\(\int \frac {(d-e x^2)^{11/2}}{(d^2-e^2 x^4)^{5/2}} \, dx\) [165]

Optimal result

Integrand size = 29, antiderivative size = 150 \[ \int \frac {\left (d-e x^2\right )^{11/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {8 d^2 x \left (d-e x^2\right )^{3/2}}{3 \left (d^2-e^2 x^4\right )^{3/2}}-\frac {20 d x \sqrt {d-e x^2}}{3 \sqrt {d^2-e^2 x^4}}-\frac {x \sqrt {d^2-e^2 x^4}}{2 \sqrt {d-e x^2}}+\frac {11 d \text {arctanh}\left (\frac {\sqrt {e} x \sqrt {d-e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{2 \sqrt {e}} \] Output:

8/3*d^2*x*(-e*x^2+d)^(3/2)/(-e^2*x^4+d^2)^(3/2)-20/3*d*x*(-e*x^2+d)^(1/2)/ 
(-e^2*x^4+d^2)^(1/2)-1/2*x*(-e^2*x^4+d^2)^(1/2)/(-e*x^2+d)^(1/2)+11/2*d*ar 
ctanh(e^(1/2)*x*(-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(1/2))/e^(1/2)
 

Mathematica [A] (verified)

Time = 5.63 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.93 \[ \int \frac {\left (d-e x^2\right )^{11/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {1}{6} \left (-\frac {x \sqrt {d^2-e^2 x^4} \left (27 d^2+46 d e x^2+3 e^2 x^4\right )}{\sqrt {d-e x^2} \left (d+e x^2\right )^2}-\frac {33 d \log \left (-d+e x^2\right )}{\sqrt {e}}+\frac {33 d \log \left (d e x-e^2 x^3+\sqrt {e} \sqrt {d-e x^2} \sqrt {d^2-e^2 x^4}\right )}{\sqrt {e}}\right ) \] Input:

Integrate[(d - e*x^2)^(11/2)/(d^2 - e^2*x^4)^(5/2),x]
 

Output:

(-((x*Sqrt[d^2 - e^2*x^4]*(27*d^2 + 46*d*e*x^2 + 3*e^2*x^4))/(Sqrt[d - e*x 
^2]*(d + e*x^2)^2)) - (33*d*Log[-d + e*x^2])/Sqrt[e] + (33*d*Log[d*e*x - e 
^2*x^3 + Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d^2 - e^2*x^4]])/Sqrt[e])/6
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {1396, 315, 27, 401, 25, 27, 299, 224, 219}

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.

\(\displaystyle \int \frac {\left (d-e x^2\right )^{11/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 401

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {1}{3} \left (-\frac {\int -\frac {d e \left (7 d-19 e x^2\right )}{\sqrt {e x^2+d}}dx}{d e}-\frac {6 x \left (d-e x^2\right )}{\sqrt {d+e x^2}}\right )+\frac {2 x \left (d-e x^2\right )^2}{3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {1}{3} \left (\frac {\int \frac {d e \left (7 d-19 e x^2\right )}{\sqrt {e x^2+d}}dx}{d e}-\frac {6 x \left (d-e x^2\right )}{\sqrt {d+e x^2}}\right )+\frac {2 x \left (d-e x^2\right )^2}{3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {1}{3} \left (\int \frac {7 d-19 e x^2}{\sqrt {e x^2+d}}dx-\frac {6 x \left (d-e x^2\right )}{\sqrt {d+e x^2}}\right )+\frac {2 x \left (d-e x^2\right )^2}{3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {1}{3} \left (\frac {33}{2} d \int \frac {1}{\sqrt {e x^2+d}}dx-\frac {19}{2} x \sqrt {d+e x^2}-\frac {6 x \left (d-e x^2\right )}{\sqrt {d+e x^2}}\right )+\frac {2 x \left (d-e x^2\right )^2}{3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {1}{3} \left (\frac {33}{2} d \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}-\frac {19}{2} x \sqrt {d+e x^2}-\frac {6 x \left (d-e x^2\right )}{\sqrt {d+e x^2}}\right )+\frac {2 x \left (d-e x^2\right )^2}{3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {1}{3} \left (\frac {33 d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}-\frac {19}{2} x \sqrt {d+e x^2}-\frac {6 x \left (d-e x^2\right )}{\sqrt {d+e x^2}}\right )+\frac {2 x \left (d-e x^2\right )^2}{3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

Input:

Int[(d - e*x^2)^(11/2)/(d^2 - e^2*x^4)^(5/2),x]
 

Output:

