\(\int \frac {\sqrt {d^2-e^2 x^4}}{(d-e x^2)^5} \, dx\) [92]

Optimal result

Integrand size = 27, antiderivative size = 261 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d-e x^2\right )^5} \, dx=\frac {x \sqrt {d^2-e^2 x^4}}{7 d \left (d-e x^2\right )^4}+\frac {11 x \sqrt {d^2-e^2 x^4}}{70 d^2 \left (d-e x^2\right )^3}+\frac {41 x \sqrt {d^2-e^2 x^4}}{210 d^3 \left (d-e x^2\right )^2}+\frac {7 x \sqrt {d^2-e^2 x^4}}{20 d^4 \left (d-e x^2\right )}-\frac {7 \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{20 d^{5/2} \sqrt {e} \sqrt {d^2-e^2 x^4}}+\frac {53 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{105 d^{5/2} \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:

1/7*x*(-e^2*x^4+d^2)^(1/2)/d/(-e*x^2+d)^4+11/70*x*(-e^2*x^4+d^2)^(1/2)/d^2 
/(-e*x^2+d)^3+41/210*x*(-e^2*x^4+d^2)^(1/2)/d^3/(-e*x^2+d)^2+7/20*x*(-e^2* 
x^4+d^2)^(1/2)/d^4/(-e*x^2+d)-7/20*(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)+53/105*(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 = 10.87 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d-e x^2\right )^5} \, dx=\frac {-\frac {x \left (d+e x^2\right ) \left (-355 d^3+671 d^2 e x^2-523 d e^2 x^4+147 e^3 x^6\right )}{\left (d-e x^2\right )^3}+\frac {i d \sqrt {1-\frac {e^2 x^4}{d^2}} \left (147 E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )-212 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )\right )}{\sqrt {-\frac {e}{d}}}}{420 d^4 \sqrt {d^2-e^2 x^4}} \] Input:

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

Output:

(-((x*(d + e*x^2)*(-355*d^3 + 671*d^2*e*x^2 - 523*d*e^2*x^4 + 147*e^3*x^6) 
)/(d - e*x^2)^3) + (I*d*Sqrt[1 - (e^2*x^4)/d^2]*(147*EllipticE[I*ArcSinh[S 
qrt[-(e/d)]*x], -1] - 212*EllipticF[I*ArcSinh[Sqrt[-(e/d)]*x], -1]))/Sqrt[ 
-(e/d)])/(420*d^4*Sqrt[d^2 - e^2*x^4])
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.27, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {1396, 314, 25, 402, 27, 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.

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

\(\Big \downarrow \) 1396

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \int \frac {\sqrt {e x^2+d}}{\left (d-e x^2\right )^{9/2}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 314

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\int \frac {d e \left (33 e x^2+49 d\right )}{\left (d-e x^2\right )^{5/2} \sqrt {e x^2+d}}dx}{10 d^2 e}+\frac {11 x \sqrt {d+e x^2}}{10 d \left (d-e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d+e x^2}}{7 d \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\int \frac {33 e x^2+49 d}{\left (d-e x^2\right )^{5/2} \sqrt {e x^2+d}}dx}{10 d}+\frac {11 x \sqrt {d+e x^2}}{10 d \left (d-e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d+e x^2}}{7 d \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\int \frac {2 d e \left (41 e x^2+106 d\right )}{\left (d-e x^2\right )^{3/2} \sqrt {e x^2+d}}dx}{6 d^2 e}+\frac {41 x \sqrt {d+e x^2}}{3 d \left (d-e x^2\right )^{3/2}}}{10 d}+\frac {11 x \sqrt {d+e x^2}}{10 d \left (d-e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d+e x^2}}{7 d \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\int \frac {41 e x^2+106 d}{\left (d-e x^2\right )^{3/2} \sqrt {e x^2+d}}dx}{3 d}+\frac {41 x \sqrt {d+e x^2}}{3 d \left (d-e x^2\right )^{3/2}}}{10 d}+\frac {11 x \sqrt {d+e x^2}}{10 d \left (d-e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d+e x^2}}{7 d \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {\int \frac {d e \left (65 d-147 e x^2\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d^2 e}+\frac {147 x \sqrt {d+e x^2}}{2 d \sqrt {d-e x^2}}}{3 d}+\frac {41 x \sqrt {d+e x^2}}{3 d \left (d-e x^2\right )^{3/2}}}{10 d}+\frac {11 x \sqrt {d+e x^2}}{10 d \left (d-e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d+e x^2}}{7 d \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {\int \frac {65 d-147 e x^2}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d}+\frac {147 x \sqrt {d+e x^2}}{2 d \sqrt {d-e x^2}}}{3 d}+\frac {41 x \sqrt {d+e x^2}}{3 d \left (d-e x^2\right )^{3/2}}}{10 d}+\frac {11 x \sqrt {d+e x^2}}{10 d \left (d-e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d+e x^2}}{7 d \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {212 d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx-147 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx}{2 d}+\frac {147 x \sqrt {d+e x^2}}{2 d \sqrt {d-e x^2}}}{3 d}+\frac {41 x \sqrt {d+e x^2}}{3 d \left (d-e x^2\right )^{3/2}}}{10 d}+\frac {11 x \sqrt {d+e x^2}}{10 d \left (d-e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d+e x^2}}{7 d \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 289

