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

Optimal result

Integrand size = 34, antiderivative size = 274 \[ \int \frac {A+B x^2+C x^4}{\sqrt {\left (a-b x^2\right ) \left (c-d x^2\right )}} \, dx=\frac {C x \sqrt {a c-(b c+a d) x^2+b d x^4}}{3 b d}-\frac {a \sqrt {c} (2 b c C+3 b B d+2 a C d) \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b c}{a d}\right )}{3 b^2 d^{3/2} \sqrt {a c-(b c+a d) x^2+b d x^4}}+\frac {\sqrt {c} \left (3 A b^2 d+2 a^2 C d+a b (c C+3 B d)\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b c}{a d}\right )}{3 b^2 d^{3/2} \sqrt {a c-(b c+a d) x^2+b d x^4}} \] Output:

1/3*C*x*(a*c-(a*d+b*c)*x^2+b*d*x^4)^(1/2)/b/d-1/3*a*c^(1/2)*(3*B*b*d+2*C*a 
*d+2*C*b*c)*(1-b*x^2/a)^(1/2)*(1-d*x^2/c)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2 
),(b*c/a/d)^(1/2))/b^2/d^(3/2)/(a*c-(a*d+b*c)*x^2+b*d*x^4)^(1/2)+1/3*c^(1/ 
2)*(3*A*b^2*d+2*a^2*C*d+a*b*(3*B*d+C*c))*(1-b*x^2/a)^(1/2)*(1-d*x^2/c)^(1/ 
2)*EllipticF(d^(1/2)*x/c^(1/2),(b*c/a/d)^(1/2))/b^2/d^(3/2)/(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 = 8.32 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x^2+C x^4}{\sqrt {\left (a-b x^2\right ) \left (c-d x^2\right )}} \, dx=\frac {-\sqrt {-\frac {b}{a}} C d x \left (a-b x^2\right ) \left (-c+d x^2\right )+i c (2 b c C+3 b B d+2 a C d) \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i \left (a c C d+b \left (2 c^2 C+3 B c d+3 A d^2\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 b \sqrt {-\frac {b}{a}} d^2 \sqrt {\left (a-b x^2\right ) \left (c-d x^2\right )}} \] 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 [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(562\) vs. \(2(274)=548\).

Time = 0.76 (sec) , antiderivative size = 562, normalized size of antiderivative = 2.05, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, 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

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1511

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 1509

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

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]*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]*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))/(S 
qrt[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]*EllipticF 
[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])/(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 = 5.59 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.06

method	result	size
elliptic	\(\frac {C x \sqrt {b d \,x^{4}-x^{2} d a -b c \,x^{2}+a c}}{3 b d}+\frac {\left (A -\frac {C a c}{3 b d}\right ) \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {b d \,x^{4}-x^{2} d a -b c \,x^{2}+a c}}+\frac {\left (B -\frac {C \left (-2 a d -2 b c \right )}{3 b d}\right ) a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {b d \,x^{4}-x^{2} d a -b c \,x^{2}+a c}\, b}\)	\(291\)
risch	\(\frac {C x \left (-b \,x^{2}+a \right ) \left (-d \,x^{2}+c \right )}{3 b d \sqrt {\left (b \,x^{2}-a \right ) \left (d \,x^{2}-c \right )}}+\frac {\frac {\left (3 B b d +2 a C d +2 C c b \right ) a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {b d \,x^{4}-x^{2} d a -b c \,x^{2}+a c}\, b}+\frac {3 A b d \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {b d \,x^{4}-x^{2} d a -b c \,x^{2}+a c}}-\frac {C a c \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {b d \,x^{4}-x^{2} d a -b c \,x^{2}+a c}}}{3 b d}\)	\(393\)
default	\(\frac {A \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {b d \,x^{4}-x^{2} d a -b c \,x^{2}+a c}}+\frac {B a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {b d \,x^{4}-x^{2} d a -b c \,x^{2}+a c}\, b}+C \left (\frac {x \sqrt {b d \,x^{4}-x^{2} d a -b c \,x^{2}+a c}}{3 b d}-\frac {a c \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )}{3 b d \sqrt {\frac {d}{c}}\, \sqrt {b d \,x^{4}-x^{2} d a -b c \,x^{2}+a c}}-\frac {\left (-2 a d -2 b c \right ) a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )\right )}{3 b^{2} d \sqrt {\frac {d}{c}}\, \sqrt {b d \,x^{4}-x^{2} d a -b c \,x^{2}+a c}}\right )\)	\(504\)

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.78 \[ \int \frac {A+B x^2+C x^4}{\sqrt {\left (a-b x^2\right ) \left (c-d x^2\right )}} \, dx=\frac {{\left (2 \, C a^{2} b c + {\left (2 \, C a^{3} + 3 \, B a^{2} b\right )} d\right )} \sqrt {b d} x \sqrt {\frac {a}{b}} E(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,\frac {b c}{a d}) - {\left ({\left (2 \, C a^{2} b - C a b^{2}\right )} c + {\left (2 \, C a^{3} + 3 \, B a^{2} b + 3 \, A b^{3}\right )} d\right )} \sqrt {b d} x \sqrt {\frac {a}{b}} F(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,\frac {b c}{a d}) + {\left (C a b^{2} d x^{2} + 2 \, C a b^{2} c + {\left (2 \, C a^{2} b + 3 \, B a b^{2}\right )} d\right )} \sqrt {b d x^{4} - {\left (b c + a d\right )} x^{2} + a c}}{3 \, a b^{3} d^{2} x} \] Input:

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

Output:

1/3*((2*C*a^2*b*c + (2*C*a^3 + 3*B*a^2*b)*d)*sqrt(b*d)*x*sqrt(a/b)*ellipti 
c_e(arcsin(sqrt(a/b)/x), b*c/(a*d)) - ((2*C*a^2*b - C*a*b^2)*c + (2*C*a^3 
+ 3*B*a^2*b + 3*A*b^3)*d)*sqrt(b*d)*x*sqrt(a/b)*elliptic_f(arcsin(sqrt(a/b 
)/x), b*c/(a*d)) + (C*a*b^2*d*x^2 + 2*C*a*b^2*c + (2*C*a^2*b + 3*B*a*b^2)* 
d)*sqrt(b*d*x^4 - (b*c + a*d)*x^2 + a*c))/(a*b^3*d^2*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="maxi 
ma")
 

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