\(\int \frac {A+B x^2+C x^4}{\sqrt {(a+b x^2) (c+d x^2)}} \, dx\) [1]

Optimal result

Integrand size = 32, antiderivative size = 324 \[ \int \frac {A+B x^2+C x^4}{\sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=-\frac {(2 b c C-3 b B d+2 a C d) x \left (c+d x^2\right )}{3 b d^2 \sqrt {a c+(b c+a d) x^2+b d x^4}}+\frac {C x \sqrt {a c+(b c+a d) x^2+b d x^4}}{3 b d}+\frac {c (2 b c C-3 b B d+2 a C d) \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 \sqrt {a} b^{3/2} d^2 \sqrt {a c+(b c+a d) x^2+b d x^4}}-\frac {(a c C-3 A b d) \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {a} b^{3/2} d \sqrt {a c+(b c+a d) x^2+b d x^4}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.99 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.69 \[ \int \frac {A+B x^2+C x^4}{\sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=\frac {\sqrt {\frac {b}{a}} C d x \left (a+b x^2\right ) \left (c+d x^2\right )+i c (2 b c C-3 b B d+2 a C d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i \left (a c C d+b \left (2 c^2 C-3 B c d+3 A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 b \sqrt {\frac {b}{a}} d^2 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.72, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2048, 2207, 25, 1511, 27, 1416, 1509}

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.

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

\(\Big \downarrow \) 2048

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

\(\Big \downarrow \) 2207

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1511

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 1509

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

Input:

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

Output:

(C*x*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4])/(3*b*d) - (-(((2*b*c*C - 3*b*B 
*d + 2*a*C*d)*(-((x*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4])/(Sqrt[a]*Sqrt[c 
] + Sqrt[b]*Sqrt[d]*x^2)) + (a^(1/4)*c^(1/4)*(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sq 
rt[d]*x^2)*Sqrt[(a*c + (b*c + a*d)*x^2 + b*d*x^4)/(Sqrt[a]*Sqrt[c] + Sqrt[ 
b]*Sqrt[d]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*d^(1/4)*x)/(a^(1/4)*c^(1/4) 
)], (2 - (b*c + a*d)/(Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d]))/4])/(b^(1/4)*d^(1/ 
4)*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4])))/(Sqrt[b]*Sqrt[d])) + (a^(1/4)* 
c^(1/4)*(2*b*c*C - 3*b*B*d + 2*a*C*d + (Sqrt[b]*Sqrt[d]*(a*c*C - 3*A*b*d)) 
/(Sqrt[a]*Sqrt[c]))*(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)*Sqrt[(a*c + (b 
*c + a*d)*x^2 + b*d*x^4)/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)^2]*Ellipt 
icF[2*ArcTan[(b^(1/4)*d^(1/4)*x)/(a^(1/4)*c^(1/4))], (2 - (b*c + a*d)/(Sqr 
t[a]*Sqrt[b]*Sqrt[c]*Sqrt[d]))/4])/(2*b^(3/4)*d^(3/4)*Sqrt[a*c + (b*c + a* 
d)*x^2 + b*d*x^4]))/(3*b*d)
 

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 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1509

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

rule 1511

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

rule 2048

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_) 
, x_Symbol] :> Int[u*(a*c*e + (b*c + a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; F 
reeQ[{a, b, c, d, e, n, p}, x]
 

rule 2207

Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{n = 
 Expon[Px, x^2], e = Coeff[Px, x^2, Expon[Px, x^2]]}, Simp[e*x^(2*n - 3)*(( 
a + b*x^2 + c*x^4)^(p + 1)/(c*(2*n + 4*p + 1))), x] + Simp[1/(c*(2*n + 4*p 
+ 1))   Int[(a + b*x^2 + c*x^4)^p*ExpandToSum[c*(2*n + 4*p + 1)*Px - a*e*(2 
*n - 3)*x^(2*n - 4) - b*e*(2*n + 2*p - 1)*x^(2*n - 2) - c*e*(2*n + 4*p + 1) 
*x^(2*n), x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Px, x^2] && Expon[ 
Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] &&  !LtQ[p, -1]
 

Maple [A] (verified)

Time = 5.36 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.86

method	result	size
elliptic	\(\frac {C x \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{3 b d}+\frac {\left (A -\frac {C a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {\left (B -\frac {C \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}\)	\(278\)
risch	\(\frac {C x \left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}{3 b d \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}+\frac {-\frac {\left (3 B b d -2 a C d -2 C c b \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}+\frac {3 A b d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {C a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}}{3 b d}\)	\(371\)
default	\(\frac {A \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {B c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}+C \left (\frac {x \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{3 b d}-\frac {a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 b d \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}+\frac {\left (2 a d +2 b c \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{3 b \,d^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}\right )\)	\(479\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.64 \[ \int \frac {A+B x^2+C x^4}{\sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=\frac {{\left (2 \, C b c^{3} + {\left (2 \, C a - 3 \, B b\right )} c^{2} d\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, C b c^{3} + C a c d^{2} - 3 \, A b d^{3} + {\left (2 \, C a - 3 \, B b\right )} c^{2} d\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (C b c d^{2} x^{2} - 2 \, C b c^{2} d - {\left (2 \, C a - 3 \, B b\right )} c d^{2}\right )} \sqrt {b d x^{4} + {\left (b c + a d\right )} x^{2} + a c}}{3 \, b^{2} c d^{3} x} \] Input:

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

Output:

1/3*((2*C*b*c^3 + (2*C*a - 3*B*b)*c^2*d)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e 
(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (2*C*b*c^3 + C*a*c*d^2 - 3*A*b*d^3 + ( 
2*C*a - 3*B*b)*c^2*d)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/ 
x), a*d/(b*c)) + (C*b*c*d^2*x^2 - 2*C*b*c^2*d - (2*C*a - 3*B*b)*c*d^2)*sqr 
t(b*d*x^4 + (b*c + a*d)*x^2 + a*c))/(b^2*c*d^3*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x^2+C x^4}{\sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, c x -2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a c d +3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} d -2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b \,c^{2}+3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b d -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,c^{2}}{3 b d} \] Input:

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

Output:

(sqrt(c + d*x**2)*sqrt(a + b*x**2)*c*x - 2*int((sqrt(c + d*x**2)*sqrt(a + 
b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*c*d + 3*int((sqr 
t(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4 
),x)*b**2*d - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x* 
*2 + b*c*x**2 + b*d*x**4),x)*b*c**2 + 3*int((sqrt(c + d*x**2)*sqrt(a + b*x 
**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*b*d - int((sqrt(c + d*x* 
*2)*sqrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*c**2)/(3 
*b*d)