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {\frac {212 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-147 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx}{2 d}+\frac {147 x \sqrt {d+e x^2}}{2 d \sqrt {d-e x^2}}}{3 d}+\frac {41 x \sqrt {d+e x^2}}{3 d \left (d-e x^2\right )^{3/2}}}{10 d}+\frac {11 x \sqrt {d+e x^2}}{10 d \left (d-e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d+e x^2}}{7 d \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 329

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {\frac {212 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {147 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {147 x \sqrt {d+e x^2}}{2 d \sqrt {d-e x^2}}}{3 d}+\frac {41 x \sqrt {d+e x^2}}{3 d \left (d-e x^2\right )^{3/2}}}{10 d}+\frac {11 x \sqrt {d+e x^2}}{10 d \left (d-e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d+e x^2}}{7 d \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {\frac {212 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {147 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {147 x \sqrt {d+e x^2}}{2 d \sqrt {d-e x^2}}}{3 d}+\frac {41 x \sqrt {d+e x^2}}{3 d \left (d-e x^2\right )^{3/2}}}{10 d}+\frac {11 x \sqrt {d+e x^2}}{10 d \left (d-e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d+e x^2}}{7 d \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {\frac {212 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {147 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {147 x \sqrt {d+e x^2}}{2 d \sqrt {d-e x^2}}}{3 d}+\frac {41 x \sqrt {d+e x^2}}{3 d \left (d-e x^2\right )^{3/2}}}{10 d}+\frac {11 x \sqrt {d+e x^2}}{10 d \left (d-e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d+e x^2}}{7 d \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\frac {\frac {\frac {\frac {212 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {147 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {147 x \sqrt {d+e x^2}}{2 d \sqrt {d-e x^2}}}{3 d}+\frac {41 x \sqrt {d+e x^2}}{3 d \left (d-e x^2\right )^{3/2}}}{10 d}+\frac {11 x \sqrt {d+e x^2}}{10 d \left (d-e x^2\right )^{5/2}}}{7 d}+\frac {x \sqrt {d+e x^2}}{7 d \left (d-e x^2\right )^{7/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

Input:

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

Output:

(Sqrt[d^2 - e^2*x^4]*((x*Sqrt[d + e*x^2])/(7*d*(d - e*x^2)^(7/2)) + ((11*x 
*Sqrt[d + e*x^2])/(10*d*(d - e*x^2)^(5/2)) + ((41*x*Sqrt[d + e*x^2])/(3*d* 
(d - e*x^2)^(3/2)) + ((147*x*Sqrt[d + e*x^2])/(2*d*Sqrt[d - e*x^2]) + ((-1 
47*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]) + (212*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))/(10*d))/(7*d)))/(Sqrt[d - e*x^2]*Sq 
rt[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 = 3.90 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.13

