\(\int \frac {\sqrt {d^2-e^2 x^4}}{(d-e x^2)^4} \, dx\) [91]

Optimal result

Integrand size = 27, antiderivative size = 227 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d-e x^2\right )^4} \, dx=\frac {x \sqrt {d^2-e^2 x^4}}{5 d \left (d-e x^2\right )^3}+\frac {7 x \sqrt {d^2-e^2 x^4}}{30 d^2 \left (d-e x^2\right )^2}+\frac {2 x \sqrt {d^2-e^2 x^4}}{5 d^3 \left (d-e x^2\right )}-\frac {2 \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{5 d^{3/2} \sqrt {e} \sqrt {d^2-e^2 x^4}}+\frac {17 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{30 d^{3/2} \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:

1/5*x*(-e^2*x^4+d^2)^(1/2)/d/(-e*x^2+d)^3+7/30*x*(-e^2*x^4+d^2)^(1/2)/d^2/ 
(-e*x^2+d)^2+2/5*x*(-e^2*x^4+d^2)^(1/2)/d^3/(-e*x^2+d)-2/5*(1-e^2*x^4/d^2) 
^(1/2)*EllipticE(e^(1/2)*x/d^(1/2),I)/d^(3/2)/e^(1/2)/(-e^2*x^4+d^2)^(1/2) 
+17/30*(1-e^2*x^4/d^2)^(1/2)*EllipticF(e^(1/2)*x/d^(1/2),I)/d^(3/2)/e^(1/2 
)/(-e^2*x^4+d^2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.78 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d-e x^2\right )^4} \, dx=\frac {\sqrt {-\frac {e}{d}} x \left (25 d^3-6 d^2 e x^2-19 d e^2 x^4+12 e^3 x^6\right )+12 i d \left (d-e x^2\right )^2 \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )-17 i d \left (d-e x^2\right )^2 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )}{30 d^3 \sqrt {-\frac {e}{d}} \left (d-e x^2\right )^2 \sqrt {d^2-e^2 x^4}} \] Input:

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

Output:

(Sqrt[-(e/d)]*x*(25*d^3 - 6*d^2*e*x^2 - 19*d*e^2*x^4 + 12*e^3*x^6) + (12*I 
)*d*(d - e*x^2)^2*Sqrt[1 - (e^2*x^4)/d^2]*EllipticE[I*ArcSinh[Sqrt[-(e/d)] 
*x], -1] - (17*I)*d*(d - e*x^2)^2*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[I*ArcS 
inh[Sqrt[-(e/d)]*x], -1])/(30*d^3*Sqrt[-(e/d)]*(d - e*x^2)^2*Sqrt[d^2 - e^ 
2*x^4])
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.26, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {1396, 314, 25, 402, 27, 402, 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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 314

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\int \frac {d e \left (7 e x^2+17 d\right )}{\left (d-e x^2\right )^{3/2} \sqrt {e x^2+d}}dx}{6 d^2 e}+\frac {7 x \sqrt {d+e x^2}}{6 d \left (d-e x^2\right )^{3/2}}}{5 d}+\frac {x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\int \frac {7 e x^2+17 d}{\left (d-e x^2\right )^{3/2} \sqrt {e x^2+d}}dx}{6 d}+\frac {7 x \sqrt {d+e x^2}}{6 d \left (d-e x^2\right )^{3/2}}}{5 d}+\frac {x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\int \frac {2 d e \left (5 d-12 e x^2\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d^2 e}+\frac {12 x \sqrt {d+e x^2}}{d \sqrt {d-e x^2}}}{6 d}+\frac {7 x \sqrt {d+e x^2}}{6 d \left (d-e x^2\right )^{3/2}}}{5 d}+\frac {x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\int \frac {5 d-12 e x^2}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{d}+\frac {12 x \sqrt {d+e x^2}}{d \sqrt {d-e x^2}}}{6 d}+\frac {7 x \sqrt {d+e x^2}}{6 d \left (d-e x^2\right )^{3/2}}}{5 d}+\frac {x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {17 d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx-12 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx}{d}+\frac {12 x \sqrt {d+e x^2}}{d \sqrt {d-e x^2}}}{6 d}+\frac {7 x \sqrt {d+e x^2}}{6 d \left (d-e x^2\right )^{3/2}}}{5 d}+\frac {x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 289

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {17 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-12 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx}{d}+\frac {12 x \sqrt {d+e x^2}}{d \sqrt {d-e x^2}}}{6 d}+\frac {7 x \sqrt {d+e x^2}}{6 d \left (d-e x^2\right )^{3/2}}}{5 d}+\frac {x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 329

