3.78 \(\int \frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx\)

Optimal. Leaf size=206 \[ -\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac{e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}-\frac{e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-3 d e^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{15}{16} d e^7 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )-\frac{3 e^6 (16 d-5 e x) \sqrt{d^2-e^2 x^2}}{16 x}+\frac{e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3} \]

[Out]

(-3*e^6*(16*d - 5*e*x)*Sqrt[d^2 - e^2*x^2])/(16*x) + (e^4*(16*d + 5*e*x)*(d^2 -
e^2*x^2)^(3/2))/(16*x^3) - (e^2*(24*d + 5*e*x)*(d^2 - e^2*x^2)^(5/2))/(40*x^5) -
(d*(d^2 - e^2*x^2)^(7/2))/(7*x^7) - (e*(d^2 - e^2*x^2)^(7/2))/(2*x^6) - 3*d*e^7
*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - (15*d*e^7*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/1
6

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Rubi [A]  time = 0.621374, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac{e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}-\frac{e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-3 d e^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{15}{16} d e^7 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )-\frac{3 e^6 (16 d-5 e x) \sqrt{d^2-e^2 x^2}}{16 x}+\frac{e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^8,x]

[Out]

(-3*e^6*(16*d - 5*e*x)*Sqrt[d^2 - e^2*x^2])/(16*x) + (e^4*(16*d + 5*e*x)*(d^2 -
e^2*x^2)^(3/2))/(16*x^3) - (e^2*(24*d + 5*e*x)*(d^2 - e^2*x^2)^(5/2))/(40*x^5) -
(d*(d^2 - e^2*x^2)^(7/2))/(7*x^7) - (e*(d^2 - e^2*x^2)^(7/2))/(2*x^6) - 3*d*e^7
*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - (15*d*e^7*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/1
6

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Rubi in Sympy [A]  time = 97.3131, size = 245, normalized size = 1.19 \[ - \frac{d^{7} \sqrt{d^{2} - e^{2} x^{2}}}{7 x^{7}} - \frac{d^{6} e \sqrt{d^{2} - e^{2} x^{2}}}{2 x^{6}} - \frac{6 d^{5} e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{35 x^{5}} + \frac{11 d^{4} e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{8 x^{4}} + \frac{62 d^{3} e^{4} \sqrt{d^{2} - e^{2} x^{2}}}{35 x^{3}} - \frac{15 d^{2} e^{5} \sqrt{d^{2} - e^{2} x^{2}}}{16 x^{2}} - 3 d e^{7} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )} - \frac{15 d e^{7} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{16} - \frac{156 d e^{6} \sqrt{d^{2} - e^{2} x^{2}}}{35 x} + e^{7} \sqrt{d^{2} - e^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(-e**2*x**2+d**2)**(5/2)/x**8,x)

[Out]

-d**7*sqrt(d**2 - e**2*x**2)/(7*x**7) - d**6*e*sqrt(d**2 - e**2*x**2)/(2*x**6) -
6*d**5*e**2*sqrt(d**2 - e**2*x**2)/(35*x**5) + 11*d**4*e**3*sqrt(d**2 - e**2*x*
*2)/(8*x**4) + 62*d**3*e**4*sqrt(d**2 - e**2*x**2)/(35*x**3) - 15*d**2*e**5*sqrt
(d**2 - e**2*x**2)/(16*x**2) - 3*d*e**7*atan(e*x/sqrt(d**2 - e**2*x**2)) - 15*d*
e**7*atanh(sqrt(d**2 - e**2*x**2)/d)/16 - 156*d*e**6*sqrt(d**2 - e**2*x**2)/(35*
x) + e**7*sqrt(d**2 - e**2*x**2)

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Mathematica [A]  time = 0.273002, size = 161, normalized size = 0.78 \[ -\frac{15}{16} d e^7 \log \left (\sqrt{d^2-e^2 x^2}+d\right )-3 d e^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{\sqrt{d^2-e^2 x^2} \left (80 d^7+280 d^6 e x+96 d^5 e^2 x^2-770 d^4 e^3 x^3-992 d^3 e^4 x^4+525 d^2 e^5 x^5+2496 d e^6 x^6-560 e^7 x^7\right )}{560 x^7}+\frac{15}{16} d e^7 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^8,x]

[Out]

