223692800*a^24*b^16*x
^17 + 10424834756444160*a^23*b^17*x^18 + 28992809667133440*a^22*b^18*x^19 + 72550320596582400*a^21*b^19*x^20 +
163574499948625920*a^20*b^20*x^21 + 332558077054156800*a^19*b^21*x^22 + 609823365393285120*a^18*b^22*x^23 + 1
008320668741140480*a^17*b^23*x^24 + 1502053114105036800*a^16*b^24*x^25 + 2013014245653872640*a^15*b^25*x^26 +
2422115453317939200*a^14*b^26*x^27 + 2609386331050082304*a^13*b^27*x^28 + 2508071013917392896*a^12*b^28*x^29 +
2141176316727132160*a^11*b^29*x^30 + 1614481075604553728*a^10*b^30*x^31 + 1067623041791426560*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 + 4406
1004337774592*a^5*b^35*x^36 + 12367444228177920*a^4*b^36*x^37 + 2694902999678976*a^3*b^37*x^38 + 4277100232048
64*a^2*b^38*x^39 + 43980465111040*a*b^39*x^40 + 2199023255552*b^40*x^41) + 3*a^3*(-32*a^39*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 - 523950
3360*a^33*b^9*x^9 - 47347610112*a^32*b^10*x^10 - 362822270976*a^31*b^11*x^11 - 2392686985216*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 - 1224391649918
976*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 - 496132577002782720*a^20
*b^22*x^22 - 942381442447441920*a^19*b^23*x^23 - 1618144034134425600*a^18*b^24*x^24 - 2510373782846177280*a^17
*b^25*x^25 - 3515067359758909440*a^16*b^26*x^26 - 4435129698971811840*a^15*b^27*x^27 - 5031501784368021504*a^1
4*b^28*x^28 - 5117457344967475200*a^13*b^29*x^29 - 4649247330644525056*a^12*b^30*x^30 - 3755657392331685888*a^
11*b^31*x^31 - 2682104117395980288*a^10*b^32*x^32 - 1681307107418177536*a^9*b^33*x^33 - 916854056020869120*a^8
*b^34*x^34 - 430004841410985984*a^7*b^35*x^35 - 170895855354642432*a^6*b^36*x^36 - 56428448565952512*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 - 4617
9488366592*a*b^41*x^41 - 2199023255552*b^42*x^42)) - (2*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 = 0.40, size = 54, normalized size = 0.29 \begin {gather*} \frac {6 \, b^{3} x^{3} \log \left (b x + a\right ) - 6 \, b^{3} x^{3} \log \relax (x) - 6 \, a b^{2} x^{2} + 3 \, a^{2} b x - 2 \, a^{3}}{6 \, a^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^4/((b*x+a)^2)^(1/2),x, algorithm="fricas")
[Out]
1/6*(6*b^3*x^3*log(b*x + a) - 6*b^3*x^3*log(x) - 6*a*b^2*x^2 + 3*a^2*b*x - 2*a^3)/(a^4*x^3)
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giac [A] time = 0.16, size = 65, normalized size = 0.35 \begin {gather*} \frac {1}{6} \, {\left (\frac {6 \, b^{3} \log \left ({\left | b x + a \right |}\right )}{a^{4}} - \frac {6 \, b^{3} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {6 \, a b^{2} x^{2} - 3 \, a^{2} b x + 2 \, a^{3}}{a^{4} x^{3}}\right )} \mathrm {sgn}\left (b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^4/((b*x+a)^2)^(1/2),x, algorithm="giac")
[Out]
1/6*(6*b^3*log(abs(b*x + a))/a^4 - 6*b^3*log(abs(x))/a^4 - (6*a*b^2*x^2 - 3*a^2*b*x + 2*a^3)/(a^4*x^3))*sgn(b*
x + a)
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maple [A] time = 0.06, size = 69, normalized size = 0.38 \begin {gather*} \frac {\left (b x +a \right ) \left (-6 b^{3} x^{3} \ln \relax (x )+6 b^{3} x^{3} \ln \left (b x +a \right )-6 a \,b^{2} x^{2}+3 a^{2} b x -2 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(1/x^4/((b*x+a)^2)^(1/2),x)
[Out]
1/6*(b*x+a)*(6*b^3*ln(b*x+a)*x^3-6*b^3*ln(x)*x^3-6*a*b^2*x^2+3*a^2*b*x-2*a^3)/((b*x+a)^2)^(1/2)/a^4/x^3
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maxima [A] time = 1.46, size = 123, normalized size = 0.67 \begin {gather*} \frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} b^{3} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{4}} - \frac {11 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}}{6 \, a^{4} x} + \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b}{6 \, a^{3} x^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}}}{3 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^4/((b*x+a)^2)^(1/2),x, algorithm="maxima")
[Out]
(-1)^(2*a*b*x + 2*a^2)*b^3*log(2*a*b*x/abs(x) + 2*a^2/abs(x))/a^4 - 11/6*sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2/(a^
4*x) + 5/6*sqrt(b^2*x^2 + 2*a*b*x + a^2)*b/(a^3*x^2) - 1/3*sqrt(b^2*x^2 + 2*a*b*x + a^2)/(a^2*x^3)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{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(1/(x^4*((a + b*x)^2)^(1/2)),x)
[Out]
int(1/(x^4*((a + b*x)^2)^(1/2)), x)
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sympy [A] time = 0.26, size = 44, normalized size = 0.24 \begin {gather*} \frac {- 2 a^{2} + 3 a b x - 6 b^{2} x^{2}}{6 a^{3} x^{3}} + \frac {b^{3} \left (- \log {\relax (x )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**4/((b*x+a)**2)**(1/2),x)
[Out]
(-2*a**2 + 3*a*b*x - 6*b**2*x**2)/(6*a**3*x**3) + b**3*(-log(x) + log(a/b + x))/a**4
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