(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*((2*x*(d - e*x^2)^2)/(3*(d + e*x^2)^(3/2) 
) + ((-6*x*(d - e*x^2))/Sqrt[d + e*x^2] - (19*x*Sqrt[d + e*x^2])/2 + (33*d 
*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*Sqrt[e]))/3))/Sqrt[d^2 - e^2*x^4 
]
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 299

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

rule 315

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), 
x] - Simp[1/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S 
imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) 
*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 
1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
 

rule 401

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(a + b*x^2)^(p + 1)*( 
c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + 
(b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L 
tQ[p, -1] && GtQ[q, 0]
 

rule 1396

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x 
_Symbol] :> Simp[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d 
+ c*(x^n/e))^FracPart[p])   Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, 
x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a* 
e^2, 0] &&  !IntegerQ[p] &&  !(EqQ[q, 1] && EqQ[n, 2])
 

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.85

method	result	size
default	\(\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, \left (-3 e^{\frac {5}{2}} x^{5}+33 \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right ) d e \,x^{2} \sqrt {e \,x^{2}+d}-46 d \,e^{\frac {3}{2}} x^{3}+33 \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right ) d^{2} \sqrt {e \,x^{2}+d}-27 d^{2} x \sqrt {e}\right )}{6 \sqrt {-e \,x^{2}+d}\, \left (e \,x^{2}+d \right )^{2} \sqrt {e}}\)	\(128\)
risch	\(\frac {x \sqrt {e \,x^{2}+d}\, \sqrt {\frac {\left (-e \,x^{2}+d \right ) \left (-e^{2} x^{4}+d^{2}\right )}{\left (e \,x^{2}-d \right )^{2}}}\, \left (e \,x^{2}-d \right )}{2 \sqrt {-e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {\left (\frac {11 d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}+\frac {2 d^{2} \sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}}{3 e \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )^{2}}-\frac {10 d \sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}}{3 e \left (x -\frac {\sqrt {-d e}}{e}\right )}-\frac {2 d^{2} \sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}}{3 e \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )^{2}}-\frac {10 d \sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}}{3 e \left (x +\frac {\sqrt {-d e}}{e}\right )}\right ) \sqrt {\frac {\left (-e \,x^{2}+d \right ) \left (-e^{2} x^{4}+d^{2}\right )}{\left (e \,x^{2}-d \right )^{2}}}\, \left (e \,x^{2}-d \right )}{\sqrt {-e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(431\)

Input:

int((-e*x^2+d)^(11/2)/(-e^2*x^4+d^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/6*(-e^2*x^4+d^2)^(1/2)*(-3*e^(5/2)*x^5+33*ln(x*e^(1/2)+(e*x^2+d)^(1/2))* 
d*e*x^2*(e*x^2+d)^(1/2)-46*d*e^(3/2)*x^3+33*ln(x*e^(1/2)+(e*x^2+d)^(1/2))* 
d^2*(e*x^2+d)^(1/2)-27*d^2*x*e^(1/2))/(-e*x^2+d)^(1/2)/(e*x^2+d)^2/e^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.45 \[ \int \frac {\left (d-e x^2\right )^{11/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\left [\frac {33 \, {\left (d e^{3} x^{6} + d^{2} e^{2} x^{4} - d^{3} e x^{2} - d^{4}\right )} \sqrt {e} \log \left (\frac {2 \, e^{2} x^{4} - d e x^{2} - 2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {e} x - d^{2}}{e x^{2} - d}\right ) + 2 \, {\left (3 \, e^{3} x^{5} + 46 \, d e^{2} x^{3} + 27 \, d^{2} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d}}{12 \, {\left (e^{4} x^{6} + d e^{3} x^{4} - d^{2} e^{2} x^{2} - d^{3} e\right )}}, \frac {33 \, {\left (d e^{3} x^{6} + d^{2} e^{2} x^{4} - d^{3} e x^{2} - d^{4}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {-e} x}{e^{2} x^{4} - d^{2}}\right ) + {\left (3 \, e^{3} x^{5} + 46 \, d e^{2} x^{3} + 27 \, d^{2} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d}}{6 \, {\left (e^{4} x^{6} + d e^{3} x^{4} - d^{2} e^{2} x^{2} - d^{3} e\right )}}\right ] \] Input:

integrate((-e*x^2+d)^(11/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="fricas")
 

Output:

[1/12*(33*(d*e^3*x^6 + d^2*e^2*x^4 - d^3*e*x^2 - d^4)*sqrt(e)*log((2*e^2*x 
^4 - d*e*x^2 - 2*sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d)*sqrt(e)*x - d^2)/(e 
*x^2 - d)) + 2*(3*e^3*x^5 + 46*d*e^2*x^3 + 27*d^2*e*x)*sqrt(-e^2*x^4 + d^2 
)*sqrt(-e*x^2 + d))/(e^4*x^6 + d*e^3*x^4 - d^2*e^2*x^2 - d^3*e), 1/6*(33*( 
d*e^3*x^6 + d^2*e^2*x^4 - d^3*e*x^2 - d^4)*sqrt(-e)*arctan(sqrt(-e^2*x^4 + 
 d^2)*sqrt(-e*x^2 + d)*sqrt(-e)*x/(e^2*x^4 - d^2)) + (3*e^3*x^5 + 46*d*e^2 
*x^3 + 27*d^2*e*x)*sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d))/(e^4*x^6 + d*e^3 
*x^4 - d^2*e^2*x^2 - d^3*e)]
 

Sympy [F]

\[ \int \frac {\left (d-e x^2\right )^{11/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {\left (d - e x^{2}\right )^{\frac {11}{2}}}{\left (- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((-e*x**2+d)**(11/2)/(-e**2*x**4+d**2)**(5/2),x)
 

Output:

Integral((d - e*x**2)**(11/2)/(-(-d + e*x**2)*(d + e*x**2))**(5/2), x)
 

Maxima [F]

\[ \int \frac {\left (d-e x^2\right )^{11/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int { \frac {{\left (-e x^{2} + d\right )}^{\frac {11}{2}}}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((-e*x^2+d)^(11/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="maxima")
 

Output:

integrate((-e*x^2 + d)^(11/2)/(-e^2*x^4 + d^2)^(5/2), x)
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.41 \[ \int \frac {\left (d-e x^2\right )^{11/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=-\frac {1}{2} \, \sqrt {e x^{2} + d} x - \frac {11 \, d \log \left ({\left | -\sqrt {e} x + \sqrt {e x^{2} + d} \right |}\right )}{2 \, \sqrt {e}} - \frac {4 \, {\left (5 \, d e x^{2} + 3 \, d^{2}\right )} x}{3 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \] Input:

integrate((-e*x^2+d)^(11/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="giac")
 

Output:

-1/2*sqrt(e*x^2 + d)*x - 11/2*d*log(abs(-sqrt(e)*x + sqrt(e*x^2 + d)))/sqr 
t(e) - 4/3*(5*d*e*x^2 + 3*d^2)*x/(e*x^2 + d)^(3/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-e x^2\right )^{11/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {{\left (d-e\,x^2\right )}^{11/2}}{{\left (d^2-e^2\,x^4\right )}^{5/2}} \,d x \] Input:

int((d - e*x^2)^(11/2)/(d^2 - e^2*x^4)^(5/2),x)
 

Output:

int((d - e*x^2)^(11/2)/(d^2 - e^2*x^4)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.26 \[ \int \frac {\left (d-e x^2\right )^{11/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {-54 \sqrt {e \,x^{2}+d}\, d^{2} e x -92 \sqrt {e \,x^{2}+d}\, d \,e^{2} x^{3}-6 \sqrt {e \,x^{2}+d}\, e^{3} x^{5}+66 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d^{3}+132 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d^{2} e \,x^{2}+66 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d \,e^{2} x^{4}-15 \sqrt {e}\, d^{3}-30 \sqrt {e}\, d^{2} e \,x^{2}-15 \sqrt {e}\, d \,e^{2} x^{4}}{12 e \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

int((-e*x^2+d)^(11/2)/(-e^2*x^4+d^2)^(5/2),x)
 

Output:

( - 54*sqrt(d + e*x**2)*d**2*e*x - 92*sqrt(d + e*x**2)*d*e**2*x**3 - 6*sqr 
t(d + e*x**2)*e**3*x**5 + 66*sqrt(e)*log((sqrt(d + e*x**2) + sqrt(e)*x)/sq 
rt(d))*d**3 + 132*sqrt(e)*log((sqrt(d + e*x**2) + sqrt(e)*x)/sqrt(d))*d**2 
*e*x**2 + 66*sqrt(e)*log((sqrt(d + e*x**2) + sqrt(e)*x)/sqrt(d))*d*e**2*x* 
*4 - 15*sqrt(e)*d**3 - 30*sqrt(e)*d**2*e*x**2 - 15*sqrt(e)*d*e**2*x**4)/(1 
2*e*(d**2 + 2*d*e*x**2 + e**2*x**4))