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

Optimal. Leaf size=209 \[ -\frac{3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac{e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac{45}{8} d^2 e^4 (d-e x) \sqrt{d^2-e^2 x^2}+\frac{15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}+\frac{45}{8} d^4 e^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{45}{8} d^4 e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

[Out]

(-45*d^2*e^4*(d - e*x)*Sqrt[d^2 - e^2*x^2])/8 + (15*d*e^3*(2*d - e*x)*(d^2 - e^2
*x^2)^(3/2))/(8*x) - (3*e^2*(3*d + 2*e*x)*(d^2 - e^2*x^2)^(5/2))/(8*x^2) - (d*(d
^2 - e^2*x^2)^(7/2))/(4*x^4) - (e*(d^2 - e^2*x^2)^(7/2))/x^3 + (45*d^4*e^4*ArcTa
n[(e*x)/Sqrt[d^2 - e^2*x^2]])/8 + (45*d^4*e^4*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/8

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Rubi [A]  time = 0.621992, antiderivative size = 209, 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 \[ -\frac{3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac{e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac{45}{8} d^2 e^4 (d-e x) \sqrt{d^2-e^2 x^2}+\frac{15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}+\frac{45}{8} d^4 e^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{45}{8} d^4 e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-45*d^2*e^4*(d - e*x)*Sqrt[d^2 - e^2*x^2])/8 + (15*d*e^3*(2*d - e*x)*(d^2 - e^2
*x^2)^(3/2))/(8*x) - (3*e^2*(3*d + 2*e*x)*(d^2 - e^2*x^2)^(5/2))/(8*x^2) - (d*(d
^2 - e^2*x^2)^(7/2))/(4*x^4) - (e*(d^2 - e^2*x^2)^(7/2))/x^3 + (45*d^4*e^4*ArcTa
n[(e*x)/Sqrt[d^2 - e^2*x^2]])/8 + (45*d^4*e^4*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/8

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Rubi in Sympy [A]  time = 83.7857, size = 238, normalized size = 1.14 \[ - \frac{d^{7} \sqrt{d^{2} - e^{2} x^{2}}}{4 x^{4}} - \frac{d^{6} e \sqrt{d^{2} - e^{2} x^{2}}}{x^{3}} - \frac{3 d^{5} e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{8 x^{2}} + \frac{45 d^{4} e^{4} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{8} + \frac{45 d^{4} e^{4} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{8} + \frac{6 d^{4} e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{x} - 5 d^{3} e^{4} \sqrt{d^{2} - e^{2} x^{2}} + \frac{3 d^{2} e^{5} x \sqrt{d^{2} - e^{2} x^{2}}}{8} - d e^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}} + \frac{e^{7} x^{3} \sqrt{d^{2} - e^{2} x^{2}}}{4} \]

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**5,x)

[Out]

-d**7*sqrt(d**2 - e**2*x**2)/(4*x**4) - d**6*e*sqrt(d**2 - e**2*x**2)/x**3 - 3*d
**5*e**2*sqrt(d**2 - e**2*x**2)/(8*x**2) + 45*d**4*e**4*atan(e*x/sqrt(d**2 - e**
2*x**2))/8 + 45*d**4*e**4*atanh(sqrt(d**2 - e**2*x**2)/d)/8 + 6*d**4*e**3*sqrt(d
**2 - e**2*x**2)/x - 5*d**3*e**4*sqrt(d**2 - e**2*x**2) + 3*d**2*e**5*x*sqrt(d**
2 - e**2*x**2)/8 - d*e**4*(d**2 - e**2*x**2)**(3/2) + e**7*x**3*sqrt(d**2 - e**2
*x**2)/4

