\(\int \frac {A+B x^2+C x^4}{(d+e x^2)^2 \sqrt {a-c x^4}} \, dx\) [45]

Optimal result

Integrand size = 34, antiderivative size = 401 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^2 \sqrt {a-c x^4}} \, dx=-\frac {\left (C d^2-B d e+A e^2\right ) x \sqrt {a-c x^4}}{2 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )}-\frac {a^{3/4} \sqrt [4]{c} \left (C d^2-B d e+A e^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 d e \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (2 \sqrt {a} C d e+\sqrt {c} \left (C d^2+e (B d-A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{c} d e^2 \left (\sqrt {c} d+\sqrt {a} e\right ) \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \left (c d^2 \left (C d^2+e (B d-3 A e)\right )-a e^2 \left (3 C d^2-e (B d+A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{c} d^2 e^2 \left (c d^2-a e^2\right ) \sqrt {a-c x^4}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 13.48 (sec) , antiderivative size = 1223, normalized size of antiderivative = 3.05 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^2 \sqrt {a-c x^4}} \, dx =\text {Too large to display} \] Input:

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

Output:

(-(a*Sqrt[-(Sqrt[c]/Sqrt[a])]*C*d^3*e^2*x) + a*B*Sqrt[-(Sqrt[c]/Sqrt[a])]* 
d^2*e^3*x - a*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*d*e^4*x + Sqrt[-(Sqrt[c]/Sqrt[a]) 
]*c*C*d^3*e^2*x^5 - B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d^2*e^3*x^5 + A*Sqrt[-(Sq 
rt[c]/Sqrt[a])]*c*d*e^4*x^5 + I*Sqrt[a]*Sqrt[c]*d*e*(C*d^2 + e*(-(B*d) + A 
*e))*(d + e*x^2)*Sqrt[1 - (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sq 
rt[a])]*x], -1] + I*d*(-(Sqrt[c]*d) + Sqrt[a]*e)*(2*Sqrt[a]*C*d*e + Sqrt[c 
]*(C*d^2 + B*d*e - A*e^2))*(d + e*x^2)*Sqrt[1 - (c*x^4)/a]*EllipticF[I*Arc 
Sinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + I*c*C*d^5*Sqrt[1 - (c*x^4)/a]*Elli 
pticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], 
-1] + I*B*c*d^4*e*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d) 
), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - (3*I)*A*c*d^3*e^2*Sqrt[1 - 
 (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c 
]/Sqrt[a])]*x], -1] - (3*I)*a*C*d^3*e^2*Sqrt[1 - (c*x^4)/a]*EllipticPi[-(( 
Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + I*a* 
B*d^2*e^3*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*Arc 
Sinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + I*a*A*d*e^4*Sqrt[1 - (c*x^4)/a]*El 
lipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x] 
, -1] + I*c*C*d^4*e*x^2*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt 
[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + I*B*c*d^3*e^2*x^2*Sq 
rt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt...
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {2211, 2235, 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.

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

\(\Big \downarrow \) 2211

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

\(\Big \downarrow \) 2235

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

\(\Big \downarrow \) 1513

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1390

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

\(\Big \downarrow \) 1389

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 1543

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

\(\Big \downarrow \) 1542

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

Input:

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

Output:

-1/2*((C*d^2 - B*d*e + A*e^2)*x*Sqrt[a - c*x^4])/(d*(c*d^2 - a*e^2)*(d + e 
*x^2)) + (-(((a^(3/4)*c^(1/4)*e*(C*d^2 - B*d*e + A*e^2)*Sqrt[1 - (c*x^4)/a 
]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/Sqrt[a - c*x^4] - (a^(1/4)*( 
Sqrt[c]*d - Sqrt[a]*e)*(2*Sqrt[a]*C*d*e + Sqrt[c]*(C*d^2 + e*(B*d - A*e))) 
*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)* 
Sqrt[a - c*x^4]))/e^2) - (a^(1/4)*(c*(C*d^4 + d^2*e*(B*d - 3*A*e)) - a*e^2 
*(3*C*d^2 - e*(B*d + A*e)))*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*e^2*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 2211

Int[((P4x_)*((d_) + (e_.)*(x_)^2)^(q_))/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol 
] :> With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4] 
}, Simp[(-(C*d^2 - B*d*e + A*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)/Sqrt[a + c*x^4])*Simp[a*d*(C*d - B*e) + A*(a*e^2*( 
2*q + 3) + 2*c*d^2*(q + 1)) + 2*d*(B*c*d - A*c*e + a*C*e)*(q + 1)*x^2 + c*( 
C*d^2 - B*d*e + A*e^2)*(2*q + 5)*x^4, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && PolyQ[P4x, x^2] && LeQ[Expon[P4x, x], 4] && 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. 717 vs. \(2 (343 ) = 686\).

Time = 1.06 (sec) , antiderivative size = 718, normalized size of antiderivative = 1.79

method	result	size
default	\(\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 )}{e^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\left (B e -2 C d \right ) \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 )}{e^{2} d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\left (A \,e^{2}-B d e +C \,d^{2}\right ) \left (\frac {e^{2} x \sqrt {-c \,x^{4}+a}}{2 \left (a \,e^{2}-c \,d^{2}\right ) d \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 {e \sqrt {c}\, \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 {e \sqrt {c}\, \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}}\right )}{e^{2}}\)	\(718\)
elliptic	\(\text {Expression too large to display}\)	\(1580\)

Input:

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

Output:

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

Fricas [F(-1)]

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

integrate((C*x^4+B*x^2+A)/(e*x^2+d)^2/(-c*x^4+a)^(1/2),x, algorithm="frica 
s")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((C*x^4+B*x^2+A)/(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)*a + int((sqrt(a - c*x**4)*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)*c 
 + int((sqrt(a - c*x**4)*x**2)/(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)*b