\(\int \frac {1}{(d+e x^2)^2 \sqrt {a-c x^4}} \, dx\) [410]

Optimal result

Integrand size = 22, antiderivative size = 299 \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a-c x^4}} \, dx=-\frac {e^2 x \sqrt {a-c x^4}}{2 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )}-\frac {a^{3/4} \sqrt [4]{c} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 d \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 d \left (\sqrt {c} d+\sqrt {a} e\right ) \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (3 c d^2-a e^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{c} d^2 \left (c d^2-a e^2\right ) \sqrt {a-c x^4}} \] Output:

-1/2*e^2*x*(-c*x^4+a)^(1/2)/d/(-a*e^2+c*d^2)/(e*x^2+d)-1/2*a^(3/4)*c^(1/4) 
*e*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/d/(-a*e^2+c*d^2)/(-c*x 
^4+a)^(1/2)-1/2*a^(1/4)*c^(1/4)*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1 
/4),I)/d/(c^(1/2)*d+a^(1/2)*e)/(-c*x^4+a)^(1/2)+1/2*a^(1/4)*(-a*e^2+3*c*d^ 
2)*(1-c*x^4/a)^(1/2)*EllipticPi(c^(1/4)*x/a^(1/4),-a^(1/2)*e/c^(1/2)/d,I)/ 
c^(1/4)/d^2/(-a*e^2+c*d^2)/(-c*x^4+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.73 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.70 \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a-c x^4}} \, dx=\frac {-a \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d e^2 x+\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d e^2 x^5+i \sqrt {a} \sqrt {c} d e \left (d+e x^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-i \sqrt {c} d \left (-\sqrt {c} d+\sqrt {a} e\right ) \left (d+e x^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-3 i c d^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+i a d e^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-3 i c d^2 e x^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+i a e^3 x^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{2 \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d^2 \left (c d^2-a e^2\right ) \left (d+e x^2\right ) \sqrt {a-c x^4}} \] Input:

Integrate[1/((d + e*x^2)^2*Sqrt[a - c*x^4]),x]
 

Output:

(-(a*Sqrt[-(Sqrt[c]/Sqrt[a])]*d*e^2*x) + Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d*e^2* 
x^5 + I*Sqrt[a]*Sqrt[c]*d*e*(d + e*x^2)*Sqrt[1 - (c*x^4)/a]*EllipticE[I*Ar 
cSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - I*Sqrt[c]*d*(-(Sqrt[c]*d) + Sqrt[ 
a]*e)*(d + e*x^2)*Sqrt[1 - (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/S 
qrt[a])]*x], -1] - (3*I)*c*d^3*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e 
)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + I*a*d*e^2*Sqr 
t[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(S 
qrt[c]/Sqrt[a])]*x], -1] - (3*I)*c*d^2*e*x^2*Sqrt[1 - (c*x^4)/a]*EllipticP 
i[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + 
 I*a*e^3*x^2*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I* 
ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(2*Sqrt[-(Sqrt[c]/Sqrt[a])]*d^2* 
(c*d^2 - a*e^2)*(d + e*x^2)*Sqrt[a - c*x^4])
 

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.94, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {1552, 2235, 27, 1513, 27, 765, 762, 1390, 1389, 327, 1543, 1542}

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.

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

\(\Big \downarrow \) 1552

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

\(\Big \downarrow \) 2235

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1513

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

\(\displaystyle \frac {\left (3 c d^2-a e^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1390

\(\displaystyle \frac {\left (3 c d^2-a e^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {e \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1389

\(\displaystyle \frac {\left (3 c d^2-a e^2\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {\sqrt {a} e \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {\frac {\sqrt {c} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 1543

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

\(\Big \downarrow \) 1542

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

Input:

Int[1/((d + e*x^2)^2*Sqrt[a - c*x^4]),x]
 

Output:

-1/2*(e^2*x*Sqrt[a - c*x^4])/(d*(c*d^2 - a*e^2)*(d + e*x^2)) + (-(c*((a^(3 
/4)*e*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^( 
3/4)*Sqrt[a - c*x^4]) + (a^(1/4)*(d - (Sqrt[a]*e)/Sqrt[c])*Sqrt[1 - (c*x^4 
)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*Sqrt[a - c*x^4]) 
)) + (a^(1/4)*(3*c*d^2 - a*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]* 
e)/(Sqrt[c]*d)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*d*Sqrt[a - c*x 
^4]))/(2*d*(c*d^2 - a*e^2))
 

Defintions of rubi rules used

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 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 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 1389