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Mathematica [A]  time = 0.321692, size = 164, normalized size = 0.78 \[ \frac{1}{8} \left (-45 d^4 e^4 \log (x)+45 d^4 e^4 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+45 d^4 e^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{\sqrt{d^2-e^2 x^2} \left (-2 d^7-8 d^6 e x-3 d^5 e^2 x^2+48 d^4 e^3 x^3-48 d^3 e^4 x^4+3 d^2 e^5 x^5+8 d e^6 x^6+2 e^7 x^7\right )}{x^4}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(-2*d^7 - 8*d^6*e*x - 3*d^5*e^2*x^2 + 48*d^4*e^3*x^3 - 48*
d^3*e^4*x^4 + 3*d^2*e^5*x^5 + 8*d*e^6*x^6 + 2*e^7*x^7))/x^4 + 45*d^4*e^4*ArcTan[
(e*x)/Sqrt[d^2 - e^2*x^2]] - 45*d^4*e^4*Log[x] + 45*d^4*e^4*Log[d + Sqrt[d^2 - e
^2*x^2]])/8

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Maple [A]  time = 0.029, size = 302, normalized size = 1.4 \[ -{\frac{d}{4\,{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{9\,{e}^{2}}{8\,d{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{9\,{e}^{4}}{8\,d} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{15\,d{e}^{4}}{8} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{45\,{d}^{3}{e}^{4}}{8}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{45\,{d}^{5}{e}^{4}}{8}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+3\,{\frac{{e}^{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{7/2}}{{d}^{2}x}}+3\,{\frac{{e}^{5}x \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}{{d}^{2}}}+{\frac{15\,{e}^{5}x}{4} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{45\,{e}^{5}{d}^{2}x}{8}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{45\,{e}^{5}{d}^{4}}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{e}{{x}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \]

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^5,x)

[Out]

-1/4*d*(-e^2*x^2+d^2)^(7/2)/x^4-9/8/d*e^2/x^2*(-e^2*x^2+d^2)^(7/2)-9/8/d*e^4*(-e
^2*x^2+d^2)^(5/2)-15/8*d*e^4*(-e^2*x^2+d^2)^(3/2)-45/8*d^3*e^4*(-e^2*x^2+d^2)^(1
/2)+45/8*d^5*e^4/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)+3*
e^3/d^2/x*(-e^2*x^2+d^2)^(7/2)+3*e^5/d^2*x*(-e^2*x^2+d^2)^(5/2)+15/4*e^5*x*(-e^2
*x^2+d^2)^(3/2)+45/8*e^5*d^2*x*(-e^2*x^2+d^2)^(1/2)+45/8*e^5*d^4/(e^2)^(1/2)*arc
tan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-e*(-e^2*x^2+d^2)^(7/2)/x^3

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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^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300563, size = 994, normalized size = 4.76 \[ -\frac{16 \, d e^{15} x^{15} + 64 \, d^{2} e^{14} x^{14} - 152 \, d^{3} e^{13} x^{13} - 1040 \, d^{4} e^{12} x^{12} + 664 \, d^{5} e^{11} x^{11} + 4840 \, d^{6} e^{10} x^{10} - 4112 \, d^{7} e^{9} x^{9} - 7688 \, d^{8} e^{8} x^{8} + 13056 \, d^{9} e^{7} x^{7} + 3456 \, d^{10} e^{6} x^{6} - 17152 \, d^{11} e^{5} x^{5} + 416 \, d^{12} e^{4} x^{4} + 8704 \, d^{13} e^{3} x^{3} + 256 \, d^{14} e^{2} x^{2} - 1024 \, d^{15} e x - 256 \, d^{16} + 90 \,{\left (d^{4} e^{12} x^{12} - 32 \, d^{6} e^{10} x^{10} + 160 \, d^{8} e^{8} x^{8} - 256 \, d^{10} e^{6} x^{6} + 128 \, d^{12} e^{4} x^{4} + 8 \,{\left (d^{5} e^{10} x^{10} - 10 \, d^{7} e^{8} x^{8} + 24 \, d^{9} e^{6} x^{6} - 16 \, d^{11} e^{4} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 45 \,{\left (d^{4} e^{12} x^{12} - 32 \, d^{6} e^{10} x^{10} + 160 \, d^{8} e^{8} x^{8} - 256 \, d^{10} e^{6} x^{6} + 128 \, d^{12} e^{4} x^{4} + 8 \,{\left (d^{5} e^{10} x^{10} - 10 \, d^{7} e^{8} x^{8} + 24 \, d^{9} e^{6} x^{6} - 16 \, d^{11} e^{4} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (2 \, e^{15} x^{15} + 8 \, d e^{14} x^{14} - 61 \, d^{2} e^{13} x^{13} - 304 \, d^{3} e^{12} x^{12} + 272 \, d^{4} e^{11} x^{11} + 2429 \, d^{5} e^{10} x^{10} - 1576 \, d^{6} e^{9} x^{9} - 5794 \, d^{7} e^{8} x^{8} + 7424 \, d^{8} e^{7} x^{7} + 3680 \, d^{9} e^{6} x^{6} - 13184 \, d^{10} e^{5} x^{5} + 448 \, d^{11} e^{4} x^{4} + 8192 \, d^{12} e^{3} x^{3} + 128 \, d^{13} e^{2} x^{2} - 1024 \, d^{14} e x - 256 \, d^{15}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{8 \,{\left (e^{8} x^{12} - 32 \, d^{2} e^{6} x^{10} + 160 \, d^{4} e^{4} x^{8} - 256 \, d^{6} e^{2} x^{6} + 128 \, d^{8} x^{4} + 8 \,{\left (d e^{6} x^{10} - 10 \, d^{3} e^{4} x^{8} + 24 \, d^{5} e^{2} x^{6} - 16 \, d^{7} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