-(Sqrt[d^2 - e^2*x^2]*(80*d^7 + 280*d^6*e*x + 96*d^5*e^2*x^2 - 770*d^4*e^3*x^3 -
992*d^3*e^4*x^4 + 525*d^2*e^5*x^5 + 2496*d*e^6*x^6 - 560*e^7*x^7))/(560*x^7) -
3*d*e^7*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] + (15*d*e^7*Log[x])/16 - (15*d*e^7*Log
[d + Sqrt[d^2 - e^2*x^2]])/16

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Maple [B]  time = 0.062, size = 377, normalized size = 1.8 \[ -{\frac{d}{7\,{x}^{7}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{{e}^{3}}{8\,{d}^{2}{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{e}^{5}}{16\,{d}^{4}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{e}^{7}}{16\,{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{e}^{7}}{16\,{d}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{15\,{e}^{7}}{16}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{15\,{e}^{7}{d}^{2}}{16}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{3\,{e}^{2}}{5\,d{x}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{2\,{e}^{4}}{5\,{d}^{3}{x}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{8\,{e}^{6}}{5\,{d}^{5}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{8\,{e}^{8}x}{5\,{d}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-2\,{\frac{{e}^{8}x \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{3/2}}{{d}^{3}}}-3\,{\frac{{e}^{8}x\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}{d}}-3\,{\frac{d{e}^{8}}{\sqrt{{e}^{2}}}\arctan \left ({\frac{\sqrt{{e}^{2}}x}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) }-{\frac{e}{2\,{x}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^8,x)

[Out]

-1/7*d*(-e^2*x^2+d^2)^(7/2)/x^7-1/8*e^3/d^2/x^4*(-e^2*x^2+d^2)^(7/2)+3/16*e^5/d^
4/x^2*(-e^2*x^2+d^2)^(7/2)+3/16*e^7/d^4*(-e^2*x^2+d^2)^(5/2)+5/16*e^7/d^2*(-e^2*
x^2+d^2)^(3/2)+15/16*e^7*(-e^2*x^2+d^2)^(1/2)-15/16*e^7*d^2/(d^2)^(1/2)*ln((2*d^
2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-3/5/d*e^2/x^5*(-e^2*x^2+d^2)^(7/2)+2/5/
d^3*e^4/x^3*(-e^2*x^2+d^2)^(7/2)-8/5/d^5*e^6/x*(-e^2*x^2+d^2)^(7/2)-8/5/d^5*e^8*
x*(-e^2*x^2+d^2)^(5/2)-2/d^3*e^8*x*(-e^2*x^2+d^2)^(3/2)-3/d*e^8*x*(-e^2*x^2+d^2)
^(1/2)-3*d*e^8/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/2*e*(-e^
2*x^2+d^2)^(7/2)/x^6

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)*(e*x + d)^3/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.302656, size = 988, normalized size = 4.8 \[ -\frac{3920 \, d e^{15} x^{15} - 19968 \, d^{2} e^{14} x^{14} - 35560 \, d^{3} e^{13} x^{13} + 227584 \, d^{4} e^{12} x^{12} + 115080 \, d^{5} e^{11} x^{11} - 766976 \, d^{6} e^{10} x^{10} - 248640 \, d^{7} e^{9} x^{9} + 1076352 \, d^{8} e^{8} x^{8} + 402080 \, d^{9} e^{7} x^{7} - 656000 \, d^{10} e^{6} x^{6} - 389760 \, d^{11} e^{5} x^{5} + 135936 \, d^{12} e^{4} x^{4} + 188160 \, d^{13} e^{3} x^{3} + 13312 \, d^{14} e^{2} x^{2} - 35840 \, d^{15} e x - 10240 \, d^{16} - 3360 \,{\left (d e^{15} x^{15} - 32 \, d^{3} e^{13} x^{13} + 160 \, d^{5} e^{11} x^{11} - 256 \, d^{7} e^{9} x^{9} + 128 \, d^{9} e^{7} x^{7} + 8 \,{\left (d^{2} e^{13} x^{13} - 10 \, d^{4} e^{11} x^{11} + 24 \, d^{6} e^{9} x^{9} - 16 \, d^{8} e^{7} x^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) - 525 \,{\left (d e^{15} x^{15} - 32 \, d^{3} e^{13} x^{13} + 160 \, d^{5} e^{11} x^{11} - 256 \, d^{7} e^{9} x^{9} + 128 \, d^{9} e^{7} x^{7} + 8 \,{\left (d^{2} e^{13} x^{13} - 10 \, d^{4} e^{11} x^{11} + 24 \, d^{6} e^{9} x^{9} - 16 \, d^{8} e^{7} x^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (560 \, e^{15} x^{15} - 2496 \, d e^{14} x^{14} - 13965 \, d^{2} e^{13} x^{13} + 80864 \, d^{3} e^{12} x^{12} + 62370 \, d^{4} e^{11} x^{11} - 431200 \, d^{5} e^{10} x^{10} - 144760 \, d^{6} e^{9} x^{9} + 800688 \, d^{7} e^{8} x^{8} + 266560 \, d^{8} e^{7} x^{7} - 586240 \, d^{9} e^{6} x^{6} - 309120 \, d^{10} e^{5} x^{5} + 138752 \, d^{11} e^{4} x^{4} + 170240 \, d^{12} e^{3} x^{3} + 8192 \, d^{13} e^{2} x^{2} - 35840 \, d^{14} e x - 10240 \, d^{15}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{560 \,{\left (e^{8} x^{15} - 32 \, d^{2} e^{6} x^{13} + 160 \, d^{4} e^{4} x^{11} - 256 \, d^{6} e^{2} x^{9} + 128 \, d^{8} x^{7} + 8 \,{\left (d e^{6} x^{13} - 10 \, d^{3} e^{4} x^{11} + 24 \, d^{5} e^{2} x^{9} - 16 \, d^{7} x^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)*(e*x + d)^3/x^8,x, algorithm="fricas")

