\(\int \frac {1}{\sqrt {d-e x^2} (d^2-e^2 x^4)^{3/2}} \, dx\) [162]

Optimal result

Integrand size = 29, antiderivative size = 131 \[ \int \frac {1}{\sqrt {d-e x^2} \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {x}{4 d^2 \sqrt {d-e x^2} \sqrt {d^2-e^2 x^4}}+\frac {x \sqrt {d-e x^2}}{8 d^3 \sqrt {d^2-e^2 x^4}}+\frac {5 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x \sqrt {d-e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{8 \sqrt {2} d^3 \sqrt {e}} \] Output:

1/4*x/d^2/(-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(1/2)+1/8*x*(-e*x^2+d)^(1/2)/d^3 
/(-e^2*x^4+d^2)^(1/2)+5/16*arctanh(2^(1/2)*e^(1/2)*x*(-e*x^2+d)^(1/2)/(-e^ 
2*x^4+d^2)^(1/2))*2^(1/2)/d^3/e^(1/2)
 

Mathematica [A] (verified)

Time = 2.96 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\sqrt {d-e x^2} \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {\sqrt {d^2-e^2 x^4} \left (-2 \sqrt {e} x \left (-3 d+e x^2\right ) \sqrt {d+e x^2}+5 \sqrt {2} \left (d^2-e^2 x^4\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )\right )}{16 d^3 \sqrt {e} \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}} \] Input:

Integrate[1/(Sqrt[d - e*x^2]*(d^2 - e^2*x^4)^(3/2)),x]
 

Output:

(Sqrt[d^2 - e^2*x^4]*(-2*Sqrt[e]*x*(-3*d + e*x^2)*Sqrt[d + e*x^2] + 5*Sqrt 
[2]*(d^2 - e^2*x^4)*ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]]))/(16*d^3 
*Sqrt[e]*(d - e*x^2)^(3/2)*(d + e*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1396, 316, 27, 402, 27, 291, 221}

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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 316

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {5}{2} \int \frac {1}{d-\frac {2 d e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}+\frac {x}{2 d \sqrt {d+e x^2}}}{4 d^2}+\frac {x}{4 d^2 \left (d-e x^2\right ) \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {5 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {2} d \sqrt {e}}+\frac {x}{2 d \sqrt {d+e x^2}}}{4 d^2}+\frac {x}{4 d^2 \left (d-e x^2\right ) \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

Input:

Int[1/(Sqrt[d - e*x^2]*(d^2 - e^2*x^4)^(3/2)),x]
 

Output:

(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*(x/(4*d^2*(d - e*x^2)*Sqrt[d + e*x^2]) + 
(x/(2*d*Sqrt[d + e*x^2]) + (5*ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]] 
)/(2*Sqrt[2]*d*Sqrt[e]))/(4*d^2)))/Sqrt[d^2 - e^2*x^4]
 

Defintions of rubi rules used

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 221

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

rule 291

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

rule 316

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

rule 402

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 + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[p, -1]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1067\) vs. \(2(105)=210\).

Time = 0.86 (sec) , antiderivative size = 1068, normalized size of antiderivative = 8.15

method	result	size
default	\(\text {Expression too large to display}\)	\(1068\)

Input:

