\(\int \frac {A+B x^2+C x^4}{(c+d x^2) \sqrt {a+c x^4}} \, dx\) [30]

Optimal result

Integrand size = 33, antiderivative size = 516 \[ \int \frac {A+B x^2+C x^4}{\left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\frac {C x \sqrt {a+c x^4}}{\sqrt {c} d \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {\left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {\sqrt {c^3+a d^2} x}{\sqrt {c} \sqrt {d} \sqrt {a+c x^4}}\right )}{2 \sqrt {c} d^{3/2} \sqrt {c^3+a d^2}}-\frac {\sqrt [4]{a} C \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{c^{3/4} d \sqrt {a+c x^4}}+\frac {\left (A c d-a C d+\sqrt {a} \sqrt {c} (2 c C-B d)\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} c^{3/4} d \left (c^{3/2}-\sqrt {a} d\right ) \sqrt {a+c x^4}}-\frac {\left (c^{3/2}+\sqrt {a} d\right ) \left (c^2 C-B c d+A d^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (c^{3/2}-\sqrt {a} d\right )^2}{4 \sqrt {a} c^{3/2} d},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} c^{5/4} d^2 \left (c^{3/2}-\sqrt {a} d\right ) \sqrt {a+c x^4}} \] Output:

C*x*(c*x^4+a)^(1/2)/c^(1/2)/d/(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^(1/2)/d^(3 
/2)/(a*d^2+c^3)^(1/2)-a^(1/4)*C*(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))/c^(3/4)/d/(c*x^4+a)^(1/2)+1/2*(A*c*d-C*a*d+a^(1/2)*c^(1/2)*(-B*d+2*C*c 
))*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*Inverse 
JacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*2^(1/2))/a^(1/4)/c^(3/4)/d/(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^(5/4)/d^2/(c^(3/2)-a^(1/2)*d)/(c*x^4+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.60 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.41 \[ \int \frac {A+B x^2+C x^4}{\left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\frac {\sqrt {1+\frac {c x^4}{a}} \left (\sqrt {a} \sqrt {c} C d E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+i \left (\left (c^2 C-B c d+i \sqrt {a} \sqrt {c} C d\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-\left (c^2 C-B c d+A d^2\right ) \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c d^2 \sqrt {a+c x^4}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {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.

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

\(\Big \downarrow \) 2233

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1510

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

\(\Big \downarrow \) 2227

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 2221

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

Input:

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

Output:

-((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)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcT 
an[(c^(1/4)*x)/a^(1/4)], 1/2])/(c^(1/4)*Sqrt[a + c*x^4])))/(Sqrt[c]*d)) + 
(((A*c*d - a*C*d + Sqrt[a]*Sqrt[c]*(2*c*C - B*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^(1/4)*(c^(3/2) - Sqrt[a]*d)*Sqrt[a + c*x^4]) 
 - (Sqrt[c]*(c^2*C - B*c*d + A*d^2)*(-1/2*((c^(3/2) - Sqrt[a]*d)*ArcTan[(S 
qrt[c^3 + a*d^2]*x)/(Sqrt[c]*Sqrt[d]*Sqrt[a + c*x^4])])/(Sqrt[c]*Sqrt[d]*S 
qrt[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)/Sqr 
t[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))/(Sqrt[c]*d)
 

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 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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.16 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.72

method	result	size
default	\(\frac {\frac {B d \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i C d \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}-\frac {C c \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}}{d^{2}}+\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{d^{2} c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\)	\(370\)
elliptic	\(\frac {\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 ) B}{d \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 {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right ) C c}{d^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i C \sqrt {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}\, \sqrt {c}}-\frac {i C \sqrt {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 )}{d \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}+\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 ) A}{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 ) B}{d \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {c \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^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\)	\(626\)

Input:

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

Output:

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

integrate((C*x^4+B*x^2+A)/(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)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(a + c*x**4)/(a*c + a*d*x**2 + c**2*x**4 + c*d*x**6),x)*a + int((s 
qrt(a + c*x**4)*x**4)/(a*c + a*d*x**2 + c**2*x**4 + c*d*x**6),x)*c + int(( 
sqrt(a + c*x**4)*x**2)/(a*c + a*d*x**2 + c**2*x**4 + c*d*x**6),x)*b