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

Optimal. Leaf size=252 \[ -\frac{1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac{3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}+\frac{41 d^{12} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{1024 e^4}+\frac{41 d^{10} x \sqrt{d^2-e^2 x^2}}{1024 e^3}+\frac{41 d^8 x \left (d^2-e^2 x^2\right )^{3/2}}{1536 e^3}+\frac{41 d^6 x \left (d^2-e^2 x^2\right )^{5/2}}{1920 e^3}-\frac{d^4 (14720 d+28413 e x) \left (d^2-e^2 x^2\right )^{7/2}}{221760 e^4}-\frac{23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2} \]

[Out]

(41*d^10*x*Sqrt[d^2 - e^2*x^2])/(1024*e^3) + (41*d^8*x*(d^2 - e^2*x^2)^(3/2))/(1
536*e^3) + (41*d^6*x*(d^2 - e^2*x^2)^(5/2))/(1920*e^3) - (23*d^3*x^2*(d^2 - e^2*
x^2)^(7/2))/(99*e^2) - (41*d^2*x^3*(d^2 - e^2*x^2)^(7/2))/(120*e) - (3*d*x^4*(d^
2 - e^2*x^2)^(7/2))/11 - (e*x^5*(d^2 - e^2*x^2)^(7/2))/12 - (d^4*(14720*d + 2841
3*e*x)*(d^2 - e^2*x^2)^(7/2))/(221760*e^4) + (41*d^12*ArcTan[(e*x)/Sqrt[d^2 - e^
2*x^2]])/(1024*e^4)

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Rubi [A]  time = 0.670631, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac{3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}+\frac{41 d^{12} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{1024 e^4}+\frac{41 d^{10} x \sqrt{d^2-e^2 x^2}}{1024 e^3}+\frac{41 d^8 x \left (d^2-e^2 x^2\right )^{3/2}}{1536 e^3}+\frac{41 d^6 x \left (d^2-e^2 x^2\right )^{5/2}}{1920 e^3}-\frac{d^4 (14720 d+28413 e x) \left (d^2-e^2 x^2\right )^{7/2}}{221760 e^4}-\frac{23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2} \]

Antiderivative was successfully verified.

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

[Out]

(41*d^10*x*Sqrt[d^2 - e^2*x^2])/(1024*e^3) + (41*d^8*x*(d^2 - e^2*x^2)^(3/2))/(1
536*e^3) + (41*d^6*x*(d^2 - e^2*x^2)^(5/2))/(1920*e^3) - (23*d^3*x^2*(d^2 - e^2*
x^2)^(7/2))/(99*e^2) - (41*d^2*x^3*(d^2 - e^2*x^2)^(7/2))/(120*e) - (3*d*x^4*(d^
2 - e^2*x^2)^(7/2))/11 - (e*x^5*(d^2 - e^2*x^2)^(7/2))/12 - (d^4*(14720*d + 2841
3*e*x)*(d^2 - e^2*x^2)^(7/2))/(221760*e^4) + (41*d^12*ArcTan[(e*x)/Sqrt[d^2 - e^
2*x^2]])/(1024*e^4)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.223069, size = 190, normalized size = 0.75 \[ \frac{41 d^{12} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{1024 e^4}+\sqrt{d^2-e^2 x^2} \left (-\frac{46 d^{11}}{693 e^4}-\frac{41 d^{10} x}{1024 e^3}-\frac{23 d^9 x^2}{693 e^2}-\frac{41 d^8 x^3}{1536 e}+\frac{52 d^7 x^4}{231}+\frac{1111 d^6 e x^5}{1920}+\frac{130}{693} d^5 e^2 x^6-\frac{207}{320} d^4 e^3 x^7-\frac{58}{99} d^3 e^4 x^8+\frac{11}{120} d^2 e^5 x^9+\frac{3}{11} d e^6 x^{10}+\frac{e^7 x^{11}}{12}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[d^2 - e^2*x^2]*((-46*d^11)/(693*e^4) - (41*d^10*x)/(1024*e^3) - (23*d^9*x^2
)/(693*e^2) - (41*d^8*x^3)/(1536*e) + (52*d^7*x^4)/231 + (1111*d^6*e*x^5)/1920 +
(130*d^5*e^2*x^6)/693 - (207*d^4*e^3*x^7)/320 - (58*d^3*e^4*x^8)/99 + (11*d^2*e
^5*x^9)/120 + (3*d*e^6*x^10)/11 + (e^7*x^11)/12) + (41*d^12*ArcTan[(e*x)/Sqrt[d^
2 - e^2*x^2]])/(1024*e^4)

