\(\int \frac {\sqrt {d^2-e^2 x^4}}{(d+e x^2)^3} \, dx\) [40]

Optimal result

Integrand size = 26, antiderivative size = 191 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^3} \, dx=\frac {x \sqrt {d^2-e^2 x^4}}{3 d \left (d+e x^2\right )^2}+\frac {x \sqrt {d^2-e^2 x^4}}{2 d^2 \left (d+e x^2\right )}+\frac {\sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{2 \sqrt {d} \sqrt {e} \sqrt {d^2-e^2 x^4}}-\frac {\sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{3 \sqrt {d} \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:

1/3*x*(-e^2*x^4+d^2)^(1/2)/d/(e*x^2+d)^2+1/2*x*(-e^2*x^4+d^2)^(1/2)/d^2/(e 
*x^2+d)+1/2*(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)-1/3*(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] (verified)

Result contains complex when optimal does not.

Time = 10.68 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^3} \, dx=\frac {\sqrt {-\frac {e}{d}} x \left (5 d^2-2 d e x^2-3 e^2 x^4\right )-3 i d \left (d+e x^2\right ) \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )+2 i d \left (d+e x^2\right ) \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )}{6 d^2 \sqrt {-\frac {e}{d}} \left (d+e x^2\right ) \sqrt {d^2-e^2 x^4}} \] Input:

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

Output:

(Sqrt[-(e/d)]*x*(5*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]) 
/(6*d^2*Sqrt[-(e/d)]*(d + e*x^2)*Sqrt[d^2 - e^2*x^4])
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.32, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {1396, 314, 25, 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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 314

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 289

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

\(\Big \downarrow \) 329

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

Input:

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

Output:

(Sqrt[d^2 - e^2*x^4]*((x*Sqrt[d - e*x^2])/(3*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]*Sq 
rt[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)))/(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 = 2.41 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.16

method	result	size
default	\(\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{3 d \,e^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {\left (-e^{2} x^{2}+d e \right ) x}{2 d^{2} e \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{6 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(222\)
elliptic	\(\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{3 d \,e^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {\left (-e^{2} x^{2}+d e \right ) x}{2 d^{2} e \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{6 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(222\)

Input:

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

Output:

1/3/d*x/e^2*(-e^2*x^4+d^2)^(1/2)/(x^2+d/e)^2+1/2*(-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)-1/2/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 = 166, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^3} \, dx=\frac {3 \, {\left (e^{3} x^{4} + 2 \, d e^{2} x^{2} + d^{2} e\right )} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left ({\left (d e^{2} - 3 \, e^{3}\right )} x^{4} + d^{3} - 3 \, d^{2} e + 2 \, {\left (d^{2} e - 3 \, d e^{2}\right )} x^{2}\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + \sqrt {-e^{2} x^{4} + d^{2}} {\left (3 \, e^{2} x^{3} + 5 \, d e x\right )}}{6 \, {\left (d^{2} e^{3} x^{4} + 2 \, d^{3} e^{2} x^{2} + d^{4} e\right )}} \] Input:

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

Output:

1/6*(3*(e^3*x^4 + 2*d*e^2*x^2 + d^2*e)*sqrt(e/d)*elliptic_e(arcsin(x*sqrt( 
e/d)), -1) + ((d*e^2 - 3*e^3)*x^4 + d^3 - 3*d^2*e + 2*(d^2*e - 3*d*e^2)*x^ 
2)*sqrt(e/d)*elliptic_f(arcsin(x*sqrt(e/d)), -1) + sqrt(-e^2*x^4 + d^2)*(3 
*e^2*x^3 + 5*d*e*x))/(d^2*e^3*x^4 + 2*d^3*e^2*x^2 + d^4*e)
 

Sympy [F]

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

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

Output:

Integral(sqrt(-(-d + e*x**2)*(d + e*x**2))/(d + e*x**2)**3, x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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