3.217 \(\int \frac{1}{x^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=271 \[ -\frac{4 e (13 d-24 e x)}{143 d^2 \left (d^2-e^2 x^2\right )^{11/2}}-\frac{8 e (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac{e (36036 d-52175 e x)}{9009 d^{12} \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^{12} x}+\frac{4 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^{12}}-\frac{e (12012 d-21583 e x)}{9009 d^{10} \left (d^2-e^2 x^2\right )^{3/2}}-\frac{e (12012 d-23225 e x)}{15015 d^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e (5148 d-10111 e x)}{9009 d^6 \left (d^2-e^2 x^2\right )^{7/2}}-\frac{e (572 d-1103 e x)}{1287 d^4 \left (d^2-e^2 x^2\right )^{9/2}} \]

[Out]

(-8*e*(d - e*x))/(13*(d^2 - e^2*x^2)^(13/2)) - (4*e*(13*d - 24*e*x))/(143*d^2*(d
^2 - e^2*x^2)^(11/2)) - (e*(572*d - 1103*e*x))/(1287*d^4*(d^2 - e^2*x^2)^(9/2))
- (e*(5148*d - 10111*e*x))/(9009*d^6*(d^2 - e^2*x^2)^(7/2)) - (e*(12012*d - 2322
5*e*x))/(15015*d^8*(d^2 - e^2*x^2)^(5/2)) - (e*(12012*d - 21583*e*x))/(9009*d^10
*(d^2 - e^2*x^2)^(3/2)) - (e*(36036*d - 52175*e*x))/(9009*d^12*Sqrt[d^2 - e^2*x^
2]) - Sqrt[d^2 - e^2*x^2]/(d^12*x) + (4*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^12

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Rubi [A]  time = 1.03711, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{4 e (13 d-24 e x)}{143 d^2 \left (d^2-e^2 x^2\right )^{11/2}}-\frac{8 e (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac{e (36036 d-52175 e x)}{9009 d^{12} \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^{12} x}+\frac{4 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^{12}}-\frac{e (12012 d-21583 e x)}{9009 d^{10} \left (d^2-e^2 x^2\right )^{3/2}}-\frac{e (12012 d-23225 e x)}{15015 d^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e (5148 d-10111 e x)}{9009 d^6 \left (d^2-e^2 x^2\right )^{7/2}}-\frac{e (572 d-1103 e x)}{1287 d^4 \left (d^2-e^2 x^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(-8*e*(d - e*x))/(13*(d^2 - e^2*x^2)^(13/2)) - (4*e*(13*d - 24*e*x))/(143*d^2*(d
^2 - e^2*x^2)^(11/2)) - (e*(572*d - 1103*e*x))/(1287*d^4*(d^2 - e^2*x^2)^(9/2))
- (e*(5148*d - 10111*e*x))/(9009*d^6*(d^2 - e^2*x^2)^(7/2)) - (e*(12012*d - 2322
5*e*x))/(15015*d^8*(d^2 - e^2*x^2)^(5/2)) - (e*(12012*d - 21583*e*x))/(9009*d^10
*(d^2 - e^2*x^2)^(3/2)) - (e*(36036*d - 52175*e*x))/(9009*d^12*Sqrt[d^2 - e^2*x^
2]) - Sqrt[d^2 - e^2*x^2]/(d^12*x) + (4*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^12

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Rubi in Sympy [A]  time = 117.476, size = 296, normalized size = 1.09 \[ - \frac{e}{13 d^{3} \left (d + e x\right )^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{35 e}{143 d^{4} \left (d + e x\right )^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{709 e}{1287 d^{5} \left (d + e x\right )^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{10111 e}{9009 d^{6} \left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{1}{d^{6} x \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{4 e}{5 d^{7} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{7648 e^{2} x}{3003 d^{8} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{4 e}{3 d^{9} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{30592 e^{2} x}{9009 d^{10} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{4 e}{d^{11} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{61184 e^{2} x}{9009 d^{12} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{4 e \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(e*x+d)**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

-e/(13*d**3*(d + e*x)**4*(d**2 - e**2*x**2)**(5/2)) - 35*e/(143*d**4*(d + e*x)**
3*(d**2 - e**2*x**2)**(5/2)) - 709*e/(1287*d**5*(d + e*x)**2*(d**2 - e**2*x**2)*
*(5/2)) - 10111*e/(9009*d**6*(d + e*x)*(d**2 - e**2*x**2)**(5/2)) - 1/(d**6*x*(d
**2 - e**2*x**2)**(5/2)) - 4*e/(5*d**7*(d**2 - e**2*x**2)**(5/2)) + 7648*e**2*x/
(3003*d**8*(d**2 - e**2*x**2)**(5/2)) - 4*e/(3*d**9*(d**2 - e**2*x**2)**(3/2)) +
30592*e**2*x/(9009*d**10*(d**2 - e**2*x**2)**(3/2)) - 4*e/(d**11*sqrt(d**2 - e*
*2*x**2)) + 61184*e**2*x/(9009*d**12*sqrt(d**2 - e**2*x**2)) + 4*e*atanh(sqrt(d*
*2 - e**2*x**2)/d)/d**12

