\(\int \frac {(d^2-e^2 x^4)^{3/2}}{(d+e x^2)^5} \, dx\) [51]

Optimal result

Integrand size = 26, antiderivative size = 221 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^5} \, dx=\frac {2 x \sqrt {d^2-e^2 x^4}}{5 \left (d+e x^2\right )^3}+\frac {2 x \sqrt {d^2-e^2 x^4}}{15 d \left (d+e x^2\right )^2}+\frac {3 x \sqrt {d^2-e^2 x^4}}{10 d^2 \left (d+e x^2\right )}+\frac {3 \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{10 \sqrt {d} \sqrt {e} \sqrt {d^2-e^2 x^4}}-\frac {2 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{15 \sqrt {d} \sqrt {e} \sqrt {d^2-e^2 x^4}} \] 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)-2/15 
*(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 = 11.10 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.85 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^5} \, dx=\frac {\sqrt {-\frac {e}{d}} x \left (25 d^3-3 d^2 e x^2-13 d e^2 x^4-9 e^3 x^6\right )-9 i d \left (d+e x^2\right )^2 \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )+4 i d \left (d+e x^2\right )^2 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )}{30 d^2 \sqrt {-\frac {e}{d}} \left (d+e x^2\right )^2 \sqrt {d^2-e^2 x^4}} \] 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] + (4*I)*d*(d + e*x^2)^2*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[I*ArcSinh 
[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 = 0.79 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.29, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {1396, 315, 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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 289

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

\(\Big \downarrow \) 329

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

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[(Sqrt 
[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x^2]) - (4*d^(3/ 
2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sq 
rt[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 = 4.23 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.14

method	result	size
default	\(\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{5 e^{3} \left (x^{2}+\frac {d}{e}\right )^{3}}+\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{15 d \,e^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {3 \left (-e^{2} x^{2}+d e \right ) x}{10 d^{2} e \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{6 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {3 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{10 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(253\)
elliptic	\(\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{5 e^{3} \left (x^{2}+\frac {d}{e}\right )^{3}}+\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{15 d \,e^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {3 \left (-e^{2} x^{2}+d e \right ) x}{10 d^{2} e \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{6 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {3 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{10 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(253\)

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 = 0.09 (sec) , antiderivative size = 225, 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)*elliptic 
_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)*ellipt 
ic_f(arcsin(x*sqrt(e/d)), -1) + (9*e^3*x^5 + 22*d*e^2*x^3 + 25*d^2*e*x)*sq 
rt(-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]

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

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

Output:

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