\(\int \frac {1}{(d-e x^2)^{3/2} (d^2-e^2 x^4)^{3/2}} \, dx\) [163]

Optimal result

Integrand size = 29, antiderivative size = 167 \[ \int \frac {1}{\left (d-e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {x}{8 d^2 \left (d-e x^2\right )^{3/2} \sqrt {d^2-e^2 x^4}}+\frac {11 x}{32 d^3 \sqrt {d-e x^2} \sqrt {d^2-e^2 x^4}}-\frac {5 x \sqrt {d-e x^2}}{64 d^4 \sqrt {d^2-e^2 x^4}}+\frac {39 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x \sqrt {d-e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{64 \sqrt {2} d^4 \sqrt {e}} \] Output:

1/8*x/d^2/(-e*x^2+d)^(3/2)/(-e^2*x^4+d^2)^(1/2)+11/32*x/d^3/(-e*x^2+d)^(1/ 
2)/(-e^2*x^4+d^2)^(1/2)-5/64*x*(-e*x^2+d)^(1/2)/d^4/(-e^2*x^4+d^2)^(1/2)+3 
9/128*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^4/e^(1/2)
 

Mathematica [A] (verified)

Time = 4.81 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (d-e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {\sqrt {d^2-e^2 x^4} \left (-2 \sqrt {e} x \sqrt {d+e x^2} \left (-25 d^2+12 d e x^2+5 e^2 x^4\right )+39 \sqrt {2} \left (d-e x^2\right )^2 \left (d+e x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )\right )}{128 d^4 \sqrt {e} \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}} \] Input:

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

Output:

(Sqrt[d^2 - e^2*x^4]*(-2*Sqrt[e]*x*Sqrt[d + e*x^2]*(-25*d^2 + 12*d*e*x^2 + 
 5*e^2*x^4) + 39*Sqrt[2]*(d - e*x^2)^2*(d + e*x^2)*ArcTanh[(Sqrt[2]*Sqrt[e 
]*x)/Sqrt[d + e*x^2]]))/(128*d^4*Sqrt[e]*(d - e*x^2)^(5/2)*(d + e*x^2)^(3/ 
2))
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {1396, 316, 27, 402, 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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 316

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {e \left (4 e x^2+7 d\right )}{\left (d-e x^2\right )^2 \left (e x^2+d\right )^{3/2}}dx}{8 d^2 e}+\frac {x}{8 d^2 \left (d-e x^2\right )^2 \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {4 e x^2+7 d}{\left (d-e x^2\right )^2 \left (e x^2+d\right )^{3/2}}dx}{8 d^2}+\frac {x}{8 d^2 \left (d-e x^2\right )^2 \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\int \frac {d e \left (22 e x^2+17 d\right )}{\left (d-e x^2\right ) \left (e x^2+d\right )^{3/2}}dx}{4 d^2 e}+\frac {11 x}{4 d \left (d-e x^2\right ) \sqrt {d+e x^2}}}{8 d^2}+\frac {x}{8 d^2 \left (d-e x^2\right )^2 \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\int \frac {22 e x^2+17 d}{\left (d-e x^2\right ) \left (e x^2+d\right )^{3/2}}dx}{4 d}+\frac {11 x}{4 d \left (d-e x^2\right ) \sqrt {d+e x^2}}}{8 d^2}+\frac {x}{8 d^2 \left (d-e x^2\right )^2 \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {-\frac {\int -\frac {39 d^2 e}{\left (d-e x^2\right ) \sqrt {e x^2+d}}dx}{2 d^2 e}-\frac {5 x}{2 d \sqrt {d+e x^2}}}{4 d}+\frac {11 x}{4 d \left (d-e x^2\right ) \sqrt {d+e x^2}}}{8 d^2}+\frac {x}{8 d^2 \left (d-e x^2\right )^2 \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {39}{2} \int \frac {1}{\left (d-e x^2\right ) \sqrt {e x^2+d}}dx-\frac {5 x}{2 d \sqrt {d+e x^2}}}{4 d}+\frac {11 x}{4 d \left (d-e x^2\right ) \sqrt {d+e x^2}}}{8 d^2}+\frac {x}{8 d^2 \left (d-e x^2\right )^2 \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {39}{2} \int \frac {1}{d-\frac {2 d e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}-\frac {5 x}{2 d \sqrt {d+e x^2}}}{4 d}+\frac {11 x}{4 d \left (d-e x^2\right ) \sqrt {d+e x^2}}}{8 d^2}+\frac {x}{8 d^2 \left (d-e x^2\right )^2 \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {39 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {2} d \sqrt {e}}-\frac {5 x}{2 d \sqrt {d+e x^2}}}{4 d}+\frac {11 x}{4 d \left (d-e x^2\right ) \sqrt {d+e x^2}}}{8 d^2}+\frac {x}{8 d^2 \left (d-e x^2\right )^2 \sqrt {d+e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

