\(\int \frac {1}{(d+e x)^2 (d^2-e^2 x^2)^{7/2}} \, dx\) [145]

Optimal result

Integrand size = 24, antiderivative size = 127 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {x}{9 d^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2}{9 e (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}+\frac {2 x}{15 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {8 x}{45 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{45 d^8 \sqrt {d^2-e^2 x^2}} \] Output:

1/9*x/d^2/(-e^2*x^2+d^2)^(7/2)-2/9/e/(e*x+d)/(-e^2*x^2+d^2)^(7/2)+2/15*x/d 
^4/(-e^2*x^2+d^2)^(5/2)+8/45*x/d^6/(-e^2*x^2+d^2)^(3/2)+16/45*x/d^8/(-e^2* 
x^2+d^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-10 d^7+25 d^6 e x+60 d^5 e^2 x^2-10 d^4 e^3 x^3-80 d^3 e^4 x^4-24 d^2 e^5 x^5+32 d e^6 x^6+16 e^7 x^7\right )}{45 d^8 e (d-e x)^3 (d+e x)^5} \] Input:

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

Output:

(Sqrt[d^2 - e^2*x^2]*(-10*d^7 + 25*d^6*e*x + 60*d^5*e^2*x^2 - 10*d^4*e^3*x 
^3 - 80*d^3*e^4*x^4 - 24*d^2*e^5*x^5 + 32*d*e^6*x^6 + 16*e^7*x^7))/(45*d^8 
*e*(d - e*x)^3*(d + e*x)^5)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.28, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {461, 470, 209, 209, 208}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 461

\(\displaystyle \frac {7 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}}dx}{9 d}-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 470

\(\displaystyle \frac {7 \left (\frac {6 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}}dx}{7 d}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}\right )}{9 d}-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 209

\(\displaystyle \frac {7 \left (\frac {6 \left (\frac {4 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d^2}+\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{7 d}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}\right )}{9 d}-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 209

\(\displaystyle \frac {7 \left (\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}+\frac {x}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}\right )}{5 d^2}+\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{7 d}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}\right )}{9 d}-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 208

\(\displaystyle \frac {7 \left (\frac {6 \left (\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 \left (\frac {x}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{3 d^4 \sqrt {d^2-e^2 x^2}}\right )}{5 d^2}\right )}{7 d}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}\right )}{9 d}-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}\)

Input:

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

Output:

-1/9*1/(d*e*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) + (7*(-1/7*1/(d*e*(d + e*x) 
*(d^2 - e^2*x^2)^(5/2)) + (6*(x/(5*d^2*(d^2 - e^2*x^2)^(5/2)) + (4*(x/(3*d 
^2*(d^2 - e^2*x^2)^(3/2)) + (2*x)/(3*d^4*Sqrt[d^2 - e^2*x^2])))/(5*d^2)))/ 
(7*d)))/(9*d)
 

Defintions of rubi rules used

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 209

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
 

rule 461

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*c*(n + p + 1))), x] + Simp[Simpl 
ify[n + 2*p + 2]/(2*c*(n + p + 1))   Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x 
], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[Simp 
lify[n + 2*p + 2], 0] && (LtQ[n, -1] || GtQ[n + p, 0])
 

rule 470

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*c*(n + p + 1))), x] + Simp[(n + 
2*p + 2)/(2*c*(n + p + 1))   Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] /; 
 FreeQ[{a, b, c, d, p}, x] && EqQ[b*c^2 + a*d^2, 0] && LtQ[n, 0] && NeQ[n + 
 p + 1, 0] && IntegerQ[2*p]
 

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.87

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (-16 e^{7} x^{7}-32 d \,e^{6} x^{6}+24 d^{2} e^{5} x^{5}+80 d^{3} e^{4} x^{4}+10 d^{4} e^{3} x^{3}-60 d^{5} e^{2} x^{2}-25 d^{6} e x +10 d^{7}\right )}{45 \left (e x +d \right ) d^{8} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\)	\(110\)
orering	\(-\frac {\left (-e x +d \right ) \left (-16 e^{7} x^{7}-32 d \,e^{6} x^{6}+24 d^{2} e^{5} x^{5}+80 d^{3} e^{4} x^{4}+10 d^{4} e^{3} x^{3}-60 d^{5} e^{2} x^{2}-25 d^{6} e x +10 d^{7}\right )}{45 \left (e x +d \right ) d^{8} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\)	\(110\)
trager	\(-\frac {\left (-16 e^{7} x^{7}-32 d \,e^{6} x^{6}+24 d^{2} e^{5} x^{5}+80 d^{3} e^{4} x^{4}+10 d^{4} e^{3} x^{3}-60 d^{5} e^{2} x^{2}-25 d^{6} e x +10 d^{7}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{45 d^{8} \left (e x +d \right )^{5} \left (-e x +d \right )^{3} e}\)	\(112\)
default	\(\frac {-\frac {1}{9 d e \left (x +\frac {d}{e}\right )^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {7 e \left (-\frac {1}{7 d e \left (x +\frac {d}{e}\right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {6 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{10 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {-\frac {2 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {4 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 e^{2} d^{4} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}}{d^{2}}\right )}{7 d}\right )}{9 d}}{e^{2}}\)	\(275\)