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Mathematica [A]  time = 0.180735, size = 183, normalized size = 0.68 \[ -\frac{4 e \log (x)}{d^{12}}+\frac{4 e \log \left (\sqrt{d^2-e^2 x^2}+d\right )}{d^{12}}+\frac{\sqrt{d^2-e^2 x^2} \left (45045 d^{10}+546316 d^9 e x+1014094 d^8 e^2 x^2-700504 d^7 e^3 x^3-3157776 d^6 e^4 x^4-1301264 d^5 e^5 x^5+2748320 d^4 e^6 x^6+2496180 d^3 e^7 x^7-350000 d^2 e^8 x^8-1043500 d e^9 x^9-305920 e^{10} x^{10}\right )}{45045 d^{12} x (e x-d)^3 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(45045*d^10 + 546316*d^9*e*x + 1014094*d^8*e^2*x^2 - 700504
*d^7*e^3*x^3 - 3157776*d^6*e^4*x^4 - 1301264*d^5*e^5*x^5 + 2748320*d^4*e^6*x^6 +
2496180*d^3*e^7*x^7 - 350000*d^2*e^8*x^8 - 1043500*d*e^9*x^9 - 305920*e^10*x^10
))/(45045*d^12*x*(-d + e*x)^3*(d + e*x)^7) - (4*e*Log[x])/d^12 + (4*e*Log[d + Sq
rt[d^2 - e^2*x^2]])/d^12

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Maple [B]  time = 0.021, size = 484, normalized size = 1.8 \[ -{\frac{1}{13\,{d}^{3}{e}^{3}} \left ( x+{\frac{d}{e}} \right ) ^{-4} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{-{\frac{5}{2}}}}-{\frac{35}{143\,{d}^{4}{e}^{2}} \left ( x+{\frac{d}{e}} \right ) ^{-3} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{-{\frac{5}{2}}}}-{\frac{709}{1287\,{d}^{5}e} \left ( x+{\frac{d}{e}} \right ) ^{-2} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{-{\frac{5}{2}}}}+{\frac{20222\,{e}^{2}x}{15015\,{d}^{8}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{-{\frac{5}{2}}}}+{\frac{80888\,{e}^{2}x}{45045\,{d}^{10}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{-{\frac{3}{2}}}}+{\frac{161776\,{e}^{2}x}{45045\,{d}^{12}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}}+{\frac{6\,{e}^{2}x}{5\,{d}^{8}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{8\,{e}^{2}x}{5\,{d}^{10}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{16\,{e}^{2}x}{5\,{d}^{12}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}+4\,{\frac{e}{{d}^{11}\sqrt{{d}^{2}}}\ln \left ({\frac{2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}{x}} \right ) }-{\frac{10111}{9009\,{d}^{6}} \left ( x+{\frac{d}{e}} \right ) ^{-1} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{-{\frac{5}{2}}}}-{\frac{1}{{d}^{6}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{4\,e}{5\,{d}^{7}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{4\,e}{3\,{d}^{9}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-4\,{\frac{e}{{d}^{11}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/13/d^3/e^3/(x+d/e)^4/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)-35/143/d^4/e^2/(x+d
/e)^3/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)-709/1287/d^5/e/(x+d/e)^2/(-(x+d/e)^2*
e^2+2*d*e*(x+d/e))^(5/2)+20222/15015/d^8*e^2/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2
)*x+80888/45045/d^10*e^2/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)*x+161776/45045/d^1
2*e^2/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)*x+6/5/d^8*e^2*x/(-e^2*x^2+d^2)^(5/2)+
8/5/d^10*e^2*x/(-e^2*x^2+d^2)^(3/2)+16/5/d^12*e^2*x/(-e^2*x^2+d^2)^(1/2)+4/d^11*
e/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-10111/9009/d^6/(x
+d/e)/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)-1/d^6/x/(-e^2*x^2+d^2)^(5/2)-4/5/d^7*
e/(-e^2*x^2+d^2)^(5/2)-4/3/d^9*e/(-e^2*x^2+d^2)^(3/2)-4/d^11*e/(-e^2*x^2+d^2)^(1
/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}{\left (e x + d\right )}^{4} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^4*x^2),x, algorithm="maxima")

[Out]

integrate(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^4*x^2), x)

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Fricas [A]  time = 0.947354, size = 1769, normalized size = 6.53 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^4*x^2),x, algorithm="fricas")

[Out]