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^5,x, algorithm="fricas")

[Out]

-1/8*(16*d*e^15*x^15 + 64*d^2*e^14*x^14 - 152*d^3*e^13*x^13 - 1040*d^4*e^12*x^12
+ 664*d^5*e^11*x^11 + 4840*d^6*e^10*x^10 - 4112*d^7*e^9*x^9 - 7688*d^8*e^8*x^8
+ 13056*d^9*e^7*x^7 + 3456*d^10*e^6*x^6 - 17152*d^11*e^5*x^5 + 416*d^12*e^4*x^4
+ 8704*d^13*e^3*x^3 + 256*d^14*e^2*x^2 - 1024*d^15*e*x - 256*d^16 + 90*(d^4*e^12
*x^12 - 32*d^6*e^10*x^10 + 160*d^8*e^8*x^8 - 256*d^10*e^6*x^6 + 128*d^12*e^4*x^4
+ 8*(d^5*e^10*x^10 - 10*d^7*e^8*x^8 + 24*d^9*e^6*x^6 - 16*d^11*e^4*x^4)*sqrt(-e
^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 45*(d^4*e^12*x^12 - 3
2*d^6*e^10*x^10 + 160*d^8*e^8*x^8 - 256*d^10*e^6*x^6 + 128*d^12*e^4*x^4 + 8*(d^5
*e^10*x^10 - 10*d^7*e^8*x^8 + 24*d^9*e^6*x^6 - 16*d^11*e^4*x^4)*sqrt(-e^2*x^2 +
d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (2*e^15*x^15 + 8*d*e^14*x^14 - 61*d^2
*e^13*x^13 - 304*d^3*e^12*x^12 + 272*d^4*e^11*x^11 + 2429*d^5*e^10*x^10 - 1576*d
^6*e^9*x^9 - 5794*d^7*e^8*x^8 + 7424*d^8*e^7*x^7 + 3680*d^9*e^6*x^6 - 13184*d^10
*e^5*x^5 + 448*d^11*e^4*x^4 + 8192*d^12*e^3*x^3 + 128*d^13*e^2*x^2 - 1024*d^14*e
*x - 256*d^15)*sqrt(-e^2*x^2 + d^2))/(e^8*x^12 - 32*d^2*e^6*x^10 + 160*d^4*e^4*x
^8 - 256*d^6*e^2*x^6 + 128*d^8*x^4 + 8*(d*e^6*x^10 - 10*d^3*e^4*x^8 + 24*d^5*e^2
*x^6 - 16*d^7*x^4)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 50.9258, size = 1028, normalized size = 4.92 \[ \text{result too large to display} \]

