\(\int \frac {A+B x^2+C x^4}{x^2 (c+d x^2) \sqrt {a+c x^4}} \, dx\) [31]

Optimal result

Integrand size = 36, antiderivative size = 534 \[ \int \frac {A+B x^2+C x^4}{x^2 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=-\frac {A \sqrt {a+c x^4}}{a c x}+\frac {A x \sqrt {a+c x^4}}{a \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {\sqrt {c^3+a d^2} x}{\sqrt {c} \sqrt {d} \sqrt {a+c x^4}}\right )}{2 c^{3/2} \sqrt {d} \sqrt {c^3+a d^2}}-\frac {A \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} c^{3/4} \sqrt {a+c x^4}}+\frac {\left (A c^{3/2}-a \sqrt {c} C+\sqrt {a} (B c-2 A d)\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 a^{3/4} c^{3/4} \left (c^{3/2}-\sqrt {a} d\right ) \sqrt {a+c x^4}}+\frac {\left (c^{3/2}+\sqrt {a} d\right ) \left (c^2 C-B c d+A d^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (c^{3/2}-\sqrt {a} d\right )^2}{4 \sqrt {a} c^{3/2} d},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} c^{9/4} d \left (c^{3/2}-\sqrt {a} d\right ) \sqrt {a+c x^4}} \] Output:

-A*(c*x^4+a)^(1/2)/a/c/x+A*x*(c*x^4+a)^(1/2)/a/c^(1/2)/(a^(1/2)+c^(1/2)*x^ 
2)-1/2*(A*d^2-B*c*d+C*c^2)*arctan((a*d^2+c^3)^(1/2)*x/c^(1/2)/d^(1/2)/(c*x 
^4+a)^(1/2))/c^(3/2)/d^(1/2)/(a*d^2+c^3)^(1/2)-A*(a^(1/2)+c^(1/2)*x^2)*((c 
*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x/a^ 
(1/4))),1/2*2^(1/2))/a^(3/4)/c^(3/4)/(c*x^4+a)^(1/2)+1/2*(A*c^(3/2)-a*c^(1 
/2)*C+a^(1/2)*(-2*A*d+B*c))*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1 
/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*2^(1/2)) 
/a^(3/4)/c^(3/4)/(c^(3/2)-a^(1/2)*d)/(c*x^4+a)^(1/2)+1/4*(c^(3/2)+a^(1/2)* 
d)*(A*d^2-B*c*d+C*c^2)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x 
^2)^2)^(1/2)*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),-1/4*(c^(3/2)-a^( 
1/2)*d)^2/a^(1/2)/c^(3/2)/d,1/2*2^(1/2))/a^(1/4)/c^(9/4)/d/(c^(3/2)-a^(1/2 
)*d)/(c*x^4+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.94 (sec) , antiderivative size = 422, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x^2+C x^4}{x^2 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\frac {-a A \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c d-A \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^2 d x^4+\sqrt {a} A c^{3/2} d x \sqrt {1+\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-i \sqrt {a} c^{3/2} \left (\sqrt {a} \sqrt {c} C-i A d\right ) x \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+i a c^2 C x \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-i a B c d x \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+i a A d^2 x \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^2 d x \sqrt {a+c x^4}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 554, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2245, 25, 2233, 27, 1510, 2227, 27, 761, 2221}

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.

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

\(\Big \downarrow \) 2245

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2233

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1510

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

\(\Big \downarrow \) 2227

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 2221

\(\displaystyle \frac {\sqrt {a} \left (\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (\sqrt {a} (B c-2 A d)-a \sqrt {c} C+A c^{3/2}\right )}{2 \sqrt [4]{a} \left (c^{3/2}-\sqrt {a} d\right ) \sqrt {a+c x^4}}+\frac {\sqrt {a} \left (A d^2-B c d+c^2 C\right ) \left (\frac {\left (\sqrt {a} d+c^{3/2}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {c^{3/2}}{\sqrt {a}}-d\right )^2}{4 c^{3/2} d},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} c^{5/4} d \sqrt {a+c x^4}}-\frac {\left (c^{3/2}-\sqrt {a} d\right ) \arctan \left (\frac {x \sqrt {a d^2+c^3}}{\sqrt {c} \sqrt {d} \sqrt {a+c x^4}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+c^3}}\right )}{c^{3/2}-\sqrt {a} d}\right )-A \sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{a c}-\frac {A \sqrt {a+c x^4}}{a c x}\)

