\(\int \frac {(d^2-e^2 x^4)^{3/2}}{(d+e x^2)^{7/2}} \, dx\) [143]

Optimal result

Integrand size = 28, antiderivative size = 123 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\frac {x \sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^{3/2}}+\frac {\arctan \left (\frac {\sqrt {e} x \sqrt {d+e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{\sqrt {e}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e} x \sqrt {d+e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{\sqrt {2} \sqrt {e}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.86 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.27 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\frac {x \sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^{3/2}}-\frac {\sqrt {d^2-e^2 x^4} \arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d-e x^2}}\right )}{\sqrt {2} \sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {i \log \left (-2 i \sqrt {e} x+\frac {2 \sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}}\right )}{\sqrt {e}} \] Input:

Integrate[(d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^(7/2),x]
 

Output:

(x*Sqrt[d^2 - e^2*x^4])/(d + e*x^2)^(3/2) - (Sqrt[d^2 - e^2*x^4]*ArcTan[(S 
qrt[2]*Sqrt[e]*x)/Sqrt[d - e*x^2]])/(Sqrt[2]*Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[ 
d + e*x^2]) + (I*Log[(-2*I)*Sqrt[e]*x + (2*Sqrt[d^2 - e^2*x^4])/Sqrt[d + e 
*x^2]])/Sqrt[e]
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1396, 315, 27, 385, 224, 216, 291, 218}

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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 385

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

Input:

Int[(d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^(7/2),x]
 

Output:

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

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 218

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

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 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 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 385

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2), 
x_Symbol] :> Simp[e^2/b   Int[(e*x)^(m - 2)*(c + d*x^2)^q, x], x] - Simp[a* 
(e^2/b)   Int[(e*x)^(m - 2)*((c + d*x^2)^q/(a + b*x^2)), x], x] /; FreeQ[{a 
, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LeQ[2, m, 3] && IntBinomial 
Q[a, b, c, d, e, m, 2, -1, q, x]
 

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. \(372\) vs. \(2(99)=198\).

Time = 0.46 (sec) , antiderivative size = 373, normalized size of antiderivative = 3.03

method	result	size
default	\(-\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, \sqrt {e}\, \left (-\ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}-\sqrt {-d e}\, x +d \right )}{e x -\sqrt {-d e}}\right ) \sqrt {2}\, e^{\frac {3}{2}} x^{2} \sqrt {d}+\ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}+\sqrt {-d e}\, x +d \right )}{e x +\sqrt {-d e}}\right ) \sqrt {2}\, e^{\frac {3}{2}} x^{2} \sqrt {d}-\sqrt {e}\, d^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}-\sqrt {-d e}\, x +d \right )}{e x -\sqrt {-d e}}\right )+\ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}+\sqrt {-d e}\, x +d \right )}{e x +\sqrt {-d e}}\right ) \sqrt {e}\, d^{\frac {3}{2}} \sqrt {2}-4 \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right ) e \,x^{2} \sqrt {-d e}-4 \sqrt {-d e}\, \sqrt {-e \,x^{2}+d}\, \sqrt {e}\, x -4 \sqrt {-d e}\, \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right ) d \right )}{4 \sqrt {e \,x^{2}+d}\, \sqrt {-e \,x^{2}+d}\, \left (e x -\sqrt {-d e}\right ) \left (e x +\sqrt {-d e}\right ) \sqrt {-d e}}\)	\(373\)

Input:

int((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 442, normalized size of antiderivative = 3.59 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\left [\frac {4 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} e x - \sqrt {2} {\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt {-e} \log \left (-\frac {3 \, e^{2} x^{4} + 2 \, d e x^{2} + 2 \, \sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {-e} x - d^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\right ) - 2 \, {\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt {-e} \log \left (-\frac {2 \, e^{2} x^{4} + d e x^{2} - 2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {-e} x - d^{2}}{e x^{2} + d}\right )}{4 \, {\left (e^{3} x^{4} + 2 \, d e^{2} x^{2} + d^{2} e\right )}}, \frac {2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} e x + \sqrt {2} {\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {e} x}{e^{2} x^{4} - d^{2}}\right ) - 2 \, {\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt {e} \arctan \left (\frac {\sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {e} x}{e^{2} x^{4} - d^{2}}\right )}{2 \, {\left (e^{3} x^{4} + 2 \, d e^{2} x^{2} + d^{2} e\right )}}\right ] \] Input:

integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(7/2),x, algorithm="fricas")
 

Output:

[1/4*(4*sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d)*e*x - sqrt(2)*(e^2*x^4 + 2*d* 
e*x^2 + d^2)*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)) - 
2*(e^2*x^4 + 2*d*e*x^2 + d^2)*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)))/(e^3*x^4 + 
 2*d*e^2*x^2 + d^2*e), 1/2*(2*sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d)*e*x + s 
qrt(2)*(e^2*x^4 + 2*d*e*x^2 + d^2)*sqrt(e)*arctan(sqrt(2)*sqrt(-e^2*x^4 + 
d^2)*sqrt(e*x^2 + d)*sqrt(e)*x/(e^2*x^4 - d^2)) - 2*(e^2*x^4 + 2*d*e*x^2 + 
 d^2)*sqrt(e)*arctan(sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d)*sqrt(e)*x/(e^2*x 
^4 - d^2)))/(e^3*x^4 + 2*d*e^2*x^2 + d^2*e)]
 

Sympy [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {\left (- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {3}{2}}}{\left (d + e x^{2}\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((-e**2*x**4+d**2)**(3/2)/(e*x**2+d)**(7/2),x)
 

Output:

Integral((-(-d + e*x**2)*(d + e*x**2))**(3/2)/(d + e*x**2)**(7/2), x)
 

Maxima [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(7/2),x, algorithm="maxima")
 

Output:

integrate((-e^2*x^4 + d^2)^(3/2)/(e*x^2 + d)^(7/2), x)
 

Giac [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(7/2),x, algorithm="giac")
 

Output:

integrate((-e^2*x^4 + d^2)^(3/2)/(e*x^2 + d)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {{\left (d^2-e^2\,x^4\right )}^{3/2}}{{\left (e\,x^2+d\right )}^{7/2}} \,d x \] Input:

int((d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^(7/2),x)
                                                                                    
                                                                                    
 

Output:

int((d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^(7/2), x)
 

Reduce [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{7/2}} \, dx=\frac {2 \sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x +3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{4} x^{8}-2 d \,e^{3} x^{6}+2 d^{3} e \,x^{2}+d^{4}}d x \right ) d^{2} e^{2}+6 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{4} x^{8}-2 d \,e^{3} x^{6}+2 d^{3} e \,x^{2}+d^{4}}d x \right ) d \,e^{3} x^{2}+3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{4} x^{8}-2 d \,e^{3} x^{6}+2 d^{3} e \,x^{2}+d^{4}}d x \right ) e^{4} x^{4}+\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}}{-e^{4} x^{8}-2 d \,e^{3} x^{6}+2 d^{3} e \,x^{2}+d^{4}}d x \right ) d^{4}+2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}}{-e^{4} x^{8}-2 d \,e^{3} x^{6}+2 d^{3} e \,x^{2}+d^{4}}d x \right ) d^{3} e \,x^{2}+\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}}{-e^{4} x^{8}-2 d \,e^{3} x^{6}+2 d^{3} e \,x^{2}+d^{4}}d x \right ) d^{2} e^{2} x^{4}}{3 e^{2} x^{4}+6 d e \,x^{2}+3 d^{2}} \] Input:

int((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^(7/2),x)
 

Output:

(2*sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*x + 3*int((sqrt(d + e*x**2)*sqr 
t(d**2 - e**2*x**4)*x**4)/(d**4 + 2*d**3*e*x**2 - 2*d*e**3*x**6 - e**4*x** 
8),x)*d**2*e**2 + 6*int((sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*x**4)/(d* 
*4 + 2*d**3*e*x**2 - 2*d*e**3*x**6 - e**4*x**8),x)*d*e**3*x**2 + 3*int((sq 
rt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*x**4)/(d**4 + 2*d**3*e*x**2 - 2*d*e* 
*3*x**6 - e**4*x**8),x)*e**4*x**4 + int((sqrt(d + e*x**2)*sqrt(d**2 - e**2 
*x**4))/(d**4 + 2*d**3*e*x**2 - 2*d*e**3*x**6 - e**4*x**8),x)*d**4 + 2*int 
((sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4))/(d**4 + 2*d**3*e*x**2 - 2*d*e** 
3*x**6 - e**4*x**8),x)*d**3*e*x**2 + int((sqrt(d + e*x**2)*sqrt(d**2 - e** 
2*x**4))/(d**4 + 2*d**3*e*x**2 - 2*d*e**3*x**6 - e**4*x**8),x)*d**2*e**2*x 
**4)/(3*(d**2 + 2*d*e*x**2 + e**2*x**4))