\(\int \frac {1}{(d+e x^2) (d^2-e^2 x^4)^{3/2}} \, dx\) [65]

Optimal result

Integrand size = 26, antiderivative size = 192 \[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {x \left (5 d-3 e x^2\right )}{12 d^4 \sqrt {d^2-e^2 x^4}}+\frac {x}{6 d^2 \left (d+e x^2\right ) \sqrt {d^2-e^2 x^4}}+\frac {\sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{4 d^{5/2} \sqrt {e} \sqrt {d^2-e^2 x^4}}+\frac {\sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{6 d^{5/2} \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:

1/12*x*(-3*e*x^2+5*d)/d^4/(-e^2*x^4+d^2)^(1/2)+1/6*x/d^2/(e*x^2+d)/(-e^2*x 
^4+d^2)^(1/2)+1/4*(1-e^2*x^4/d^2)^(1/2)*EllipticE(e^(1/2)*x/d^(1/2),I)/d^( 
5/2)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)+1/6*(1-e^2*x^4/d^2)^(1/2)*EllipticF(e^(1 
/2)*x/d^(1/2),I)/d^(5/2)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.48 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {\sqrt {-\frac {e}{d}} x \left (7 d^2+2 d e x^2-3 e^2 x^4\right )-3 i d \left (d+e x^2\right ) \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )-2 i d \left (d+e x^2\right ) \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )}{12 d^4 \sqrt {-\frac {e}{d}} \left (d+e x^2\right ) \sqrt {d^2-e^2 x^4}} \] Input:

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

Output:

(Sqrt[-(e/d)]*x*(7*d^2 + 2*d*e*x^2 - 3*e^2*x^4) - (3*I)*d*(d + e*x^2)*Sqrt 
[1 - (e^2*x^4)/d^2]*EllipticE[I*ArcSinh[Sqrt[-(e/d)]*x], -1] - (2*I)*d*(d 
+ e*x^2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[I*ArcSinh[Sqrt[-(e/d)]*x], -1]) 
/(12*d^4*Sqrt[-(e/d)]*(d + e*x^2)*Sqrt[d^2 - e^2*x^4])
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.52, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {1396, 316, 27, 402, 27, 402, 25, 27, 399, 289, 329, 327, 765, 762}

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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 316

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 289

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

\(\Big \downarrow \) 329

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\frac {\frac {2 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {3 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {3 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{3 d}-\frac {x \sqrt {d-e x^2}}{3 d \left (d+e x^2\right )^{3/2}}}{2 d^2}+\frac {x}{2 d^2 \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

Input:

Int[1/((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/(2*d^2*Sqrt[d - e*x^2]*(d + e*x^2)^(3/ 
2)) + (-1/3*(x*Sqrt[d - e*x^2])/(d*(d + e*x^2)^(3/2)) + ((3*x*Sqrt[d - e*x 
^2])/(2*d*Sqrt[d + e*x^2]) + ((3*d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticE 
[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x^2 
]) + (2*d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[ArcSin[(Sqrt[e]*x)/Sqrt[ 
d]], -1])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x^2]))/(2*d))/(3*d))/(2*d^2) 
))/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 289

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Sim 
p[(a + b*x^2)^FracPart[p]*((c + d*x^2)^FracPart[p]/(a*c + b*d*x^4)^FracPart 
[p])   Int[(a*c + b*d*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b* 
c + a*d, 0] &&  !IntegerQ[p]
 

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 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 329

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
a*(Sqrt[1 - b^2*(x^4/a^2)]/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]))   Int[Sqrt[1 
+ b*(x^2/a)]/Sqrt[1 - b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b 
*c + a*d, 0] &&  !(LtQ[a*c, 0] && GtQ[a*b, 0])
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

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 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 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 = 1.34 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.41

method	result	size
default	\(\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{12 d^{3} e^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {3 \left (-e^{2} x^{2}+d e \right ) x}{8 e \,d^{4} \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}-\frac {\left (-e^{2} x^{2}-d e \right ) x}{8 d^{4} e \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {5 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{12 d^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{4 d^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(270\)
elliptic	\(\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{12 d^{3} e^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {3 \left (-e^{2} x^{2}+d e \right ) x}{8 e \,d^{4} \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}-\frac {\left (-e^{2} x^{2}-d e \right ) x}{8 d^{4} e \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {5 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{12 d^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{4 d^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(270\)

Input:

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

Output:

1/12/d^3/e^2*x*(-e^2*x^4+d^2)^(1/2)/(x^2+d/e)^2+3/8*(-e^2*x^2+d*e)/e/d^4*x 
/((x^2+d/e)*(-e^2*x^2+d*e))^(1/2)-1/8*(-e^2*x^2-d*e)/d^4*x/e/((x^2-d/e)*(- 
e^2*x^2-d*e))^(1/2)+5/12/d^3/(e/d)^(1/2)*(1-e*x^2/d)^(1/2)*(1+e*x^2/d)^(1/ 
2)/(-e^2*x^4+d^2)^(1/2)*EllipticF(x*(e/d)^(1/2),I)-1/4/d^3/(e/d)^(1/2)*(1- 
e*x^2/d)^(1/2)*(1+e*x^2/d)^(1/2)/(-e^2*x^4+d^2)^(1/2)*(EllipticF(x*(e/d)^( 
1/2),I)-EllipticE(x*(e/d)^(1/2),I))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {3 \, {\left (e^{4} x^{6} + d e^{3} x^{4} - d^{2} e^{2} x^{2} - d^{3} e\right )} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left ({\left (5 \, d e^{3} - 3 \, e^{4}\right )} x^{6} + {\left (5 \, d^{2} e^{2} - 3 \, d e^{3}\right )} x^{4} - 5 \, d^{4} + 3 \, d^{3} e - {\left (5 \, d^{3} e - 3 \, d^{2} e^{2}\right )} x^{2}\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left (3 \, e^{3} x^{5} - 2 \, d e^{2} x^{3} - 7 \, d^{2} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{12 \, {\left (d^{4} e^{4} x^{6} + d^{5} e^{3} x^{4} - d^{6} e^{2} x^{2} - d^{7} e\right )}} \] Input:

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

Output:

1/12*(3*(e^4*x^6 + d*e^3*x^4 - d^2*e^2*x^2 - d^3*e)*sqrt(e/d)*elliptic_e(a 
rcsin(x*sqrt(e/d)), -1) + ((5*d*e^3 - 3*e^4)*x^6 + (5*d^2*e^2 - 3*d*e^3)*x 
^4 - 5*d^4 + 3*d^3*e - (5*d^3*e - 3*d^2*e^2)*x^2)*sqrt(e/d)*elliptic_f(arc 
sin(x*sqrt(e/d)), -1) + (3*e^3*x^5 - 2*d*e^2*x^3 - 7*d^2*e*x)*sqrt(-e^2*x^ 
4 + d^2))/(d^4*e^4*x^6 + d^5*e^3*x^4 - d^6*e^2*x^2 - d^7*e)
 

Sympy [F]

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

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

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{e^{5} x^{10}+d \,e^{4} x^{8}-2 d^{2} e^{3} x^{6}-2 d^{3} e^{2} x^{4}+d^{4} e \,x^{2}+d^{5}}d x \] Input:

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

Output:

int(sqrt(d**2 - e**2*x**4)/(d**5 + d**4*e*x**2 - 2*d**3*e**2*x**4 - 2*d**2 
*e**3*x**6 + d*e**4*x**8 + e**5*x**10),x)