Input:

int(1/(e*x+d)^2/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-1/45*(-e*x+d)*(-16*e^7*x^7-32*d*e^6*x^6+24*d^2*e^5*x^5+80*d^3*e^4*x^4+10* 
d^4*e^3*x^3-60*d^5*e^2*x^2-25*d^6*e*x+10*d^7)/(e*x+d)/d^8/e/(-e^2*x^2+d^2) 
^(7/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (107) = 214\).

Time = 0.23 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.95 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {10 \, e^{8} x^{8} + 20 \, d e^{7} x^{7} - 20 \, d^{2} e^{6} x^{6} - 60 \, d^{3} e^{5} x^{5} + 60 \, d^{5} e^{3} x^{3} + 20 \, d^{6} e^{2} x^{2} - 20 \, d^{7} e x - 10 \, d^{8} + {\left (16 \, e^{7} x^{7} + 32 \, d e^{6} x^{6} - 24 \, d^{2} e^{5} x^{5} - 80 \, d^{3} e^{4} x^{4} - 10 \, d^{4} e^{3} x^{3} + 60 \, d^{5} e^{2} x^{2} + 25 \, d^{6} e x - 10 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{45 \, {\left (d^{8} e^{9} x^{8} + 2 \, d^{9} e^{8} x^{7} - 2 \, d^{10} e^{7} x^{6} - 6 \, d^{11} e^{6} x^{5} + 6 \, d^{13} e^{4} x^{3} + 2 \, d^{14} e^{3} x^{2} - 2 \, d^{15} e^{2} x - d^{16} e\right )}} \] Input:

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

Output:

-1/45*(10*e^8*x^8 + 20*d*e^7*x^7 - 20*d^2*e^6*x^6 - 60*d^3*e^5*x^5 + 60*d^ 
5*e^3*x^3 + 20*d^6*e^2*x^2 - 20*d^7*e*x - 10*d^8 + (16*e^7*x^7 + 32*d*e^6* 
x^6 - 24*d^2*e^5*x^5 - 80*d^3*e^4*x^4 - 10*d^4*e^3*x^3 + 60*d^5*e^2*x^2 + 
25*d^6*e*x - 10*d^7)*sqrt(-e^2*x^2 + d^2))/(d^8*e^9*x^8 + 2*d^9*e^8*x^7 - 
2*d^10*e^7*x^6 - 6*d^11*e^6*x^5 + 6*d^13*e^4*x^3 + 2*d^14*e^3*x^2 - 2*d^15 
*e^2*x - d^16*e)
 

Sympy [F]

\[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )^{2}}\, dx \] Input:

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

Output:

Integral(1/((-(-d + e*x)*(d + e*x))**(7/2)*(d + e*x)**2), x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.39 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {1}{9 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{3} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e\right )}} - \frac {1}{9 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e\right )}} + \frac {2 \, x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}} + \frac {8 \, x}{45 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6}} + \frac {16 \, x}{45 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{8}} \] Input:

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

Output:

-1/9/((-e^2*x^2 + d^2)^(5/2)*d*e^3*x^2 + 2*(-e^2*x^2 + d^2)^(5/2)*d^2*e^2* 
x + (-e^2*x^2 + d^2)^(5/2)*d^3*e) - 1/9/((-e^2*x^2 + d^2)^(5/2)*d^2*e^2*x 
+ (-e^2*x^2 + d^2)^(5/2)*d^3*e) + 2/15*x/((-e^2*x^2 + d^2)^(5/2)*d^4) + 8/ 
45*x/((-e^2*x^2 + d^2)^(3/2)*d^6) + 16/45*x/(sqrt(-e^2*x^2 + d^2)*d^8)
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.18 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.31 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {e^{7} {\left (\frac {3 \, {\left (315 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{2} + \frac {70 \, d}{e x + d} - 32\right )}}{d^{8} e^{7} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )} - \frac {5 \, d^{64} e^{56} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {9}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{8} \mathrm {sgn}\left (e\right )^{8} + 45 \, d^{64} e^{56} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {7}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{8} \mathrm {sgn}\left (e\right )^{8} + 189 \, d^{64} e^{56} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{8} \mathrm {sgn}\left (e\right )^{8} + 525 \, d^{64} e^{56} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{8} \mathrm {sgn}\left (e\right )^{8} + 1575 \, d^{64} e^{56} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{8} \mathrm {sgn}\left (e\right )^{8}}{d^{72} e^{63} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{9} \mathrm {sgn}\left (e\right )^{9}}\right )} + \frac {2048 i \, \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{8}}}{5760 \, {\left | e \right |}} \] Input:

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

Output:

1/5760*(e^7*(3*(315*(2*d/(e*x + d) - 1)^2 + 70*d/(e*x + d) - 32)/(d^8*e^7* 
(2*d/(e*x + d) - 1)^(5/2)*sgn(1/(e*x + d))*sgn(e)) - (5*d^64*e^56*(2*d/(e* 
x + d) - 1)^(9/2)*sgn(1/(e*x + d))^8*sgn(e)^8 + 45*d^64*e^56*(2*d/(e*x + d 
) - 1)^(7/2)*sgn(1/(e*x + d))^8*sgn(e)^8 + 189*d^64*e^56*(2*d/(e*x + d) - 
1)^(5/2)*sgn(1/(e*x + d))^8*sgn(e)^8 + 525*d^64*e^56*(2*d/(e*x + d) - 1)^( 
3/2)*sgn(1/(e*x + d))^8*sgn(e)^8 + 1575*d^64*e^56*sqrt(2*d/(e*x + d) - 1)* 
sgn(1/(e*x + d))^8*sgn(e)^8)/(d^72*e^63*sgn(1/(e*x + d))^9*sgn(e)^9)) + 20 
48*I*sgn(1/(e*x + d))*sgn(e)/d^8)/abs(e)
 

Mupad [B] (verification not implemented)

Time = 6.63 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.45 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {31\,x}{120\,d^4}-\frac {5}{24\,d^3\,e}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {8\,x}{45\,d^6}+\frac {5}{144\,d^5\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}}{72\,d^4\,e\,{\left (d+e\,x\right )}^5}-\frac {5\,\sqrt {d^2-e^2\,x^2}}{144\,d^5\,e\,{\left (d+e\,x\right )}^4}+\frac {16\,x\,\sqrt {d^2-e^2\,x^2}}{45\,d^8\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \] Input:

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

Output:

((d^2 - e^2*x^2)^(1/2)*((31*x)/(120*d^4) - 5/(24*d^3*e)))/((d + e*x)^3*(d 
- e*x)^3) + ((d^2 - e^2*x^2)^(1/2)*((8*x)/(45*d^6) + 5/(144*d^5*e)))/((d + 
 e*x)^2*(d - e*x)^2) - (d^2 - e^2*x^2)^(1/2)/(72*d^4*e*(d + e*x)^5) - (5*( 
d^2 - e^2*x^2)^(1/2))/(144*d^5*e*(d + e*x)^4) + (16*x*(d^2 - e^2*x^2)^(1/2 
))/(45*d^8*(d + e*x)*(d - e*x))
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 312, normalized size of antiderivative = 2.46 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {25 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{6}+50 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{5} e x -25 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} e^{2} x^{2}-100 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} e^{3} x^{3}-25 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e^{4} x^{4}+50 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{5} x^{5}+25 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{6} x^{6}-20 d^{7}+50 d^{6} e x +120 d^{5} e^{2} x^{2}-20 d^{4} e^{3} x^{3}-160 d^{3} e^{4} x^{4}-48 d^{2} e^{5} x^{5}+64 d \,e^{6} x^{6}+32 e^{7} x^{7}}{90 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{8} e \left (e^{6} x^{6}+2 d \,e^{5} x^{5}-d^{2} e^{4} x^{4}-4 d^{3} e^{3} x^{3}-d^{4} e^{2} x^{2}+2 d^{5} e x +d^{6}\right )} \] Input:

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

Output:

(25*sqrt(d**2 - e**2*x**2)*d**6 + 50*sqrt(d**2 - e**2*x**2)*d**5*e*x - 25* 
sqrt(d**2 - e**2*x**2)*d**4*e**2*x**2 - 100*sqrt(d**2 - e**2*x**2)*d**3*e* 
*3*x**3 - 25*sqrt(d**2 - e**2*x**2)*d**2*e**4*x**4 + 50*sqrt(d**2 - e**2*x 
**2)*d*e**5*x**5 + 25*sqrt(d**2 - e**2*x**2)*e**6*x**6 - 20*d**7 + 50*d**6 
*e*x + 120*d**5*e**2*x**2 - 20*d**4*e**3*x**3 - 160*d**3*e**4*x**4 - 48*d* 
*2*e**5*x**5 + 64*d*e**6*x**6 + 32*e**7*x**7)/(90*sqrt(d**2 - e**2*x**2)*d 
**8*e*(d**6 + 2*d**5*e*x - d**4*e**2*x**2 - 4*d**3*e**3*x**3 - d**2*e**4*x 
**4 + 2*d*e**5*x**5 + e**6*x**6))