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**5,x)

[Out]

d**7*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), Abs(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*sq
rt(-d**2/(e**2*x**2) + 1)) - I*e**4*asin(d/(e*x))/(8*d**3), True)) + 3*d**6*e*Pi
ecewise((-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**5*e**2*Piecewise((
-d**2/(2*e*x**3*sqrt(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)) - 5*d**4*e**3*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*as
in(e*x/d) + e**2*x/(d*sqrt(1 - e**2*x**2/d**2)), True)) - 5*d**3*e**4*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)) + d**2*e**5*P
iecewise((-I*d**2*acosh(e*x/d)/(2*e) - I*d*x/(2*sqrt(-1 + e**2*x**2/d**2)) + I*e
**2*x**3/(2*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (d**2*asin(e
*x/d)/(2*e) + d*x*sqrt(1 - e**2*x**2/d**2)/2, True)) + 3*d*e**6*Piecewise((x**2*
sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*e**2), True)) + e**7*
Piecewise((-I*d**4*acosh(e*x/d)/(8*e**3) + I*d**3*x/(8*e**2*sqrt(-1 + e**2*x**2/
d**2)) - 3*I*d*x**3/(8*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**5/(4*d*sqrt(-1 + e
**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (d**4*asin(e*x/d)/(8*e**3) - d**3*x/(
8*e**2*sqrt(1 - e**2*x**2/d**2)) + 3*d*x**3/(8*sqrt(1 - e**2*x**2/d**2)) - e**2*
x**5/(4*d*sqrt(1 - e**2*x**2/d**2)), True))

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GIAC/XCAS [A]  time = 0.295445, size = 505, normalized size = 2.42 \[ \frac{45}{8} \, d^{4} \arcsin \left (\frac{x e}{d}\right ) e^{4}{\rm sign}\left (d\right ) + \frac{45}{8} \, d^{4} e^{4}{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right ) + \frac{{\left (d^{4} e^{10} + \frac{8 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{4} e^{8}}{x} + \frac{8 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4} e^{6}}{x^{2}} - \frac{184 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{4} e^{4}}{x^{3}}\right )} x^{4} e^{2}}{64 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4}} + \frac{1}{64} \,{\left (\frac{184 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{4} e^{26}}{x} - \frac{8 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4} e^{24}}{x^{2}} - \frac{8 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{4} e^{22}}{x^{3}} - \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{4} e^{20}}{x^{4}}\right )} e^{\left (-24\right )} - \frac{1}{8} \,{\left (48 \, d^{3} e^{4} -{\left (3 \, d^{2} e^{5} + 2 \,{\left (x e^{7} + 4 \, d e^{6}\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \]

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^5,x, algorithm="giac")

[Out]

45/8*d^4*arcsin(x*e/d)*e^4*sign(d) + 45/8*d^4*e^4*ln(1/2*abs(-2*d*e - 2*sqrt(-x^
2*e^2 + d^2)*e)*e^(-2)/abs(x)) + 1/64*(d^4*e^10 + 8*(d*e + sqrt(-x^2*e^2 + d^2)*
e)*d^4*e^8/x + 8*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^4*e^6/x^2 - 184*(d*e + sqrt(
-x^2*e^2 + d^2)*e)^3*d^4*e^4/x^3)*x^4*e^2/(d*e + sqrt(-x^2*e^2 + d^2)*e)^4 + 1/6
4*(184*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^4*e^26/x - 8*(d*e + sqrt(-x^2*e^2 + d^2)
*e)^2*d^4*e^24/x^2 - 8*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^4*e^22/x^3 - (d*e + sq
rt(-x^2*e^2 + d^2)*e)^4*d^4*e^20/x^4)*e^(-24) - 1/8*(48*d^3*e^4 - (3*d^2*e^5 + 2
*(x*e^7 + 4*d*e^6)*x)*x)*sqrt(-x^2*e^2 + d^2)