\(\int \frac {1}{(d-e x^2)^{3/2} (d^2-e^2 x^4)^{5/2}} \, dx\) [172]

Optimal result

Integrand size = 29, antiderivative size = 239 \[ \int \frac {1}{\left (d-e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {x}{12 d^2 \left (d-e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{3/2}}+\frac {19 x}{96 d^3 \sqrt {d-e x^2} \left (d^2-e^2 x^4\right )^{3/2}}+\frac {61 x \sqrt {d-e x^2}}{128 d^4 \left (d^2-e^2 x^4\right )^{3/2}}-\frac {71 x \left (d-e x^2\right )^{3/2}}{256 d^5 \left (d^2-e^2 x^4\right )^{3/2}}-\frac {9 x \sqrt {d-e x^2}}{512 d^6 \sqrt {d^2-e^2 x^4}}+\frac {275 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x \sqrt {d-e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{512 \sqrt {2} d^6 \sqrt {e}} \] Output:

1/12*x/d^2/(-e*x^2+d)^(3/2)/(-e^2*x^4+d^2)^(3/2)+19/96*x/d^3/(-e*x^2+d)^(1 
/2)/(-e^2*x^4+d^2)^(3/2)+61/128*x*(-e*x^2+d)^(1/2)/d^4/(-e^2*x^4+d^2)^(3/2 
)-71/256*x*(-e*x^2+d)^(3/2)/d^5/(-e^2*x^4+d^2)^(3/2)-9/512*x*(-e*x^2+d)^(1 
/2)/d^6/(-e^2*x^4+d^2)^(1/2)+275/1024*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^6/e^(1/2)
 

Mathematica [A] (verified)

Time = 5.43 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\left (d-e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {\sqrt {d^2-e^2 x^4} \left (2 \sqrt {e} x \sqrt {d+e x^2} \left (711 d^4-436 d^3 e x^2-546 d^2 e^2 x^4+372 d e^3 x^6+27 e^4 x^8\right )+825 \sqrt {2} \left (d-e x^2\right )^3 \left (d+e x^2\right )^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )\right )}{3072 d^6 \sqrt {e} \left (d-e x^2\right )^{7/2} \left (d+e x^2\right )^{5/2}} \] Input:

Integrate[1/((d - e*x^2)^(3/2)*(d^2 - e^2*x^4)^(5/2)),x]
 

Output:

(Sqrt[d^2 - e^2*x^4]*(2*Sqrt[e]*x*Sqrt[d + e*x^2]*(711*d^4 - 436*d^3*e*x^2 
 - 546*d^2*e^2*x^4 + 372*d*e^3*x^6 + 27*e^4*x^8) + 825*Sqrt[2]*(d - e*x^2) 
^3*(d + e*x^2)^2*ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]]))/(3072*d^6* 
Sqrt[e]*(d - e*x^2)^(7/2)*(d + e*x^2)^(5/2))
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {1396, 316, 27, 402, 27, 402, 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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 316

