\(\int \frac {(d^2-e^2 x^4)^{3/2}}{(d-e x^2)^5} \, dx\) [101]

Optimal result

Integrand size = 27, antiderivative size = 224 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^5} \, dx=\frac {2 x \sqrt {d^2-e^2 x^4}}{5 \left (d-e x^2\right )^3}+\frac {2 x \sqrt {d^2-e^2 x^4}}{15 d \left (d-e x^2\right )^2}+\frac {3 x \sqrt {d^2-e^2 x^4}}{10 d^2 \left (d-e x^2\right )}-\frac {3 \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{10 \sqrt {d} \sqrt {e} \sqrt {d^2-e^2 x^4}}+\frac {7 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{15 \sqrt {d} \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:

2/5*x*(-e^2*x^4+d^2)^(1/2)/(-e*x^2+d)^3+2/15*x*(-e^2*x^4+d^2)^(1/2)/d/(-e* 
x^2+d)^2+3/10*x*(-e^2*x^4+d^2)^(1/2)/d^2/(-e*x^2+d)-3/10*(1-e^2*x^4/d^2)^( 
1/2)*EllipticE(e^(1/2)*x/d^(1/2),I)/d^(1/2)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)+7 
/15*(1-e^2*x^4/d^2)^(1/2)*EllipticF(e^(1/2)*x/d^(1/2),I)/d^(1/2)/e^(1/2)/( 
-e^2*x^4+d^2)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.93 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.85 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^5} \, dx=\frac {\sqrt {-\frac {e}{d}} x \left (25 d^3+3 d^2 e x^2-13 d e^2 x^4+9 e^3 x^6\right )+9 i d \left (d-e x^2\right )^2 \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )-14 i d \left (d-e x^2\right )^2 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )}{30 d^2 \sqrt {-\frac {e}{d}} \left (d-e x^2\right )^2 \sqrt {d^2-e^2 x^4}} \] Input:

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

Output:

(Sqrt[-(e/d)]*x*(25*d^3 + 3*d^2*e*x^2 - 13*d*e^2*x^4 + 9*e^3*x^6) + (9*I)* 
d*(d - e*x^2)^2*Sqrt[1 - (e^2*x^4)/d^2]*EllipticE[I*ArcSinh[Sqrt[-(e/d)]*x 
], -1] - (14*I)*d*(d - e*x^2)^2*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[I*ArcSin 
h[Sqrt[-(e/d)]*x], -1])/(30*d^2*Sqrt[-(e/d)]*(d - e*x^2)^2*Sqrt[d^2 - e^2* 
x^4])
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.28, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {1396, 315, 25, 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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{5} \left (\frac {\frac {\int \frac {d e \left (5 d-9 e x^2\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d^2 e}+\frac {9 x \sqrt {d+e x^2}}{2 d \sqrt {d-e x^2}}}{3 d}+\frac {2 x \sqrt {d+e x^2}}{3 d \left (d-e x^2\right )^{3/2}}\right )+\frac {2 x \sqrt {d+e x^2}}{5 \left (d-e x^2\right )^{5/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{5} \left (\frac {\frac {\int \frac {5 d-9 e x^2}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d}+\frac {9 x \sqrt {d+e x^2}}{2 d \sqrt {d-e x^2}}}{3 d}+\frac {2 x \sqrt {d+e x^2}}{3 d \left (d-e x^2\right )^{3/2}}\right )+\frac {2 x \sqrt {d+e x^2}}{5 \left (d-e x^2\right )^{5/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{5} \left (\frac {\frac {14 d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx-9 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx}{2 d}+\frac {9 x \sqrt {d+e x^2}}{2 d \sqrt {d-e x^2}}}{3 d}+\frac {2 x \sqrt {d+e x^2}}{3 d \left (d-e x^2\right )^{3/2}}\right )+\frac {2 x \sqrt {d+e x^2}}{5 \left (d-e x^2\right )^{5/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 289

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{5} \left (\frac {\frac {\frac {14 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-9 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx}{2 d}+\frac {9 x \sqrt {d+e x^2}}{2 d \sqrt {d-e x^2}}}{3 d}+\frac {2 x \sqrt {d+e x^2}}{3 d \left (d-e x^2\right )^{3/2}}\right )+\frac {2 x \sqrt {d+e x^2}}{5 \left (d-e x^2\right )^{5/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 329