int(1/(-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/4*(-e^2*x^4+d^2)^(1/2)*e^(11/2)*(-5*2^(1/2)*ln(2*e*(2^(1/2)*d^(1/2)*(e* 
x^2+d)^(1/2)+(d*e)^(1/2)*x+d)/(e*x-(d*e)^(1/2)))*e^(7/2)*x^6*d^(1/2)+5*2^( 
1/2)*ln(2*e*(2^(1/2)*d^(1/2)*(e*x^2+d)^(1/2)-(d*e)^(1/2)*x+d)/(e*x+(d*e)^( 
1/2)))*e^(7/2)*x^6*d^(1/2)-5*2^(1/2)*ln(2*e*(2^(1/2)*d^(1/2)*(e*x^2+d)^(1/ 
2)+(d*e)^(1/2)*x+d)/(e*x-(d*e)^(1/2)))*d^(3/2)*e^(5/2)*x^4+5*2^(1/2)*ln(2* 
e*(2^(1/2)*d^(1/2)*(e*x^2+d)^(1/2)-(d*e)^(1/2)*x+d)/(e*x+(d*e)^(1/2)))*d^( 
3/2)*e^(5/2)*x^4-8*ln((e^(1/2)*(-(e*x+(-d*e)^(1/2))/e*(-e*x+(-d*e)^(1/2))) 
^(1/2)+e*x)/e^(1/2))*e^3*x^6*(d*e)^(1/2)+8*ln(((e*x^2+d)^(1/2)*e^(1/2)+e*x 
)/e^(1/2))*e^3*x^6*(d*e)^(1/2)+4*e^(5/2)*x^5*(e*x^2+d)^(1/2)*(d*e)^(1/2)-8 
*e^(5/2)*x^5*(d*e)^(1/2)*(-(e*x+(-d*e)^(1/2))/e*(-e*x+(-d*e)^(1/2)))^(1/2) 
+5*2^(1/2)*ln(2*e*(2^(1/2)*d^(1/2)*(e*x^2+d)^(1/2)+(d*e)^(1/2)*x+d)/(e*x-( 
d*e)^(1/2)))*d^(5/2)*e^(3/2)*x^2-5*2^(1/2)*ln(2*e*(2^(1/2)*d^(1/2)*(e*x^2+ 
d)^(1/2)-(d*e)^(1/2)*x+d)/(e*x+(d*e)^(1/2)))*d^(5/2)*e^(3/2)*x^2-8*ln((e^( 
1/2)*(-(e*x+(-d*e)^(1/2))/e*(-e*x+(-d*e)^(1/2)))^(1/2)+e*x)/e^(1/2))*d*e^2 
*x^4*(d*e)^(1/2)+8*ln(((e*x^2+d)^(1/2)*e^(1/2)+e*x)/e^(1/2))*d*e^2*x^4*(d* 
e)^(1/2)+8*d*e^(3/2)*x^3*(e*x^2+d)^(1/2)*(d*e)^(1/2)+5*2^(1/2)*ln(2*e*(2^( 
1/2)*d^(1/2)*(e*x^2+d)^(1/2)+(d*e)^(1/2)*x+d)/(e*x-(d*e)^(1/2)))*d^(7/2)*e 
^(1/2)-5*2^(1/2)*ln(2*e*(2^(1/2)*d^(1/2)*(e*x^2+d)^(1/2)-(d*e)^(1/2)*x+d)/ 
(e*x+(d*e)^(1/2)))*d^(7/2)*e^(1/2)+8*ln((e^(1/2)*(-(e*x+(-d*e)^(1/2))/e*(- 
e*x+(-d*e)^(1/2)))^(1/2)+e*x)/e^(1/2))*d^2*e*x^2*(d*e)^(1/2)-8*ln(((e*x...
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.81 \[ \int \frac {1}{\sqrt {d-e x^2} \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\left [\frac {5 \, \sqrt {2} {\left (e^{3} x^{6} - d e^{2} x^{4} - d^{2} e x^{2} + d^{3}\right )} \sqrt {e} \log \left (-\frac {3 \, e^{2} x^{4} - 2 \, d e x^{2} - 2 \, \sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {e} x - d^{2}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right ) - 4 \, \sqrt {-e^{2} x^{4} + d^{2}} {\left (e^{2} x^{3} - 3 \, d e x\right )} \sqrt {-e x^{2} + d}}{32 \, {\left (d^{3} e^{4} x^{6} - d^{4} e^{3} x^{4} - d^{5} e^{2} x^{2} + d^{6} e\right )}}, \frac {5 \, \sqrt {2} {\left (e^{3} x^{6} - d e^{2} x^{4} - d^{2} e x^{2} + d^{3}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {-e} x}{e^{2} x^{4} - d^{2}}\right ) - 2 \, \sqrt {-e^{2} x^{4} + d^{2}} {\left (e^{2} x^{3} - 3 \, d e x\right )} \sqrt {-e x^{2} + d}}{16 \, {\left (d^{3} e^{4} x^{6} - d^{4} e^{3} x^{4} - d^{5} e^{2} x^{2} + d^{6} e\right )}}\right ] \] Input:

integrate(1/(-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(3/2),x, algorithm="fricas")
 

Output:

[1/32*(5*sqrt(2)*(e^3*x^6 - d*e^2*x^4 - d^2*e*x^2 + d^3)*sqrt(e)*log(-(3*e 
^2*x^4 - 2*d*e*x^2 - 2*sqrt(2)*sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d)*sqrt( 
e)*x - d^2)/(e^2*x^4 - 2*d*e*x^2 + d^2)) - 4*sqrt(-e^2*x^4 + d^2)*(e^2*x^3 
 - 3*d*e*x)*sqrt(-e*x^2 + d))/(d^3*e^4*x^6 - d^4*e^3*x^4 - d^5*e^2*x^2 + d 
^6*e), 1/16*(5*sqrt(2)*(e^3*x^6 - d*e^2*x^4 - d^2*e*x^2 + d^3)*sqrt(-e)*ar 
ctan(sqrt(2)*sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d)*sqrt(-e)*x/(e^2*x^4 - d 
^2)) - 2*sqrt(-e^2*x^4 + d^2)*(e^2*x^3 - 3*d*e*x)*sqrt(-e*x^2 + d))/(d^3*e 
^4*x^6 - d^4*e^3*x^4 - d^5*e^2*x^2 + d^6*e)]
 

Sympy [F]

\[ \int \frac {1}{\sqrt {d-e x^2} \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {3}{2}} \sqrt {d - e x^{2}}}\, dx \] Input:

integrate(1/(-e*x**2+d)**(1/2)/(-e**2*x**4+d**2)**(3/2),x)
 

Output:

Integral(1/((-(-d + e*x**2)*(d + e*x**2))**(3/2)*sqrt(d - e*x**2)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {d-e x^2} \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}} \sqrt {-e x^{2} + d}} \,d x } \] Input:

integrate(1/(-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(3/2),x, algorithm="maxima")
 

Output:

integrate(1/((-e^2*x^4 + d^2)^(3/2)*sqrt(-e*x^2 + d)), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {d-e x^2} \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}} \sqrt {-e x^{2} + d}} \,d x } \] Input:

integrate(1/(-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(3/2),x, algorithm="giac")
 

Output:

integrate(1/((-e^2*x^4 + d^2)^(3/2)*sqrt(-e*x^2 + d)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d-e x^2} \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (d^2-e^2\,x^4\right )}^{3/2}\,\sqrt {d-e\,x^2}} \,d x \] Input:

int(1/((d^2 - e^2*x^4)^(3/2)*(d - e*x^2)^(1/2)),x)
 

Output:

int(1/((d^2 - e^2*x^4)^(3/2)*(d - e*x^2)^(1/2)), x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.85 \[ \int \frac {1}{\sqrt {d-e x^2} \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {12 \sqrt {e \,x^{2}+d}\, d e x -4 \sqrt {e \,x^{2}+d}\, e^{2} x^{3}-5 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d^{2}+5 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) e^{2} x^{4}+5 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d^{2}-5 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) e^{2} x^{4}+5 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d^{2}-5 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) e^{2} x^{4}-5 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d^{2}+5 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) e^{2} x^{4}-12 \sqrt {e}\, d^{2}+12 \sqrt {e}\, e^{2} x^{4}}{32 d^{3} e \left (-e^{2} x^{4}+d^{2}\right )} \] Input:

int(1/(-e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(3/2),x)
 

Output:

(12*sqrt(d + e*x**2)*d*e*x - 4*sqrt(d + e*x**2)*e**2*x**3 - 5*sqrt(e)*sqrt 
(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d) 
)*d**2 + 5*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) - sqrt( 
d) + sqrt(e)*x)/sqrt(d))*e**2*x**4 + 5*sqrt(e)*sqrt(2)*log((sqrt(d + e*x** 
2) - sqrt(d)*sqrt(2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*d**2 - 5*sqrt(e)*sqrt 
(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) + sqrt(d) + sqrt(e)*x)/sqrt(d) 
)*e**2*x**4 + 5*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) + sqrt(d)*sqrt(2) - 
sqrt(d) + sqrt(e)*x)/sqrt(d))*d**2 - 5*sqrt(e)*sqrt(2)*log((sqrt(d + e*x** 
2) + sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*e**2*x**4 - 5*sqrt(e) 
*sqrt(2)*log((sqrt(d + e*x**2) + sqrt(d)*sqrt(2) + sqrt(d) + sqrt(e)*x)/sq 
rt(d))*d**2 + 5*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) + sqrt(d)*sqrt(2) + 
sqrt(d) + sqrt(e)*x)/sqrt(d))*e**2*x**4 - 12*sqrt(e)*d**2 + 12*sqrt(e)*e** 
2*x**4)/(32*d**3*e*(d**2 - e**2*x**4))