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {e \left (8 e x^2+11 d\right )}{\left (d-e x^2\right )^3 \left (e x^2+d\right )^{5/2}}dx}{12 d^2 e}+\frac {x}{12 d^2 \left (d-e x^2\right )^3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {8 e x^2+11 d}{\left (d-e x^2\right )^3 \left (e x^2+d\right )^{5/2}}dx}{12 d^2}+\frac {x}{12 d^2 \left (d-e x^2\right )^3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\int \frac {3 d e \left (38 e x^2+23 d\right )}{\left (d-e x^2\right )^2 \left (e x^2+d\right )^{5/2}}dx}{8 d^2 e}+\frac {19 x}{8 d \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}}{12 d^2}+\frac {x}{12 d^2 \left (d-e x^2\right )^3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \int \frac {38 e x^2+23 d}{\left (d-e x^2\right )^2 \left (e x^2+d\right )^{5/2}}dx}{8 d}+\frac {19 x}{8 d \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}}{12 d^2}+\frac {x}{12 d^2 \left (d-e x^2\right )^3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\int \frac {d e \left (244 e x^2+31 d\right )}{\left (d-e x^2\right ) \left (e x^2+d\right )^{5/2}}dx}{4 d^2 e}+\frac {61 x}{4 d \left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}}\right )}{8 d}+\frac {19 x}{8 d \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}}{12 d^2}+\frac {x}{12 d^2 \left (d-e x^2\right )^3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\int \frac {244 e x^2+31 d}{\left (d-e x^2\right ) \left (e x^2+d\right )^{5/2}}dx}{4 d}+\frac {61 x}{4 d \left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}}\right )}{8 d}+\frac {19 x}{8 d \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}}{12 d^2}+\frac {x}{12 d^2 \left (d-e x^2\right )^3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {-\frac {\int -\frac {3 d e \left (142 e x^2+133 d\right )}{\left (d-e x^2\right ) \left (e x^2+d\right )^{3/2}}dx}{6 d^2 e}-\frac {71 x}{2 d \left (d+e x^2\right )^{3/2}}}{4 d}+\frac {61 x}{4 d \left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}}\right )}{8 d}+\frac {19 x}{8 d \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}}{12 d^2}+\frac {x}{12 d^2 \left (d-e x^2\right )^3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\frac {\int \frac {142 e x^2+133 d}{\left (d-e x^2\right ) \left (e x^2+d\right )^{3/2}}dx}{2 d}-\frac {71 x}{2 d \left (d+e x^2\right )^{3/2}}}{4 d}+\frac {61 x}{4 d \left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}}\right )}{8 d}+\frac {19 x}{8 d \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}}{12 d^2}+\frac {x}{12 d^2 \left (d-e x^2\right )^3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\frac {-\frac {\int -\frac {275 d^2 e}{\left (d-e x^2\right ) \sqrt {e x^2+d}}dx}{2 d^2 e}-\frac {9 x}{2 d \sqrt {d+e x^2}}}{2 d}-\frac {71 x}{2 d \left (d+e x^2\right )^{3/2}}}{4 d}+\frac {61 x}{4 d \left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}}\right )}{8 d}+\frac {19 x}{8 d \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}}{12 d^2}+\frac {x}{12 d^2 \left (d-e x^2\right )^3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\frac {\frac {275}{2} \int \frac {1}{\left (d-e x^2\right ) \sqrt {e x^2+d}}dx-\frac {9 x}{2 d \sqrt {d+e x^2}}}{2 d}-\frac {71 x}{2 d \left (d+e x^2\right )^{3/2}}}{4 d}+\frac {61 x}{4 d \left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}}\right )}{8 d}+\frac {19 x}{8 d \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}}{12 d^2}+\frac {x}{12 d^2 \left (d-e x^2\right )^3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\frac {\frac {275}{2} \int \frac {1}{d-\frac {2 d e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}-\frac {9 x}{2 d \sqrt {d+e x^2}}}{2 d}-\frac {71 x}{2 d \left (d+e x^2\right )^{3/2}}}{4 d}+\frac {61 x}{4 d \left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}}\right )}{8 d}+\frac {19 x}{8 d \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}}{12 d^2}+\frac {x}{12 d^2 \left (d-e x^2\right )^3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\frac {\frac {275 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {2} d \sqrt {e}}-\frac {9 x}{2 d \sqrt {d+e x^2}}}{2 d}-\frac {71 x}{2 d \left (d+e x^2\right )^{3/2}}}{4 d}+\frac {61 x}{4 d \left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}}\right )}{8 d}+\frac {19 x}{8 d \left (d-e x^2\right )^2 \left (d+e x^2\right )^{3/2}}}{12 d^2}+\frac {x}{12 d^2 \left (d-e x^2\right )^3 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

Input:

Int[1/((d - e*x^2)^(3/2)*(d^2 - e^2*x^4)^(5/2)),x]
 

Output:

(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*(x/(12*d^2*(d - e*x^2)^3*(d + e*x^2)^(3/2 
)) + ((19*x)/(8*d*(d - e*x^2)^2*(d + e*x^2)^(3/2)) + (3*((61*x)/(4*d*(d - 
e*x^2)*(d + e*x^2)^(3/2)) + ((-71*x)/(2*d*(d + e*x^2)^(3/2)) + ((-9*x)/(2* 
d*Sqrt[d + e*x^2]) + (275*ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2 
*Sqrt[2]*d*Sqrt[e]))/(2*d))/(4*d)))/(8*d))/(12*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. \(1324\) vs. \(2(195)=390\).