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{5} \left (\frac {\frac {\frac {14 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {9 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {9 x \sqrt {d+e x^2}}{2 d \sqrt {d-e x^2}}}{3 d}+\frac {2 x \sqrt {d+e x^2}}{3 d \left (d-e x^2\right )^{3/2}}\right )+\frac {2 x \sqrt {d+e x^2}}{5 \left (d-e x^2\right )^{5/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{5} \left (\frac {\frac {\frac {14 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {9 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {9 x \sqrt {d+e x^2}}{2 d \sqrt {d-e x^2}}}{3 d}+\frac {2 x \sqrt {d+e x^2}}{3 d \left (d-e x^2\right )^{3/2}}\right )+\frac {2 x \sqrt {d+e x^2}}{5 \left (d-e x^2\right )^{5/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{5} \left (\frac {\frac {\frac {14 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {9 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {9 x \sqrt {d+e x^2}}{2 d \sqrt {d-e x^2}}}{3 d}+\frac {2 x \sqrt {d+e x^2}}{3 d \left (d-e x^2\right )^{3/2}}\right )+\frac {2 x \sqrt {d+e x^2}}{5 \left (d-e x^2\right )^{5/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{5} \left (\frac {\frac {\frac {14 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {9 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {9 x \sqrt {d+e x^2}}{2 d \sqrt {d-e x^2}}}{3 d}+\frac {2 x \sqrt {d+e x^2}}{3 d \left (d-e x^2\right )^{3/2}}\right )+\frac {2 x \sqrt {d+e x^2}}{5 \left (d-e x^2\right )^{5/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

Input:

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

Output:

(Sqrt[d^2 - e^2*x^4]*((2*x*Sqrt[d + e*x^2])/(5*(d - e*x^2)^(5/2)) + ((2*x* 
Sqrt[d + e*x^2])/(3*d*(d - e*x^2)^(3/2)) + ((9*x*Sqrt[d + e*x^2])/(2*d*Sqr 
t[d - e*x^2]) + ((-9*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]) + (14*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))/5))/(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 315

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), 
x] - Simp[1/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S 
imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) 
*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 
1] && GtQ[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 = 4.10 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.15

method	result	size
default	\(-\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{5 e^{3} \left (x^{2}-\frac {d}{e}\right )^{3}}+\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{15 d \,e^{2} \left (x^{2}-\frac {d}{e}\right )^{2}}-\frac {3 \left (-e^{2} x^{2}-d e \right ) x}{10 d^{2} e \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{6 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {3 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{10 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(258\)
elliptic	\(-\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{5 e^{3} \left (x^{2}-\frac {d}{e}\right )^{3}}+\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{15 d \,e^{2} \left (x^{2}-\frac {d}{e}\right )^{2}}-\frac {3 \left (-e^{2} x^{2}-d e \right ) x}{10 d^{2} e \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{6 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {3 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{10 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(258\)

Input:

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

Output:

-2/5*x/e^3*(-e^2*x^4+d^2)^(1/2)/(x^2-d/e)^3+2/15/d*x/e^2*(-e^2*x^4+d^2)^(1 
/2)/(x^2-d/e)^2-3/10*(-e^2*x^2-d*e)/d^2*x/e/((x^2-d/e)*(-e^2*x^2-d*e))^(1/ 
2)+1/6/d/(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)+3/10/d/(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 = 228, normalized size of antiderivative = 1.02 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^5} \, dx=-\frac {9 \, {\left (e^{4} x^{6} - 3 \, d e^{3} x^{4} + 3 \, d^{2} e^{2} x^{2} - d^{3} e\right )} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) - {\left ({\left (5 \, d e^{3} + 9 \, e^{4}\right )} x^{6} - 3 \, {\left (5 \, d^{2} e^{2} + 9 \, d e^{3}\right )} x^{4} - 5 \, d^{4} - 9 \, d^{3} e + 3 \, {\left (5 \, d^{3} e + 9 \, d^{2} e^{2}\right )} x^{2}\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left (9 \, e^{3} x^{5} - 22 \, d e^{2} x^{3} + 25 \, d^{2} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{30 \, {\left (d^{2} e^{4} x^{6} - 3 \, d^{3} e^{3} x^{4} + 3 \, d^{4} e^{2} x^{2} - d^{5} e\right )}} \] Input:

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

Output:

-1/30*(9*(e^4*x^6 - 3*d*e^3*x^4 + 3*d^2*e^2*x^2 - d^3*e)*sqrt(e/d)*ellipti 
c_e(arcsin(x*sqrt(e/d)), -1) - ((5*d*e^3 + 9*e^4)*x^6 - 3*(5*d^2*e^2 + 9*d 
*e^3)*x^4 - 5*d^4 - 9*d^3*e + 3*(5*d^3*e + 9*d^2*e^2)*x^2)*sqrt(e/d)*ellip 
tic_f(arcsin(x*sqrt(e/d)), -1) + (9*e^3*x^5 - 22*d*e^2*x^3 + 25*d^2*e*x)*s 
qrt(-e^2*x^4 + d^2))/(d^2*e^4*x^6 - 3*d^3*e^3*x^4 + 3*d^4*e^2*x^2 - d^5*e)
 

Sympy [F]

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

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

Output:

-Integral(d**2*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) - Integral(-e 
**2*x**4*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]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d-e x^2\right )^5} \, dx =\text {Too large to display} \] Input:

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

Output:

(5*sqrt(d**2 - e**2*x**4)*x + int(sqrt(d**2 - e**2*x**4)/(d**5 - 3*d**4*e* 
x**2 + 2*d**3*e**2*x**4 + 2*d**2*e**3*x**6 - 3*d*e**4*x**8 + e**5*x**10),x 
)*d**5 - 3*int(sqrt(d**2 - e**2*x**4)/(d**5 - 3*d**4*e*x**2 + 2*d**3*e**2* 
x**4 + 2*d**2*e**3*x**6 - 3*d*e**4*x**8 + e**5*x**10),x)*d**4*e*x**2 + 3*i 
nt(sqrt(d**2 - e**2*x**4)/(d**5 - 3*d**4*e*x**2 + 2*d**3*e**2*x**4 + 2*d** 
2*e**3*x**6 - 3*d*e**4*x**8 + e**5*x**10),x)*d**3*e**2*x**4 - int(sqrt(d** 
2 - e**2*x**4)/(d**5 - 3*d**4*e*x**2 + 2*d**3*e**2*x**4 + 2*d**2*e**3*x**6 
 - 3*d*e**4*x**8 + e**5*x**10),x)*d**2*e**3*x**6 - 9*int((sqrt(d**2 - e**2 
*x**4)*x**4)/(d**5 - 3*d**4*e*x**2 + 2*d**3*e**2*x**4 + 2*d**2*e**3*x**6 - 
 3*d*e**4*x**8 + e**5*x**10),x)*d**3*e**2 + 27*int((sqrt(d**2 - e**2*x**4) 
*x**4)/(d**5 - 3*d**4*e*x**2 + 2*d**3*e**2*x**4 + 2*d**2*e**3*x**6 - 3*d*e 
**4*x**8 + e**5*x**10),x)*d**2*e**3*x**2 - 27*int((sqrt(d**2 - e**2*x**4)* 
x**4)/(d**5 - 3*d**4*e*x**2 + 2*d**3*e**2*x**4 + 2*d**2*e**3*x**6 - 3*d*e* 
*4*x**8 + e**5*x**10),x)*d*e**4*x**4 + 9*int((sqrt(d**2 - e**2*x**4)*x**4) 
/(d**5 - 3*d**4*e*x**2 + 2*d**3*e**2*x**4 + 2*d**2*e**3*x**6 - 3*d*e**4*x* 
*8 + e**5*x**10),x)*e**5*x**6 - 18*int((sqrt(d**2 - e**2*x**4)*x**2)/(d**5 
 - 3*d**4*e*x**2 + 2*d**3*e**2*x**4 + 2*d**2*e**3*x**6 - 3*d*e**4*x**8 + e 
**5*x**10),x)*d**4*e + 54*int((sqrt(d**2 - e**2*x**4)*x**2)/(d**5 - 3*d**4 
*e*x**2 + 2*d**3*e**2*x**4 + 2*d**2*e**3*x**6 - 3*d*e**4*x**8 + e**5*x**10 
),x)*d**3*e**2*x**2 - 54*int((sqrt(d**2 - e**2*x**4)*x**2)/(d**5 - 3*d*...