Input:

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

Output:

-((A*Sqrt[a + c*x^4])/(a*c*x)) + (-(A*Sqrt[c]*(-((x*Sqrt[a + c*x^4])/(Sqrt 
[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(S 
qrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(c 
^(1/4)*Sqrt[a + c*x^4]))) + Sqrt[a]*((c^(1/4)*(A*c^(3/2) - a*Sqrt[c]*C + S 
qrt[a]*(B*c - 2*A*d))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + 
Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)* 
(c^(3/2) - Sqrt[a]*d)*Sqrt[a + c*x^4]) + (Sqrt[a]*(c^2*C - B*c*d + A*d^2)* 
(-1/2*((c^(3/2) - Sqrt[a]*d)*ArcTan[(Sqrt[c^3 + a*d^2]*x)/(Sqrt[c]*Sqrt[d] 
*Sqrt[a + c*x^4])])/(Sqrt[c]*Sqrt[d]*Sqrt[c^3 + a*d^2]) + ((c^(3/2) + Sqrt 
[a]*d)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2] 
*EllipticPi[-1/4*(Sqrt[a]*(c^(3/2)/Sqrt[a] - d)^2)/(c^(3/2)*d), 2*ArcTan[( 
c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*c^(5/4)*d*Sqrt[a + c*x^4])))/(c^(3/2 
) - Sqrt[a]*d)))/(a*c)
 

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 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 1510

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

rule 2221

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) 
, x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTan[Rt[c*(d/e 
) + a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[c*(d/e) + a*(e/d), 2])), x] 
+ Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4* 
d*e*q*Sqrt[a + c*x^4]))*EllipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x 
], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && Po 
sQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[c*(d/e) + a*(e/d)]
 

rule 2227

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

rule 2233

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> 
With[{q = Rt[c/a, 2], A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff 
[P4x, x, 4]}, Simp[-C/(e*q)   Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] + Sim 
p[1/(c*e)   Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - a*e*q))*x^2)/((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] && PosQ[c/a]
 

rule 2245

Int[((Px_)*(x_)^(m_))/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_ 
Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 2], C = Coeff[Px, x, 
 4]}, Simp[A*x^(m + 1)*(Sqrt[a + c*x^4]/(a*d*(m + 1))), x] + Simp[1/(a*d*(m 
 + 1))   Int[(x^(m + 2)/((d + e*x^2)*Sqrt[a + c*x^4]))*Simp[a*B*d*(m + 1) - 
 A*a*e*(m + 1) + (a*C*d*(m + 1) - A*c*d*(m + 3))*x^2 - A*c*e*(m + 3)*x^4, x 
], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x^2, 2] && ILtQ[m/2, 0]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.86 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.59

method	result	size
default	\(\frac {C \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{d \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {A \left (-\frac {\sqrt {c \,x^{4}+a}}{a x}+\frac {i \sqrt {c}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )}{c}-\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{c^{2} d \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\)	\(317\)
risch	\(-\frac {A \sqrt {c \,x^{4}+a}}{a c x}+\frac {\frac {i A \sqrt {c}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {C c a \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{d \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) a \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{d c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}}{a c}\)	\(327\)
elliptic	\(-\frac {A \sqrt {c \,x^{4}+a}}{a c x}+\frac {C \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{d \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i A \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}-\frac {i A \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}-\frac {d \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right ) A}{c^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {\sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right ) B}{c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {\sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right ) C}{d \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\)	\(567\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(a + c*x**4)/(a*c*x**2 + a*d*x**4 + c**2*x**6 + c*d*x**8),x)*a + i 
nt(sqrt(a + c*x**4)/(a*c + a*d*x**2 + c**2*x**4 + c*d*x**6),x)*b + int((sq 
rt(a + c*x**4)*x**2)/(a*c + a*d*x**2 + c**2*x**4 + c*d*x**6),x)*c