Time = 0.52 (sec) , antiderivative size = 1325, normalized size of antiderivative = 5.54

method	result	size
default	\(\text {Expression too large to display}\)	\(1325\)

Input:

int(1/(-e*x^2+d)^(3/2)/(-e^2*x^4+d^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/96*(-e^2*x^4+d^2)^(1/2)*e^(23/2)*(960*ln(((e*x^2+d)^(1/2)*e^(1/2)+e*x)/ 
e^(1/2))*e^6*x^12*(d*e)^(1/2)-960*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^6*x^12*(d*e)^(1/2)+825*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)*d^( 
13/2)*e^(1/2)-825*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)*d^(13/2)*e^(1/2)+1132*e^(11/2)*x^11*(d*e)^(1/2 
)*(e*x^2+d)^(1/2)-1024*e^(11/2)*x^11*(d*e)^(1/2)*(-(e*x+(-d*e)^(1/2))/e*(- 
e*x+(-d*e)^(1/2)))^(1/2)+1152*d^5*(d*e)^(1/2)*e^(1/2)*(-(e*x+(-d*e)^(1/2)) 
/e*(-e*x+(-d*e)^(1/2)))^(1/2)*x+2475*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)*d^(5/2)*e^(9/2)*x^8-2475*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)*d^(5/2)*e^(9/2)*x^8-2475*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)*d^(9/2)*e^(5/2)*x^4+2475*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)*d^(9/2)*e^(5/2)*x^4-2880*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^4*e^2*x^4*(d*e)^(1/2)+700*d*e^(9/2)*x^9*( 
d*e)^(1/2)*(e*x^2+d)^(1/2)-3000*d^2*e^(7/2)*x^7*(d*e)^(1/2)*(e*x^2+d)^(1/2 
)-1880*d^3*e^(5/2)*x^5*(d*e)^(1/2)*(e*x^2+d)^(1/2)+2380*d^4*e^(3/2)*x^3*(d 
*e)^(1/2)*(e*x^2+d)^(1/2)-825*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^(13/2)*x^12*d^(1/2)+825*ln(2*...
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 568, normalized size of antiderivative = 2.38 \[ \int \frac {1}{\left (d-e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\left [\frac {825 \, \sqrt {2} {\left (e^{6} x^{12} - 2 \, d e^{5} x^{10} - d^{2} e^{4} x^{8} + 4 \, d^{3} e^{3} x^{6} - d^{4} e^{2} x^{4} - 2 \, d^{5} e x^{2} + d^{6}\right )} \sqrt {e} \log \left (-\frac {3 \, e^{2} x^{4} - 2 \, d e x^{2} - 2 \, \sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {e} x - d^{2}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right ) + 4 \, {\left (27 \, e^{5} x^{9} + 372 \, d e^{4} x^{7} - 546 \, d^{2} e^{3} x^{5} - 436 \, d^{3} e^{2} x^{3} + 711 \, d^{4} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d}}{6144 \, {\left (d^{6} e^{7} x^{12} - 2 \, d^{7} e^{6} x^{10} - d^{8} e^{5} x^{8} + 4 \, d^{9} e^{4} x^{6} - d^{10} e^{3} x^{4} - 2 \, d^{11} e^{2} x^{2} + d^{12} e\right )}}, \frac {825 \, \sqrt {2} {\left (e^{6} x^{12} - 2 \, d e^{5} x^{10} - d^{2} e^{4} x^{8} + 4 \, d^{3} e^{3} x^{6} - d^{4} e^{2} x^{4} - 2 \, d^{5} e x^{2} + d^{6}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {-e} x}{e^{2} x^{4} - d^{2}}\right ) + 2 \, {\left (27 \, e^{5} x^{9} + 372 \, d e^{4} x^{7} - 546 \, d^{2} e^{3} x^{5} - 436 \, d^{3} e^{2} x^{3} + 711 \, d^{4} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d}}{3072 \, {\left (d^{6} e^{7} x^{12} - 2 \, d^{7} e^{6} x^{10} - d^{8} e^{5} x^{8} + 4 \, d^{9} e^{4} x^{6} - d^{10} e^{3} x^{4} - 2 \, d^{11} e^{2} x^{2} + d^{12} e\right )}}\right ] \] Input:

integrate(1/(-e*x^2+d)^(3/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="fricas")
 

Output:

[1/6144*(825*sqrt(2)*(e^6*x^12 - 2*d*e^5*x^10 - d^2*e^4*x^8 + 4*d^3*e^3*x^ 
6 - d^4*e^2*x^4 - 2*d^5*e*x^2 + d^6)*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*(27*e^5*x^9 + 372*d*e^4*x^7 - 546*d^2*e^3*x^5 - 4 
36*d^3*e^2*x^3 + 711*d^4*e*x)*sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d))/(d^6* 
e^7*x^12 - 2*d^7*e^6*x^10 - d^8*e^5*x^8 + 4*d^9*e^4*x^6 - d^10*e^3*x^4 - 2 
*d^11*e^2*x^2 + d^12*e), 1/3072*(825*sqrt(2)*(e^6*x^12 - 2*d*e^5*x^10 - d^ 
2*e^4*x^8 + 4*d^3*e^3*x^6 - d^4*e^2*x^4 - 2*d^5*e*x^2 + d^6)*sqrt(-e)*arct 
an(sqrt(2)*sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d)*sqrt(-e)*x/(e^2*x^4 - d^2 
)) + 2*(27*e^5*x^9 + 372*d*e^4*x^7 - 546*d^2*e^3*x^5 - 436*d^3*e^2*x^3 + 7 
11*d^4*e*x)*sqrt(-e^2*x^4 + d^2)*sqrt(-e*x^2 + d))/(d^6*e^7*x^12 - 2*d^7*e 
^6*x^10 - d^8*e^5*x^8 + 4*d^9*e^4*x^6 - d^10*e^3*x^4 - 2*d^11*e^2*x^2 + d^ 
12*e)]
 

Sympy [F]

\[ \int \frac {1}{\left (d-e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {1}{\left (- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {5}{2}} \left (d - e x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(-e*x**2+d)**(3/2)/(-e**2*x**4+d**2)**(5/2),x)
 

Output:

Integral(1/((-(-d + e*x**2)*(d + e*x**2))**(5/2)*(d - e*x**2)**(3/2)), x)
 

Maxima [F]

\[ \int \frac {1}{\left (d-e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {5}{2}} {\left (-e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(-e*x^2+d)^(3/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="maxima")
 

Output:

integrate(1/((-e^2*x^4 + d^2)^(5/2)*(-e*x^2 + d)^(3/2)), x)
 

Giac [F]

\[ \int \frac {1}{\left (d-e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {5}{2}} {\left (-e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(-e*x^2+d)^(3/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="giac")
 

Output:

integrate(1/((-e^2*x^4 + d^2)^(5/2)*(-e*x^2 + d)^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d-e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {1}{{\left (d^2-e^2\,x^4\right )}^{5/2}\,{\left (d-e\,x^2\right )}^{3/2}} \,d x \] Input:

int(1/((d^2 - e^2*x^4)^(5/2)*(d - e*x^2)^(3/2)),x)
 

Output:

int(1/((d^2 - e^2*x^4)^(5/2)*(d - e*x^2)^(3/2)), x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 1181, normalized size of antiderivative = 4.94 \[ \int \frac {1}{\left (d-e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int(1/(-e*x^2+d)^(3/2)/(-e^2*x^4+d^2)^(5/2),x)
 

Output:

(2844*sqrt(d + e*x**2)*d**4*e*x - 1744*sqrt(d + e*x**2)*d**3*e**2*x**3 - 2 
184*sqrt(d + e*x**2)*d**2*e**3*x**5 + 1488*sqrt(d + e*x**2)*d*e**4*x**7 + 
108*sqrt(d + e*x**2)*e**5*x**9 - 825*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) 
 - sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*d**5 + 825*sqrt(e)*sqrt 
(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d) 
)*d**4*e*x**2 + 1650*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt( 
2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*d**3*e**2*x**4 - 1650*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**3*x**6 - 825*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) 
- sqrt(d) + sqrt(e)*x)/sqrt(d))*d*e**4*x**8 + 825*sqrt(e)*sqrt(2)*log((sqr 
t(d + e*x**2) - sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*e**5*x**10 
 + 825*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) + sqrt(d) + 
 sqrt(e)*x)/sqrt(d))*d**5 - 825*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) - sq 
rt(d)*sqrt(2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*d**4*e*x**2 - 1650*sqrt(e)*s 
qrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) + sqrt(d) + sqrt(e)*x)/sqrt 
(d))*d**3*e**2*x**4 + 1650*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d) 
*sqrt(2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*d**2*e**3*x**6 + 825*sqrt(e)*sqrt 
(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) + sqrt(d) + sqrt(e)*x)/sqrt(d) 
)*d*e**4*x**8 - 825*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2 
) + sqrt(d) + sqrt(e)*x)/sqrt(d))*e**5*x**10 + 825*sqrt(e)*sqrt(2)*log(...