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Maple [A]  time = 0.021, size = 241, normalized size = 1. \[ -{\frac{23\,{d}^{3}{x}^{2}}{99\,{e}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{46\,{d}^{5}}{693\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{e{x}^{5}}{12} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{41\,{d}^{2}{x}^{3}}{120\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{41\,{d}^{4}x}{320\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{41\,{d}^{6}x}{1920\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{41\,{d}^{8}x}{1536\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{41\,{d}^{10}x}{1024\,{e}^{3}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{41\,{d}^{12}}{1024\,{e}^{3}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{3\,d{x}^{4}}{11} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-23/99*d^3*x^2*(-e^2*x^2+d^2)^(7/2)/e^2-46/693*d^5/e^4*(-e^2*x^2+d^2)^(7/2)-1/12
*e*x^5*(-e^2*x^2+d^2)^(7/2)-41/120*d^2*x^3*(-e^2*x^2+d^2)^(7/2)/e-41/320/e^3*d^4
*x*(-e^2*x^2+d^2)^(7/2)+41/1920*d^6*x*(-e^2*x^2+d^2)^(5/2)/e^3+41/1536*d^8*x*(-e
^2*x^2+d^2)^(3/2)/e^3+41/1024*d^10*x*(-e^2*x^2+d^2)^(1/2)/e^3+41/1024/e^3*d^12/(
e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-3/11*d*x^4*(-e^2*x^2+d^2)^
(7/2)

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Maxima [A]  time = 0.799463, size = 315, normalized size = 1.25 \[ -\frac{1}{12} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} e x^{5} + \frac{41 \, d^{12} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{1024 \, \sqrt{e^{2}} e^{3}} + \frac{41 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{10} x}{1024 \, e^{3}} - \frac{3}{11} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d x^{4} + \frac{41 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{8} x}{1536 \, e^{3}} - \frac{41 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{2} x^{3}}{120 \, e} + \frac{41 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{6} x}{1920 \, e^{3}} - \frac{23 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{3} x^{2}}{99 \, e^{2}} - \frac{41 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{4} x}{320 \, e^{3}} - \frac{46 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{5}}{693 \, e^{4}} \]

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^3,x, algorithm="maxima")

[Out]

-1/12*(-e^2*x^2 + d^2)^(7/2)*e*x^5 + 41/1024*d^12*arcsin(e^2*x/sqrt(d^2*e^2))/(s
qrt(e^2)*e^3) + 41/1024*sqrt(-e^2*x^2 + d^2)*d^10*x/e^3 - 3/11*(-e^2*x^2 + d^2)^
(7/2)*d*x^4 + 41/1536*(-e^2*x^2 + d^2)^(3/2)*d^8*x/e^3 - 41/120*(-e^2*x^2 + d^2)
^(7/2)*d^2*x^3/e + 41/1920*(-e^2*x^2 + d^2)^(5/2)*d^6*x/e^3 - 23/99*(-e^2*x^2 +
d^2)^(7/2)*d^3*x^2/e^2 - 41/320*(-e^2*x^2 + d^2)^(7/2)*d^4*x/e^3 - 46/693*(-e^2*
x^2 + d^2)^(7/2)*d^5/e^4

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Fricas [A]  time = 0.291455, size = 1103, normalized size = 4.38 \[ \text{result too large to display} \]

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^3,x, algorithm="fricas")

[Out]

