\(\int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (A+B x^2+C x^4)}{e+f x^2} \, dx\) [9]

Optimal result

Integrand size = 44, antiderivative size = 666 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{e+f x^2} \, dx=-\frac {\left (2 a^2 C d f+b^2 \left (5 c C e+15 B d e-\frac {15 C d e^2}{f}-5 B c f+\frac {2 c^2 C f}{d}-15 A d f\right )+a b (5 C d e-2 c C f-5 B d f)\right ) x \sqrt {c+d x^2}}{15 b d f^2 \sqrt {a+b x^2}}+\frac {(a C d f-b (5 C d e+2 c C f-5 B d f)) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 b d f^2}+\frac {C x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 d f}+\frac {\sqrt {a} \left (2 a^2 C d^2 f^2+a b d f (5 C d e-2 c C f-5 B d f)+b^2 \left (5 d f (3 B d e-B c f-3 A d f)-C \left (15 d^2 e^2-5 c d e f-2 c^2 f^2\right )\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 b^{3/2} d^2 f^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} \left (a c C d f^2+5 b d f (3 B d e-2 B c f-3 A d f)-b C \left (15 d^2 e^2-10 c d e f-c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 b^{3/2} c d f^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} (d e-c f) \left (C e^2-B e f+A f^2\right ) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e f^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

-1/15*(2*a^2*C*d*f+b^2*(5*c*C*e+15*B*d*e-15*C*d*e^2/f-5*B*c*f+2*c^2*C*f/d- 
15*A*d*f)+a*b*(-5*B*d*f-2*C*c*f+5*C*d*e))*x*(d*x^2+c)^(1/2)/b/d/f^2/(b*x^2 
+a)^(1/2)+1/15*(a*C*d*f-b*(-5*B*d*f+2*C*c*f+5*C*d*e))*x*(b*x^2+a)^(1/2)*(d 
*x^2+c)^(1/2)/b/d/f^2+1/5*C*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/d/f+1/15*a^( 
1/2)*(2*a^2*C*d^2*f^2+a*b*d*f*(-5*B*d*f-2*C*c*f+5*C*d*e)+b^2*(5*d*f*(-3*A* 
d*f-B*c*f+3*B*d*e)-C*(-2*c^2*f^2-5*c*d*e*f+15*d^2*e^2)))*(d*x^2+c)^(1/2)*E 
llipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/b^(3/2)/d^ 
2/f^3/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/15*a^(3/2)*(a*c*C* 
d*f^2+5*b*d*f*(-3*A*d*f-2*B*c*f+3*B*d*e)-b*C*(-c^2*f^2-10*c*d*e*f+15*d^2*e 
^2))*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c) 
^(1/2))/b^(3/2)/c/d/f^3/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-a^ 
(3/2)*(-c*f+d*e)*(A*f^2-B*e*f+C*e^2)*(d*x^2+c)^(1/2)*EllipticPi(b^(1/2)*x/ 
a^(1/2)/(1+b*x^2/a)^(1/2),1-a*f/b/e,(1-a*d/b/c)^(1/2))/b^(1/2)/c/e/f^3/(b* 
x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.82 (sec) , antiderivative size = 541, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{e+f x^2} \, dx=\frac {i c e f \left (2 a^2 C d^2 f^2+a b d f (5 C d e-2 c C f-5 B d f)+b^2 \left (-5 d f (-3 B d e+B c f+3 A d f)+C \left (-15 d^2 e^2+5 c d e f+2 c^2 f^2\right )\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i e \left (a^2 c C d^2 f^3+a b d f \left (5 d f (-3 B d e+B c f+3 A d f)+C \left (15 d^2 e^2-5 c d e f-3 c^2 f^2\right )\right )+b^2 \left (-5 d f \left (-3 B d^2 e^2+3 A d^2 e f+B c^2 f^2\right )+C \left (-15 d^3 e^3+5 c^2 d e f^2+2 c^3 f^3\right )\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+d \left (\sqrt {\frac {b}{a}} e f^2 x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (5 b B d f+a C d f+b C \left (-5 d e+c f+3 d f x^2\right )\right )-15 i b d (-b e+a f) (-d e+c f) \left (C e^2+f (-B e+A f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{15 b \sqrt {\frac {b}{a}} d^2 e f^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(I*c*e*f*(2*a^2*C*d^2*f^2 + a*b*d*f*(5*C*d*e - 2*c*C*f - 5*B*d*f) + b^2*(- 
5*d*f*(-3*B*d*e + B*c*f + 3*A*d*f) + C*(-15*d^2*e^2 + 5*c*d*e*f + 2*c^2*f^ 
2)))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a] 
*x], (a*d)/(b*c)] - I*e*(a^2*c*C*d^2*f^3 + a*b*d*f*(5*d*f*(-3*B*d*e + B*c* 
f + 3*A*d*f) + C*(15*d^2*e^2 - 5*c*d*e*f - 3*c^2*f^2)) + b^2*(-5*d*f*(-3*B 
*d^2*e^2 + 3*A*d^2*e*f + B*c^2*f^2) + C*(-15*d^3*e^3 + 5*c^2*d*e*f^2 + 2*c 
^3*f^3)))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt 
[b/a]*x], (a*d)/(b*c)] + d*(Sqrt[b/a]*e*f^2*x*(a + b*x^2)*(c + d*x^2)*(5*b 
*B*d*f + a*C*d*f + b*C*(-5*d*e + c*f + 3*d*f*x^2)) - (15*I)*b*d*(-(b*e) + 
a*f)*(-(d*e) + c*f)*(C*e^2 + f*(-(B*e) + A*f))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 
+ (d*x^2)/c]*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]) 
)/(15*b*Sqrt[b/a]*d^2*e*f^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 1.77 (sec) , antiderivative size = 1029, normalized size of antiderivative = 1.55, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {7276, 2009}