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {17 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {12 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {12 x \sqrt {d+e x^2}}{d \sqrt {d-e x^2}}}{6 d}+\frac {7 x \sqrt {d+e x^2}}{6 d \left (d-e x^2\right )^{3/2}}}{5 d}+\frac {x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {17 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {12 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {12 x \sqrt {d+e x^2}}{d \sqrt {d-e x^2}}}{6 d}+\frac {7 x \sqrt {d+e x^2}}{6 d \left (d-e x^2\right )^{3/2}}}{5 d}+\frac {x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {17 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {12 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {12 x \sqrt {d+e x^2}}{d \sqrt {d-e x^2}}}{6 d}+\frac {7 x \sqrt {d+e x^2}}{6 d \left (d-e x^2\right )^{3/2}}}{5 d}+\frac {x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {17 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {12 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {12 x \sqrt {d+e x^2}}{d \sqrt {d-e x^2}}}{6 d}+\frac {7 x \sqrt {d+e x^2}}{6 d \left (d-e x^2\right )^{3/2}}}{5 d}+\frac {x \sqrt {d+e x^2}}{5 d \left (d-e x^2\right )^{5/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

Input:

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

Output:

(Sqrt[d^2 - e^2*x^4]*((x*Sqrt[d + e*x^2])/(5*d*(d - e*x^2)^(5/2)) + ((7*x* 
Sqrt[d + e*x^2])/(6*d*(d - e*x^2)^(3/2)) + ((12*x*Sqrt[d + e*x^2])/(d*Sqrt 
[d - e*x^2]) + ((-12*d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticE[ArcSin[(Sqr 
t[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x^2]) + (17*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]))/d)/(6*d))/(5*d)))/(Sqrt[d - e*x^ 
2]*Sqrt[d + e*x^2])
 

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 314

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] + Simp[1/(2*a* 
(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*p + 3) + d 
*(2*(p + q + 1) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && LtQ[p, -1] && LtQ[0, 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 = 2.98 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.15

method	result	size
default	\(-\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{5 d \,e^{3} \left (x^{2}-\frac {d}{e}\right )^{3}}+\frac {7 x \sqrt {-e^{2} x^{4}+d^{2}}}{30 d^{2} e^{2} \left (x^{2}-\frac {d}{e}\right )^{2}}-\frac {2 \left (-e^{2} x^{2}-d e \right ) x}{5 e \,d^{3} \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{6 d^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {2 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{5 d^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(261\)
elliptic	\(-\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{5 d \,e^{3} \left (x^{2}-\frac {d}{e}\right )^{3}}+\frac {7 x \sqrt {-e^{2} x^{4}+d^{2}}}{30 d^{2} e^{2} \left (x^{2}-\frac {d}{e}\right )^{2}}-\frac {2 \left (-e^{2} x^{2}-d e \right ) x}{5 e \,d^{3} \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{6 d^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {2 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{5 d^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(261\)

Input:

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

Output:

-1/5/d*x/e^3*(-e^2*x^4+d^2)^(1/2)/(x^2-d/e)^3+7/30/d^2*x/e^2*(-e^2*x^4+d^2 
)^(1/2)/(x^2-d/e)^2-2/5*(-e^2*x^2-d*e)/e/d^3*x/((x^2-d/e)*(-e^2*x^2-d*e))^ 
(1/2)+1/6/d^2/(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)+2/5/d^2/(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)-Ellipti 
cE(x*(e/d)^(1/2),I))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d-e x^2\right )^4} \, dx=-\frac {12 \, {\left (e^{4} x^{6} - 3 \, d e^{3} x^{4} + 3 \, d^{2} e^{2} x^{2} - d^{3} e\right )} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) - {\left ({\left (5 \, d e^{3} + 12 \, e^{4}\right )} x^{6} - 3 \, {\left (5 \, d^{2} e^{2} + 12 \, d e^{3}\right )} x^{4} - 5 \, d^{4} - 12 \, d^{3} e + 3 \, {\left (5 \, d^{3} e + 12 \, d^{2} e^{2}\right )} x^{2}\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left (12 \, e^{3} x^{5} - 31 \, d e^{2} x^{3} + 25 \, d^{2} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{30 \, {\left (d^{3} e^{4} x^{6} - 3 \, d^{4} e^{3} x^{4} + 3 \, d^{5} e^{2} x^{2} - d^{6} e\right )}} \] Input:

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

Output:

-1/30*(12*(e^4*x^6 - 3*d*e^3*x^4 + 3*d^2*e^2*x^2 - d^3*e)*sqrt(e/d)*ellipt 
ic_e(arcsin(x*sqrt(e/d)), -1) - ((5*d*e^3 + 12*e^4)*x^6 - 3*(5*d^2*e^2 + 1 
2*d*e^3)*x^4 - 5*d^4 - 12*d^3*e + 3*(5*d^3*e + 12*d^2*e^2)*x^2)*sqrt(e/d)* 
elliptic_f(arcsin(x*sqrt(e/d)), -1) + (12*e^3*x^5 - 31*d*e^2*x^3 + 25*d^2* 
e*x)*sqrt(-e^2*x^4 + d^2))/(d^3*e^4*x^6 - 3*d^4*e^3*x^4 + 3*d^5*e^2*x^2 - 
d^6*e)
 

Sympy [F]

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

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

Output:

Integral(sqrt(-(-d + e*x**2)*(d + e*x**2))/(-d + e*x**2)**4, x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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