-1/3548160*(3548160*d*e^23*x^23 + 11612160*d^2*e^22*x^22 - 82435584*d^3*e^21*x^2
1 - 307507200*d^4*e^20*x^20 + 490133952*d^5*e^19*x^19 + 2620006400*d^6*e^18*x^18
- 523597536*d^7*e^17*x^17 - 10685030400*d^8*e^16*x^16 - 4561549608*d^9*e^15*x^1
5 + 23861882880*d^10*e^14*x^14 + 20013992460*d^11*e^13*x^13 - 29731215360*d^12*e
^12*x^12 - 37348461612*d^13*e^11*x^11 + 18053038080*d^14*e^10*x^10 + 37402830696
*d^15*e^9*x^9 - 794787840*d^16*e^8*x^8 - 19429546752*d^17*e^7*x^7 - 4844421120*d
^18*e^6*x^6 + 3501679104*d^19*e^5*x^5 + 1816657920*d^20*e^4*x^4 + 824355840*d^21
*e^3*x^3 - 290949120*d^23*e*x + 284130*(d^12*e^12*x^12 - 72*d^14*e^10*x^10 + 840
*d^16*e^8*x^8 - 3584*d^18*e^6*x^6 + 6912*d^20*e^4*x^4 - 6144*d^22*e^2*x^2 + 2048
*d^24 + 4*(3*d^13*e^10*x^10 - 70*d^15*e^8*x^8 + 448*d^17*e^6*x^6 - 1152*d^19*e^4
*x^4 + 1280*d^21*e^2*x^2 - 512*d^23)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^
2*x^2 + d^2))/(e*x)) - (295680*e^23*x^23 + 967680*d*e^22*x^22 - 20963712*d^2*e^2
1*x^21 - 71751680*d^3*e^20*x^20 + 222658128*d^4*e^19*x^19 + 963184640*d^5*e^18*x
^18 - 619200120*d^6*e^17*x^17 - 5261414400*d^7*e^16*x^16 - 1197850038*d^8*e^15*x
^15 + 14640215040*d^9*e^14*x^14 + 10388814975*d^10*e^13*x^13 - 22019880960*d^11*
e^12*x^12 - 24685042536*d^12*e^11*x^11 + 16406691840*d^13*e^10*x^10 + 2917924178
4*d^14*e^9*x^9 - 2535751680*d^15*e^8*x^8 - 17460495360*d^16*e^7*x^7 - 3936092160
*d^17*e^6*x^6 + 3804751104*d^18*e^5*x^5 + 1816657920*d^19*e^4*x^4 + 678881280*d^
20*e^3*x^3 - 290949120*d^22*e*x)*sqrt(-e^2*x^2 + d^2))/(e^16*x^12 - 72*d^2*e^14*
x^10 + 840*d^4*e^12*x^8 - 3584*d^6*e^10*x^6 + 6912*d^8*e^8*x^4 - 6144*d^10*e^6*x
^2 + 2048*d^12*e^4 + 4*(3*d*e^14*x^10 - 70*d^3*e^12*x^8 + 448*d^5*e^10*x^6 - 115
2*d^7*e^8*x^4 + 1280*d^9*e^6*x^2 - 512*d^11*e^4)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 169.554, size = 1919, normalized size = 7.62 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**7*Piecewise((-2*d**4*sqrt(d**2 - e**2*x**2)/(15*e**4) - d**2*x**2*sqrt(d**2 -
e**2*x**2)/(15*e**2) + x**4*sqrt(d**2 - e**2*x**2)/5, Ne(e, 0)), (x**4*sqrt(d**
2)/4, True)) + 3*d**6*e*Piecewise((-I*d**6*acosh(e*x/d)/(16*e**5) + I*d**5*x/(16
*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x**3/(48*e**2*sqrt(-1 + e**2*x**2/d**2
)) - 5*I*d*x**5/(24*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**7/(6*d*sqrt(-1 + e**2
*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (d**6*asin(e*x/d)/(16*e**5) - d**5*x/(16
*e**4*sqrt(1 - e**2*x**2/d**2)) + d**3*x**3/(48*e**2*sqrt(1 - e**2*x**2/d**2)) +
5*d*x**5/(24*sqrt(1 - e**2*x**2/d**2)) - e**2*x**7/(6*d*sqrt(1 - e**2*x**2/d**2
)), True)) + d**5*e**2*Piecewise((-8*d**6*sqrt(d**2 - e**2*x**2)/(105*e**6) - 4*
d**4*x**2*sqrt(d**2 - e**2*x**2)/(105*e**4) - d**2*x**4*sqrt(d**2 - e**2*x**2)/(
35*e**2) + x**6*sqrt(d**2 - e**2*x**2)/7, Ne(e, 0)), (x**6*sqrt(d**2)/6, True))
- 5*d**4*e**3*Piecewise((-5*I*d**8*acosh(e*x/d)/(128*e**7) + 5*I*d**7*x/(128*e**
6*sqrt(-1 + e**2*x**2/d**2)) - 5*I*d**5*x**3/(384*e**4*sqrt(-1 + e**2*x**2/d**2)
) - I*d**3*x**5/(192*e**2*sqrt(-1 + e**2*x**2/d**2)) - 7*I*d*x**7/(48*sqrt(-1 +
e**2*x**2/d**2)) + I*e**2*x**9/(8*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d*
*2) > 1), (5*d**8*asin(e*x/d)/(128*e**7) - 5*d**7*x/(128*e**6*sqrt(1 - e**2*x**2
/d**2)) + 5*d**5*x**3/(384*e**4*sqrt(1 - e**2*x**2/d**2)) + d**3*x**5/(192*e**2*
sqrt(1 - e**2*x**2/d**2)) + 7*d*x**7/(48*sqrt(1 - e**2*x**2/d**2)) - e**2*x**9/(
8*d*sqrt(1 - e**2*x**2/d**2)), True)) - 5*d**3*e**4*Piecewise((-16*d**8*sqrt(d**
2 - e**2*x**2)/(315*e**8) - 8*d**6*x**2*sqrt(d**2 - e**2*x**2)/(315*e**6) - 2*d*
*4*x**4*sqrt(d**2 - e**2*x**2)/(105*e**4) - d**2*x**6*sqrt(d**2 - e**2*x**2)/(63
*e**2) + x**8*sqrt(d**2 - e**2*x**2)/9, Ne(e, 0)), (x**8*sqrt(d**2)/8, True)) +
d**2*e**5*Piecewise((-7*I*d**10*acosh(e*x/d)/(256*e**9) + 7*I*d**9*x/(256*e**8*s
qrt(-1 + e**2*x**2/d**2)) - 7*I*d**7*x**3/(768*e**6*sqrt(-1 + e**2*x**2/d**2)) -
7*I*d**5*x**5/(1920*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x**7/(480*e**2*sqr
t(-1 + e**2*x**2/d**2)) - 9*I*d*x**9/(80*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**
11/(10*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (7*d**10*asin(e*x
/d)/(256*e**9) - 7*d**9*x/(256*e**8*sqrt(1 - e**2*x**2/d**2)) + 7*d**7*x**3/(768
*e**6*sqrt(1 - e**2*x**2/d**2)) + 7*d**5*x**5/(1920*e**4*sqrt(1 - e**2*x**2/d**2
)) + d**3*x**7/(480*e**2*sqrt(1 - e**2*x**2/d**2)) + 9*d*x**9/(80*sqrt(1 - e**2*
x**2/d**2)) - e**2*x**11/(10*d*sqrt(1 - e**2*x**2/d**2)), True)) + 3*d*e**6*Piec
ewise((-128*d**10*sqrt(d**2 - e**2*x**2)/(3465*e**10) - 64*d**8*x**2*sqrt(d**2 -
e**2*x**2)/(3465*e**8) - 16*d**6*x**4*sqrt(d**2 - e**2*x**2)/(1155*e**6) - 8*d*
*4*x**6*sqrt(d**2 - e**2*x**2)/(693*e**4) - d**2*x**8*sqrt(d**2 - e**2*x**2)/(99
*e**2) + x**10*sqrt(d**2 - e**2*x**2)/11, Ne(e, 0)), (x**10*sqrt(d**2)/10, True)
) + e**7*Piecewise((-21*I*d**12*acosh(e*x/d)/(1024*e**11) + 21*I*d**11*x/(1024*e
**10*sqrt(-1 + e**2*x**2/d**2)) - 7*I*d**9*x**3/(1024*e**8*sqrt(-1 + e**2*x**2/d
**2)) - 7*I*d**7*x**5/(2560*e**6*sqrt(-1 + e**2*x**2/d**2)) - I*d**5*x**7/(640*e
**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x**9/(960*e**2*sqrt(-1 + e**2*x**2/d**2)
) - 11*I*d*x**11/(120*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**13/(12*d*sqrt(-1 +
e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (21*d**12*asin(e*x/d)/(1024*e**11) -
21*d**11*x/(1024*e**10*sqrt(1 - e**2*x**2/d**2)) + 7*d**9*x**3/(1024*e**8*sqrt(
1 - e**2*x**2/d**2)) + 7*d**7*x**5/(2560*e**6*sqrt(1 - e**2*x**2/d**2)) + d**5*x
**7/(640*e**4*sqrt(1 - e**2*x**2/d**2)) + d**3*x**9/(960*e**2*sqrt(1 - e**2*x**2
/d**2)) + 11*d*x**11/(120*sqrt(1 - e**2*x**2/d**2)) - e**2*x**13/(12*d*sqrt(1 -
e**2*x**2/d**2)), True))