method	result	size
default	\(\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{7 d \,e^{4} \left (x^{2}-\frac {d}{e}\right )^{4}}-\frac {11 x \sqrt {-e^{2} x^{4}+d^{2}}}{70 d^{2} e^{3} \left (x^{2}-\frac {d}{e}\right )^{3}}+\frac {41 x \sqrt {-e^{2} x^{4}+d^{2}}}{210 e^{2} d^{3} \left (x^{2}-\frac {d}{e}\right )^{2}}-\frac {7 \left (-e^{2} x^{2}-d e \right ) x}{20 d^{4} e \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {13 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{84 d^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {7 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{20 d^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(296\)
elliptic	\(\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{7 d \,e^{4} \left (x^{2}-\frac {d}{e}\right )^{4}}-\frac {11 x \sqrt {-e^{2} x^{4}+d^{2}}}{70 d^{2} e^{3} \left (x^{2}-\frac {d}{e}\right )^{3}}+\frac {41 x \sqrt {-e^{2} x^{4}+d^{2}}}{210 e^{2} d^{3} \left (x^{2}-\frac {d}{e}\right )^{2}}-\frac {7 \left (-e^{2} x^{2}-d e \right ) x}{20 d^{4} e \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {13 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{84 d^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {7 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{20 d^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(296\)

Input:

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

Output:

1/7/d*x/e^4*(-e^2*x^4+d^2)^(1/2)/(x^2-d/e)^4-11/70/d^2/e^3*x*(-e^2*x^4+d^2 
)^(1/2)/(x^2-d/e)^3+41/210/e^2/d^3*x*(-e^2*x^4+d^2)^(1/2)/(x^2-d/e)^2-7/20 
*(-e^2*x^2-d*e)/d^4*x/e/((x^2-d/e)*(-e^2*x^2-d*e))^(1/2)+13/84/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)+7/20/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 = 281, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d-e x^2\right )^5} \, dx=-\frac {147 \, {\left (e^{5} x^{8} - 4 \, d e^{4} x^{6} + 6 \, d^{2} e^{3} x^{4} - 4 \, d^{3} e^{2} x^{2} + d^{4} e\right )} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) - {\left ({\left (65 \, d e^{4} + 147 \, e^{5}\right )} x^{8} - 4 \, {\left (65 \, d^{2} e^{3} + 147 \, d e^{4}\right )} x^{6} + 65 \, d^{5} + 147 \, d^{4} e + 6 \, {\left (65 \, d^{3} e^{2} + 147 \, d^{2} e^{3}\right )} x^{4} - 4 \, {\left (65 \, d^{4} e + 147 \, d^{3} e^{2}\right )} x^{2}\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left (147 \, e^{4} x^{7} - 523 \, d e^{3} x^{5} + 671 \, d^{2} e^{2} x^{3} - 355 \, d^{3} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{420 \, {\left (d^{4} e^{5} x^{8} - 4 \, d^{5} e^{4} x^{6} + 6 \, d^{6} e^{3} x^{4} - 4 \, d^{7} e^{2} x^{2} + d^{8} e\right )}} \] Input:

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

Output:

-1/420*(147*(e^5*x^8 - 4*d*e^4*x^6 + 6*d^2*e^3*x^4 - 4*d^3*e^2*x^2 + d^4*e 
)*sqrt(e/d)*elliptic_e(arcsin(x*sqrt(e/d)), -1) - ((65*d*e^4 + 147*e^5)*x^ 
8 - 4*(65*d^2*e^3 + 147*d*e^4)*x^6 + 65*d^5 + 147*d^4*e + 6*(65*d^3*e^2 + 
147*d^2*e^3)*x^4 - 4*(65*d^4*e + 147*d^3*e^2)*x^2)*sqrt(e/d)*elliptic_f(ar 
csin(x*sqrt(e/d)), -1) + (147*e^4*x^7 - 523*d*e^3*x^5 + 671*d^2*e^2*x^3 - 
355*d^3*e*x)*sqrt(-e^2*x^4 + d^2))/(d^4*e^5*x^8 - 4*d^5*e^4*x^6 + 6*d^6*e^ 
3*x^4 - 4*d^7*e^2*x^2 + d^8*e)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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