-1/45045*(305920*e^20*x^20 + 4704860*d*e^19*x^19 - 300560*d^2*e^18*x^18 - 102268
860*d^3*e^17*x^17 - 161498240*d^4*e^16*x^16 + 562549336*d^5*e^15*x^15 + 13907710
24*d^6*e^14*x^14 - 996944104*d^7*e^13*x^13 - 4697402606*d^8*e^12*x^12 - 59670868
4*d^9*e^11*x^11 + 7949190535*d^10*e^10*x^10 + 4548721320*d^11*e^9*x^9 - 69341629
50*d^12*e^8*x^8 - 6612365760*d^13*e^7*x^7 + 2583300720*d^14*e^6*x^6 + 4345461120
*d^15*e^5*x^5 + 135014880*d^16*e^4*x^4 - 1245404160*d^17*e^3*x^3 - 288288000*d^1
8*e^2*x^2 + 92252160*d^19*e*x + 23063040*d^20 + 180180*(10*d*e^19*x^19 + 40*d^2*
e^18*x^18 - 130*d^3*e^17*x^17 - 720*d^4*e^16*x^16 + 52*d^5*e^15*x^15 + 3968*d^6*
e^14*x^14 + 3372*d^7*e^13*x^13 - 9392*d^8*e^12*x^12 - 14238*d^9*e^11*x^11 + 8920
*d^10*e^10*x^10 + 25750*d^11*e^9*x^9 + 1920*d^12*e^8*x^8 - 23360*d^13*e^7*x^7 -
10880*d^14*e^6*x^6 + 9568*d^15*e^5*x^5 + 8192*d^16*e^4*x^4 - 512*d^17*e^3*x^3 -
2048*d^18*e^2*x^2 - 512*d^19*e*x - (e^19*x^19 + 4*d*e^18*x^18 - 46*d^2*e^17*x^17
- 204*d^3*e^16*x^16 + 190*d^4*e^15*x^15 + 1796*d^5*e^14*x^14 + 984*d^6*e^13*x^1
3 - 5876*d^7*e^12*x^12 - 7399*d^8*e^11*x^11 + 7800*d^9*e^10*x^10 + 17358*d^10*e^
9*x^9 - 1088*d^11*e^8*x^8 - 18928*d^12*e^7*x^7 - 7552*d^13*e^6*x^6 + 9120*d^14*e
^5*x^5 + 7168*d^15*e^4*x^4 - 768*d^16*e^3*x^3 - 2048*d^17*e^2*x^2 - 512*d^18*e*x
)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - 2*(183068*e^19*x^19
- 797328*d*e^18*x^18 - 13638628*d^2*e^17*x^17 - 14622272*d^3*e^16*x^16 + 130743
820*d^4*e^15*x^15 + 267742608*d^5*e^14*x^14 - 376677808*d^6*e^13*x^13 - 12723310
08*d^7*e^12*x^12 + 119066948*d^8*e^11*x^11 + 2709942950*d^9*e^10*x^10 + 12540613
80*d^10*e^9*x^9 - 2837832855*d^11*e^8*x^8 - 2438916480*d^12*e^7*x^7 + 1274953680
*d^13*e^6*x^6 + 1878676800*d^14*e^5*x^5 - 240240*d^15*e^4*x^4 - 599639040*d^16*e
^3*x^3 - 138378240*d^17*e^2*x^2 + 46126080*d^18*e*x + 11531520*d^19)*sqrt(-e^2*x
^2 + d^2))/(10*d^13*e^18*x^19 + 40*d^14*e^17*x^18 - 130*d^15*e^16*x^17 - 720*d^1
6*e^15*x^16 + 52*d^17*e^14*x^15 + 3968*d^18*e^13*x^14 + 3372*d^19*e^12*x^13 - 93
92*d^20*e^11*x^12 - 14238*d^21*e^10*x^11 + 8920*d^22*e^9*x^10 + 25750*d^23*e^8*x
^9 + 1920*d^24*e^7*x^8 - 23360*d^25*e^6*x^7 - 10880*d^26*e^5*x^6 + 9568*d^27*e^4
*x^5 + 8192*d^28*e^3*x^4 - 512*d^29*e^2*x^3 - 2048*d^30*e*x^2 - 512*d^31*x - (d^
12*e^18*x^19 + 4*d^13*e^17*x^18 - 46*d^14*e^16*x^17 - 204*d^15*e^15*x^16 + 190*d
^16*e^14*x^15 + 1796*d^17*e^13*x^14 + 984*d^18*e^12*x^13 - 5876*d^19*e^11*x^12 -
7399*d^20*e^10*x^11 + 7800*d^21*e^9*x^10 + 17358*d^22*e^8*x^9 - 1088*d^23*e^7*x
^8 - 18928*d^24*e^6*x^7 - 7552*d^25*e^5*x^6 + 9120*d^26*e^4*x^5 + 7168*d^27*e^3*
x^4 - 768*d^28*e^2*x^3 - 2048*d^29*e*x^2 - 512*d^30*x)*sqrt(-e^2*x^2 + d^2))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(e*x+d)**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^4*x^2),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, undef, undef, 1]