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GIAC/XCAS [A]  time = 0.286089, size = 201, normalized size = 0.8 \[ \frac{41}{1024} \, d^{12} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-4\right )}{\rm sign}\left (d\right ) - \frac{1}{3548160} \,{\left (235520 \, d^{11} e^{\left (-4\right )} +{\left (142065 \, d^{10} e^{\left (-3\right )} + 2 \,{\left (58880 \, d^{9} e^{\left (-2\right )} +{\left (47355 \, d^{8} e^{\left (-1\right )} - 4 \,{\left (99840 \, d^{7} +{\left (256641 \, d^{6} e + 2 \,{\left (41600 \, d^{5} e^{2} - 7 \,{\left (20493 \, d^{4} e^{3} + 8 \,{\left (2320 \, d^{3} e^{4} - 3 \,{\left (121 \, d^{2} e^{5} + 10 \,{\left (11 \, x e^{7} + 36 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \]

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^3,x, algorithm="giac")

[Out]

41/1024*d^12*arcsin(x*e/d)*e^(-4)*sign(d) - 1/3548160*(235520*d^11*e^(-4) + (142
065*d^10*e^(-3) + 2*(58880*d^9*e^(-2) + (47355*d^8*e^(-1) - 4*(99840*d^7 + (2566
41*d^6*e + 2*(41600*d^5*e^2 - 7*(20493*d^4*e^3 + 8*(2320*d^3*e^4 - 3*(121*d^2*e^
5 + 10*(11*x*e^7 + 36*d*e^6)*x)*x)*x)*x)*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)