[Out]

-1/560*(3920*d*e^15*x^15 - 19968*d^2*e^14*x^14 - 35560*d^3*e^13*x^13 + 227584*d^
4*e^12*x^12 + 115080*d^5*e^11*x^11 - 766976*d^6*e^10*x^10 - 248640*d^7*e^9*x^9 +
1076352*d^8*e^8*x^8 + 402080*d^9*e^7*x^7 - 656000*d^10*e^6*x^6 - 389760*d^11*e^
5*x^5 + 135936*d^12*e^4*x^4 + 188160*d^13*e^3*x^3 + 13312*d^14*e^2*x^2 - 35840*d
^15*e*x - 10240*d^16 - 3360*(d*e^15*x^15 - 32*d^3*e^13*x^13 + 160*d^5*e^11*x^11
- 256*d^7*e^9*x^9 + 128*d^9*e^7*x^7 + 8*(d^2*e^13*x^13 - 10*d^4*e^11*x^11 + 24*d
^6*e^9*x^9 - 16*d^8*e^7*x^7)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 +
d^2))/(e*x)) - 525*(d*e^15*x^15 - 32*d^3*e^13*x^13 + 160*d^5*e^11*x^11 - 256*d^7
*e^9*x^9 + 128*d^9*e^7*x^7 + 8*(d^2*e^13*x^13 - 10*d^4*e^11*x^11 + 24*d^6*e^9*x^
9 - 16*d^8*e^7*x^7)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (
560*e^15*x^15 - 2496*d*e^14*x^14 - 13965*d^2*e^13*x^13 + 80864*d^3*e^12*x^12 + 6
2370*d^4*e^11*x^11 - 431200*d^5*e^10*x^10 - 144760*d^6*e^9*x^9 + 800688*d^7*e^8*
x^8 + 266560*d^8*e^7*x^7 - 586240*d^9*e^6*x^6 - 309120*d^10*e^5*x^5 + 138752*d^1
1*e^4*x^4 + 170240*d^12*e^3*x^3 + 8192*d^13*e^2*x^2 - 35840*d^14*e*x - 10240*d^1
5)*sqrt(-e^2*x^2 + d^2))/(e^8*x^15 - 32*d^2*e^6*x^13 + 160*d^4*e^4*x^11 - 256*d^
6*e^2*x^9 + 128*d^8*x^7 + 8*(d*e^6*x^13 - 10*d^3*e^4*x^11 + 24*d^5*e^2*x^9 - 16*
d^7*x^7)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 60.842, size = 1513, normalized size = 7.34 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*(-e**2*x**2+d**2)**(5/2)/x**8,x)

