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

Optimal result

Integrand size = 33, antiderivative size = 278 \[ \int \frac {A+B x^2+C x^4}{\sqrt {\left (a+b x^2\right ) \left (c-d x^2\right )}} \, dx=-\frac {C x \sqrt {a c+(b c-a d) x^2-b d x^4}}{3 b d}+\frac {a \sqrt {c} (2 b c C+3 b B d-2 a C d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{3 b^2 d^{3/2} \sqrt {a c+(b c-a d) x^2-b d x^4}}+\frac {\sqrt {c} \left (3 A b^2 d+2 a^2 C d-a b (c C+3 B d)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{3 b^2 d^{3/2} \sqrt {a c+(b c-a d) x^2-b d x^4}} \] 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 = 8.10 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.84 \[ \int \frac {A+B x^2+C x^4}{\sqrt {\left (a+b x^2\right ) \left (c-d x^2\right )}} \, dx=\frac {\sqrt {\frac {b}{a}} C d x \left (a+b x^2\right ) \left (-c+d x^2\right )-i c (-2 b c C-3 b B d+2 a C d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+i \left (a c C d-b \left (2 c^2 C+3 B c d+3 A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{3 b \sqrt {\frac {b}{a}} d^2 \sqrt {\left (a+b x^2\right ) \left (c-d x^2\right )}} \] 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 = 0.50 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.82, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {2048, 2207, 25, 1514, 399, 321, 327}

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

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

\(\Big \downarrow \) 2207

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1514

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[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] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

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 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 1514

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[1 + 2*c*(x^2/(b - q))]*(Sqrt 
[1 + 2*c*(x^2/(b + q))]/Sqrt[a + b*x^2 + c*x^4])   Int[(d + e*x^2)/(Sqrt[1 
+ 2*c*(x^2/(b - q))]*Sqrt[1 + 2*c*(x^2/(b + q))]), x], x]] /; FreeQ[{a, b, 
c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[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 = 5.41 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.03

method	result	size
elliptic	\(-\frac {C x \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}{3 b d}+\frac {\left (A +\frac {C a c}{3 b d}\right ) \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}-\frac {\left (B +\frac {C \left (-2 a d +2 b c \right )}{3 b d}\right ) a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}\, b}\)	\(287\)
risch	\(-\frac {C x \left (b \,x^{2}+a \right ) \left (-d \,x^{2}+c \right )}{3 b d \sqrt {-\left (b \,x^{2}+a \right ) \left (d \,x^{2}-c \right )}}+\frac {-\frac {\left (3 B b d -2 a C d +2 C c b \right ) a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}\, b}+\frac {C a c \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}+\frac {3 A b d \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}}{3 b d}\)	\(384\)
default	\(\frac {A \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}-\frac {B a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}\, b}+C \left (-\frac {x \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}{3 b d}+\frac {a c \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{3 b d \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}-\frac {\left (-2 a d +2 b c \right ) a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{3 b^{2} d \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}\right )\)	\(495\)

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 = 212, normalized size of antiderivative = 0.76 \[ \int \frac {A+B x^2+C x^4}{\sqrt {\left (a+b x^2\right ) \left (c-d x^2\right )}} \, dx=-\frac {{\left (2 \, C b c^{3} - {\left (2 \, C a - 3 \, B b\right )} c^{2} d\right )} \sqrt {-b d} x \sqrt {\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {\frac {c}{d}}}{x}\right )\,|\,-\frac {a d}{b c}) - {\left (2 \, C b c^{3} + C a c d^{2} + 3 \, A b d^{3} - {\left (2 \, C a - 3 \, B b\right )} c^{2} d\right )} \sqrt {-b d} x \sqrt {\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {\frac {c}{d}}}{x}\right )\,|\,-\frac {a d}{b c}) + {\left (C b c d^{2} x^{2} + 2 \, C b c^{2} d - {\left (2 \, C a - 3 \, B b\right )} c d^{2}\right )} \sqrt {-b d x^{4} + {\left (b c - a d\right )} x^{2} + a c}}{3 \, b^{2} c d^{3} x} \] Input:

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

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)*sq 
rt(-b*d*x^4 + (b*c - a*d)*x^2 + a*c))/(b^2*c*d^3*x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

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="maxim 
a")
 

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