Input:

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

Output:

(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*(x/(8*d^2*(d - e*x^2)^2*Sqrt[d + e*x^2]) 
+ ((11*x)/(4*d*(d - e*x^2)*Sqrt[d + e*x^2]) + ((-5*x)/(2*d*Sqrt[d + e*x^2] 
) + (39*ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*Sqrt[2]*d*Sqrt[e] 
))/(4*d))/(8*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. \(980\) vs. \(2(135)=270\).

Time = 0.88 (sec) , antiderivative size = 981, normalized size of antiderivative = 5.87

method	result	size
default	\(\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, e^{\frac {15}{2}} \left (39 \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 ) e^{\frac {9}{2}} x^{8} \sqrt {d}-39 \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 ) e^{\frac {9}{2}} x^{8} \sqrt {d}-48 \ln \left (\frac {\sqrt {e \,x^{2}+d}\, \sqrt {e}+e x}{\sqrt {e}}\right ) e^{4} x^{8} \sqrt {d e}+48 \ln \left (\frac {\sqrt {e}\, \sqrt {-\frac {\left (e x +\sqrt {-d e}\right ) \left (-e x +\sqrt {-d e}\right )}{e}}+e x}{\sqrt {e}}\right ) e^{4} x^{8} \sqrt {d e}-52 e^{\frac {7}{2}} x^{7} \sqrt {d e}\, \sqrt {e \,x^{2}+d}+32 e^{\frac {7}{2}} x^{7} \sqrt {d e}\, \sqrt {-\frac {\left (e x +\sqrt {-d e}\right ) \left (-e x +\sqrt {-d e}\right )}{e}}-78 \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 ) d^{\frac {5}{2}} e^{\frac {5}{2}} x^{4}+78 \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 ) d^{\frac {5}{2}} e^{\frac {5}{2}} x^{4}-36 d \,e^{\frac {5}{2}} x^{5} \sqrt {d e}\, \sqrt {e \,x^{2}+d}-32 d \,e^{\frac {5}{2}} \sqrt {d e}\, \sqrt {-\frac {\left (e x +\sqrt {-d e}\right ) \left (-e x +\sqrt {-d e}\right )}{e}}\, x^{5}+96 \ln \left (\frac {\sqrt {e \,x^{2}+d}\, \sqrt {e}+e x}{\sqrt {e}}\right ) d^{2} e^{2} x^{4} \sqrt {d e}-96 \ln \left (\frac {\sqrt {e}\, \sqrt {-\frac {\left (e x +\sqrt {-d e}\right ) \left (-e x +\sqrt {-d e}\right )}{e}}+e x}{\sqrt {e}}\right ) d^{2} e^{2} x^{4} \sqrt {d e}+84 d^{2} e^{\frac {3}{2}} x^{3} \sqrt {d e}\, \sqrt {e \,x^{2}+d}-32 d^{2} e^{\frac {3}{2}} \sqrt {d e}\, \sqrt {-\frac {\left (e x +\sqrt {-d e}\right ) \left (-e x +\sqrt {-d e}\right )}{e}}\, x^{3}+39 \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 ) d^{\frac {9}{2}} \sqrt {e}-39 \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 ) d^{\frac {9}{2}} \sqrt {e}+68 d^{3} x \sqrt {e}\, \sqrt {d e}\, \sqrt {e \,x^{2}+d}+32 d^{3} \sqrt {d e}\, \sqrt {e}\, \sqrt {-\frac {\left (e x +\sqrt {-d e}\right ) \left (-e x +\sqrt {-d e}\right )}{e}}\, x -48 \ln \left (\frac {\sqrt {e \,x^{2}+d}\, \sqrt {e}+e x}{\sqrt {e}}\right ) d^{4} \sqrt {d e}+48 \ln \left (\frac {\sqrt {e}\, \sqrt {-\frac {\left (e x +\sqrt {-d e}\right ) \left (-e x +\sqrt {-d e}\right )}{e}}+e x}{\sqrt {e}}\right ) d^{4} \sqrt {d e}\right )}{16 \sqrt {-e \,x^{2}+d}\, \sqrt {e \,x^{2}+d}\, \left (\sqrt {d e}-\sqrt {-d e}\right )^{4} \left (\sqrt {d e}+\sqrt {-d e}\right )^{4} \left (e x -\sqrt {d e}\right )^{2} \left (e x +\sqrt {d e}\right )^{2} \left (e x -\sqrt {-d e}\right )^{2} \left (e x +\sqrt {-d e}\right )^{2} \sqrt {d e}}\)	\(981\)