[Out]

d**7*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(7*x**6) + e**3*sqrt(d**2/(e**2*x*
*2) - 1)/(35*d**2*x**4) + 4*e**5*sqrt(d**2/(e**2*x**2) - 1)/(105*d**4*x**2) + 8*
e**7*sqrt(d**2/(e**2*x**2) - 1)/(105*d**6), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sq
rt(-d**2/(e**2*x**2) + 1)/(7*x**6) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(35*d**2
*x**4) + 4*I*e**5*sqrt(-d**2/(e**2*x**2) + 1)/(105*d**4*x**2) + 8*I*e**7*sqrt(-d
**2/(e**2*x**2) + 1)/(105*d**6), True)) + 3*d**6*e*Piecewise((-d**2/(6*e*x**7*sq
rt(d**2/(e**2*x**2) - 1)) + 5*e/(24*x**5*sqrt(d**2/(e**2*x**2) - 1)) + e**3/(48*
d**2*x**3*sqrt(d**2/(e**2*x**2) - 1)) - e**5/(16*d**4*x*sqrt(d**2/(e**2*x**2) -
1)) + e**6*acosh(d/(e*x))/(16*d**5), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(6*e*x*
*7*sqrt(-d**2/(e**2*x**2) + 1)) - 5*I*e/(24*x**5*sqrt(-d**2/(e**2*x**2) + 1)) -
I*e**3/(48*d**2*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + I*e**5/(16*d**4*x*sqrt(-d**2
/(e**2*x**2) + 1)) - I*e**6*asin(d/(e*x))/(16*d**5), True)) + d**5*e**2*Piecewis
e((3*I*d**3*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) - 4*I*d*e**
2*x**2*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2*I*e**6*x**6*
sqrt(-1 + e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - I*e**4*x**4*sqrt
(-1 + e**2*x**2/d**2)/(-15*d**3*x**5 + 15*d*e**2*x**7), Abs(e**2*x**2/d**2) > 1)
, (3*d**3*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) - 4*d*e**2*x**
2*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2*e**6*x**6*sqrt(1 -
e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - e**4*x**4*sqrt(1 - e**2*x
**2/d**2)/(-15*d**3*x**5 + 15*d*e**2*x**7), True)) - 5*d**4*e**3*Piecewise((-d**
2/(4*e*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 3*e/(8*x**3*sqrt(d**2/(e**2*x**2) - 1)
) - e**3/(8*d**2*x*sqrt(d**2/(e**2*x**2) - 1)) + e**4*acosh(d/(e*x))/(8*d**3), A
bs(d**2/(e**2*x**2)) > 1), (I*d**2/(4*e*x**5*sqrt(-d**2/(e**2*x**2) + 1)) - 3*I*
e/(8*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + I*e**3/(8*d**2*x*sqrt(-d**2/(e**2*x**2)
+ 1)) - I*e**4*asin(d/(e*x))/(8*d**3), True)) - 5*d**3*e**4*Piecewise((-e*sqrt(
d**2/(e**2*x**2) - 1)/(3*x**2) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(3*d**2), Abs(d
**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(3*x**2) + I*e**3*sqrt(
-d**2/(e**2*x**2) + 1)/(3*d**2), True)) + d**2*e**5*Piecewise((-d**2/(2*e*x**3*s
qrt(d**2/(e**2*x**2) - 1)) + e/(2*x*sqrt(d**2/(e**2*x**2) - 1)) + e**2*acosh(d/(
e*x))/(2*d), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(2*x)
- I*e**2*asin(d/(e*x))/(2*d), True)) + 3*d*e**6*Piecewise((I*d/(x*sqrt(-1 + e**
2*x**2/d**2)) + I*e*acosh(e*x/d) - I*e**2*x/(d*sqrt(-1 + e**2*x**2/d**2)), Abs(e
**2*x**2/d**2) > 1), (-d/(x*sqrt(1 - e**2*x**2/d**2)) - e*asin(e*x/d) + e**2*x/(
d*sqrt(1 - e**2*x**2/d**2)), True)) + e**7*Piecewise((d**2/(e*x*sqrt(d**2/(e**2*
x**2) - 1)) - d*acosh(d/(e*x)) - e*x/sqrt(d**2/(e**2*x**2) - 1), Abs(d**2/(e**2*
x**2)) > 1), (-I*d**2/(e*x*sqrt(-d**2/(e**2*x**2) + 1)) + I*d*asin(d/(e*x)) + I*
e*x/sqrt(-d**2/(e**2*x**2) + 1), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.298732, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)*(e*x + d)^3/x^8,x, algorithm="giac")

[Out]

Done