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq 
rt[a]   Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, 
 d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
 

rule 1390

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt 
[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], 
x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] &&  !GtQ 
[a, 0] &&  !(LtQ[a, 0] && GtQ[c, 0])
 

rule 1513

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[-c/a, 2]}, Simp[(d*q - e)/q   Int[1/Sqrt[a + c*x^4], x], x] + Simp[e/q 
  Int[(1 + q*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && Neg 
Q[c/a] && NeQ[c*d^2 + a*e^2, 0]
 

rule 1542

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ 
{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x 
], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
 

rule 1543

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ 
Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) 
]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 0]
 

rule 1552

Int[((d_) + (e_.)*(x_)^2)^(q_)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp 
[(-e^2)*x*(d + e*x^2)^(q + 1)*(Sqrt[a + c*x^4]/(2*d*(q + 1)*(c*d^2 + a*e^2) 
)), x] + Simp[1/(2*d*(q + 1)*(c*d^2 + a*e^2))   Int[((d + e*x^2)^(q + 1)/Sq 
rt[a + c*x^4])*Simp[a*e^2*(2*q + 3) + 2*c*d^2*(q + 1) - 2*e*c*d*(q + 1)*x^2 
 + c*e^2*(2*q + 5)*x^4, x], x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + 
 a*e^2, 0] && ILtQ[q, -1]
 

rule 2235

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> 
With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, Si 
mp[-(e^2)^(-1)   Int[(C*d - B*e - C*e*x^2)/Sqrt[a + c*x^4], x], x] + Simp[( 
C*d^2 - B*d*e + A*e^2)/e^2   Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x], x]] / 
; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 - a*e^2, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 522 vs. \(2 (245 ) = 490\).

Time = 0.65 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.75

method	result	size
default	\(\frac {e^{2} x \sqrt {-c \,x^{4}+a}}{2 d \left (a \,e^{2}-c \,d^{2}\right ) \left (e \,x^{2}+d \right )}+\frac {c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\sqrt {c}\, e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \left (a \,e^{2}-c \,d^{2}\right ) d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\sqrt {c}\, e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \left (a \,e^{2}-c \,d^{2}\right ) d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {e^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) a}{2 \left (a \,e^{2}-c \,d^{2}\right ) d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) c}{2 \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\)	\(523\)
elliptic	\(\frac {e^{2} x \sqrt {-c \,x^{4}+a}}{2 d \left (a \,e^{2}-c \,d^{2}\right ) \left (e \,x^{2}+d \right )}+\frac {c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\sqrt {c}\, e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \left (a \,e^{2}-c \,d^{2}\right ) d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\sqrt {c}\, e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 \left (a \,e^{2}-c \,d^{2}\right ) d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {e^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) a}{2 \left (a \,e^{2}-c \,d^{2}\right ) d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) c}{2 \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\)	\(523\)

Input:

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

Output:

1/2*e^2/d/(a*e^2-c*d^2)*x*(-c*x^4+a)^(1/2)/(e*x^2+d)+1/2*c/(a*e^2-c*d^2)/( 
1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1 
/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)-1 
/2*c^(1/2)*e/(a*e^2-c*d^2)/d*a^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2 
)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*Elli 
pticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)+1/2*c^(1/2)*e/(a*e^2-c*d^2)/d*a^(1/2) 
/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^ 
(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticE(x*(1/a^(1/2)*c^(1/2))^(1/2),I) 
+1/2/(a*e^2-c*d^2)/d^2*e^2/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)* 
x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticPi(x*( 
1/a^(1/2)*c^(1/2))^(1/2),-a^(1/2)*e/c^(1/2)/d,(-1/a^(1/2)*c^(1/2))^(1/2)/( 
1/a^(1/2)*c^(1/2))^(1/2))*a-3/2/(a*e^2-c*d^2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1 
-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^( 
1/2)*EllipticPi(x*(1/a^(1/2)*c^(1/2))^(1/2),-a^(1/2)*e/c^(1/2)/d,(-1/a^(1/ 
2)*c^(1/2))^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2))*c
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a-c x^4}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a-c x^4}} \, dx=\int \frac {1}{\sqrt {a - c x^{4}} \left (d + e x^{2}\right )^{2}}\, dx \] Input:

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

Output:

Integral(1/(sqrt(a - c*x**4)*(d + e*x**2)**2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:

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

Output:

integrate(1/(sqrt(-c*x^4 + a)*(e*x^2 + d)^2), x)
 

Giac [F]

\[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:

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

Output:

integrate(1/(sqrt(-c*x^4 + a)*(e*x^2 + d)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a-c x^4}} \, dx=\int \frac {1}{\sqrt {a-c\,x^4}\,{\left (e\,x^2+d\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a-c x^4}} \, dx=\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \] Input:

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

Output:

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