Input:

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

Output:

1/16*(-e^2*x^4+d^2)^(1/2)*e^(15/2)*(39*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^(9/2)*x^8*d^(1/2)-39*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^(9/2)*x^8*d^(1/2)-48*ln(((e*x^2+d)^(1/2)*e^(1/2)+e*x)/e^(1/2))*e 
^4*x^8*(d*e)^(1/2)+48*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^4*x^8*(d*e)^(1/2)-52*e^(7/2)*x^7*(d*e)^(1/2)*(e* 
x^2+d)^(1/2)+32*e^(7/2)*x^7*(d*e)^(1/2)*(-(e*x+(-d*e)^(1/2))/e*(-e*x+(-d*e 
)^(1/2)))^(1/2)-78*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^(5/2)*x^4+78*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^(5 
/2)*x^4-36*d*e^(5/2)*x^5*(d*e)^(1/2)*(e*x^2+d)^(1/2)-32*d*e^(5/2)*(d*e)^(1 
/2)*(-(e*x+(-d*e)^(1/2))/e*(-e*x+(-d*e)^(1/2)))^(1/2)*x^5+96*ln(((e*x^2+d) 
^(1/2)*e^(1/2)+e*x)/e^(1/2))*d^2*e^2*x^4*(d*e)^(1/2)-96*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^2*x^4*(d*e 
)^(1/2)+84*d^2*e^(3/2)*x^3*(d*e)^(1/2)*(e*x^2+d)^(1/2)-32*d^2*e^(3/2)*(d*e 
)^(1/2)*(-(e*x+(-d*e)^(1/2))/e*(-e*x+(-d*e)^(1/2)))^(1/2)*x^3+39*2^(1/2)*l 
n(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^(9/2)*e^(1/2)-39*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^(9/2)*e^(1/2)+68*d^3*x*e^(1/2)*(d*e)^(1/2) 
*(e*x^2+d)^(1/2)+32*d^3*(d*e)^(1/2)*e^(1/2)*(-(e*x+(-d*e)^(1/2))/e*(-e*...
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.38 \[ \int \frac {1}{\left (d-e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\left [\frac {39 \, \sqrt {2} {\left (e^{4} x^{8} - 2 \, d e^{3} x^{6} + 2 \, d^{3} e x^{2} - d^{4}\right )} \sqrt {e} \log \left (-\frac {3 \, e^{2} x^{4} - 2 \, d e x^{2} - 2 \, \sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {e} x - d^{2}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right ) + 4 \, {\left (5 \, e^{3} x^{5} + 12 \, d e^{2} x^{3} - 25 \, d^{2} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d}}{256 \, {\left (d^{4} e^{5} x^{8} - 2 \, d^{5} e^{4} x^{6} + 2 \, d^{7} e^{2} x^{2} - d^{8} e\right )}}, \frac {39 \, \sqrt {2} {\left (e^{4} x^{8} - 2 \, d e^{3} x^{6} + 2 \, d^{3} e x^{2} - d^{4}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {-e} x}{e^{2} x^{4} - d^{2}}\right ) + 2 \, {\left (5 \, e^{3} x^{5} + 12 \, d e^{2} x^{3} - 25 \, d^{2} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d}}{128 \, {\left (d^{4} e^{5} x^{8} - 2 \, d^{5} e^{4} x^{6} + 2 \, d^{7} e^{2} x^{2} - d^{8} e\right )}}\right ] \] Input:

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

Output:

[1/256*(39*sqrt(2)*(e^4*x^8 - 2*d*e^3*x^6 + 2*d^3*e*x^2 - d^4)*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*(5*e^3*x^5 + 12*d*e^2*x 
^3 - 25*d^2*e*x)*sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d))/(d^4*e^5*x^8 - 2*d 
^5*e^4*x^6 + 2*d^7*e^2*x^2 - d^8*e), 1/128*(39*sqrt(2)*(e^4*x^8 - 2*d*e^3* 
x^6 + 2*d^3*e*x^2 - d^4)*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*(5*e^3*x^5 + 12*d*e^2*x^3 - 2 
5*d^2*e*x)*sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d))/(d^4*e^5*x^8 - 2*d^5*e^4 
*x^6 + 2*d^7*e^2*x^2 - d^8*e)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 755, normalized size of antiderivative = 4.52 \[ \int \frac {1}{\left (d-e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(1100*sqrt(d + e*x**2)*d**2*e*x - 528*sqrt(d + e*x**2)*d*e**2*x**3 - 220*s 
qrt(d + e*x**2)*e**3*x**5 - 429*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) - sq 
rt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*d**3 + 429*sqrt(e)*sqrt(2)*l 
og((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*d** 
2*e*x**2 + 429*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) - s 
qrt(d) + sqrt(e)*x)/sqrt(d))*d*e**2*x**4 - 429*sqrt(e)*sqrt(2)*log((sqrt(d 
 + e*x**2) - sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*e**3*x**6 + 4 
29*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) + sqrt(d) + sqr 
t(e)*x)/sqrt(d))*d**3 - 429*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d 
)*sqrt(2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*d**2*e*x**2 - 429*sqrt(e)*sqrt(2 
)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) + sqrt(d) + sqrt(e)*x)/sqrt(d))* 
d*e**2*x**4 + 429*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) 
+ sqrt(d) + sqrt(e)*x)/sqrt(d))*e**3*x**6 + 429*sqrt(e)*sqrt(2)*log((sqrt( 
d + e*x**2) + sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*d**3 - 429*s 
qrt(e)*sqrt(2)*log((sqrt(d + e*x**2) + sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e) 
*x)/sqrt(d))*d**2*e*x**2 - 429*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) + sqr 
t(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*d*e**2*x**4 + 429*sqrt(e)*sqr 
t(2)*log((sqrt(d + e*x**2) + sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d 
))*e**3*x**6 - 429*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) + sqrt(d)*sqrt(2) 
 + sqrt(d) + sqrt(e)*x)/sqrt(d))*d**3 + 429*sqrt(e)*sqrt(2)*log((sqrt(d...