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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {C \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{5 f}-\frac {(C e-B f) \sqrt {b x^2+a} \sqrt {d x^2+c} x}{3 f^2}+\frac {C (b c+a d) \sqrt {b x^2+a} \sqrt {d x^2+c} x}{15 b d f}-\frac {(b c+a d) (C e-B f) \sqrt {b x^2+a} x}{3 b f^2 \sqrt {d x^2+c}}+\frac {d \left (C e^2-B f e+A f^2\right ) \sqrt {b x^2+a} x}{f^3 \sqrt {d x^2+c}}-\frac {2 C \left (b^2 c^2-a b d c+a^2 d^2\right ) \sqrt {b x^2+a} x}{15 b^2 d f \sqrt {d x^2+c}}+\frac {\sqrt {c} (b c+a d) (C e-B f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b \sqrt {d} f^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {d} \left (C e^2-B f e+A f^2\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{f^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 \sqrt {c} C \left (b^2 c^2-a b d c+a^2 d^2\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^2 d^{3/2} f \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {2 c^{3/2} (C e-B f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 \sqrt {d} f^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b c^{3/2} \left (C e^2-B f e+A f^2\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} f^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {c^{3/2} C (b c+a d) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 b d^{3/2} f \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {c^{3/2} (b e-a f) \left (C e^2-B f e+A f^2\right ) \sqrt {b x^2+a} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\)

Input:

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

Output:

(-2*C*(b^2*c^2 - a*b*c*d + a^2*d^2)*x*Sqrt[a + b*x^2])/(15*b^2*d*f*Sqrt[c 
+ d*x^2]) - ((b*c + a*d)*(C*e - B*f)*x*Sqrt[a + b*x^2])/(3*b*f^2*Sqrt[c + 
d*x^2]) + (d*(C*e^2 - B*e*f + A*f^2)*x*Sqrt[a + b*x^2])/(f^3*Sqrt[c + d*x^ 
2]) + (C*(b*c + a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15*b*d*f) - ((C*e 
 - B*f)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*f^2) + (C*x^3*Sqrt[a + b*x^2 
]*Sqrt[c + d*x^2])/(5*f) + (2*Sqrt[c]*C*(b^2*c^2 - a*b*c*d + a^2*d^2)*Sqrt 
[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b 
^2*d^(3/2)*f*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqr 
t[c]*(b*c + a*d)*(C*e - B*f)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/ 
Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b*Sqrt[d]*f^2*Sqrt[(c*(a + b*x^2))/(a*(c + 
d*x^2))]*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[d]*(C*e^2 - B*e*f + A*f^2)*Sqrt[ 
a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(f^3*S 
qrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*C*(b*c + 
a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d 
)])/(15*b*d^(3/2)*f*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) 
 - (2*c^(3/2)*(C*e - B*f)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqr 
t[c]], 1 - (b*c)/(a*d)])/(3*Sqrt[d]*f^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2 
))]*Sqrt[c + d*x^2]) + (b*c^(3/2)*(C*e^2 - B*e*f + A*f^2)*Sqrt[a + b*x^2]* 
EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*f^3*Sq 
rt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*(b*e - ...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [A] (verified)

Time = 9.10 (sec) , antiderivative size = 802, normalized size of antiderivative = 1.20

method	result	size
risch	\(\frac {x \left (3 C \,x^{2} b d f +5 b B d f +a C d f +C b f c -5 C b d e \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 b d \,f^{2}}+\frac {\left (\frac {\left (15 A a b \,d^{2} f^{3}+15 A \,b^{2} c d \,f^{3}-15 A \,b^{2} d^{2} e \,f^{2}+10 B a b c d \,f^{3}-15 B a b \,d^{2} e \,f^{2}-15 B \,b^{2} c d e \,f^{2}+15 B \,b^{2} d^{2} e^{2} f -C \,a^{2} c d \,f^{3}-C a b \,c^{2} f^{3}-10 C a b c d e \,f^{2}+15 C a b \,d^{2} e^{2} f +15 C \,b^{2} c d \,e^{2} f -15 C \,b^{2} d^{2} e^{3}\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 )}{f^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (15 A \,b^{2} d^{2} f^{2}+5 B a b \,d^{2} f^{2}+5 B \,b^{2} c \,f^{2} d -15 e B \,b^{2} f \,d^{2}-2 a^{2} C \,d^{2} f^{2}+2 C a b c d \,f^{2}-5 C a b \,d^{2} e f -2 C \,b^{2} c^{2} f^{2}-5 C c e f \,b^{2} d +15 C \,d^{2} e^{2} b^{2}\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 )}{f \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {15 \left (A a c \,f^{4}-A a d e \,f^{3}-A b c e \,f^{3}+A b d \,e^{2} f^{2}-B a c e \,f^{3}+B a d \,e^{2} f^{2}+B b c \,e^{2} f^{2}-B b d \,e^{3} f +C a c \,e^{2} f^{2}-C a d \,e^{3} f -C b c \,e^{3} f +C b d \,e^{4}\right ) b d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right )}{f^{2} e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 d b \,f^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(802\)
default	\(\text {Expression too large to display}\)	\(2787\)
elliptic	\(\text {Expression too large to display}\)	\(3571\)

Input:

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

Output:

1/15*x*(3*C*b*d*f*x^2+5*B*b*d*f+C*a*d*f+C*b*c*f-5*C*b*d*e)*(b*x^2+a)^(1/2) 
*(d*x^2+c)^(1/2)/b/d/f^2+1/15/d/b/f^2*((15*A*a*b*d^2*f^3+15*A*b^2*c*d*f^3- 
15*A*b^2*d^2*e*f^2+10*B*a*b*c*d*f^3-15*B*a*b*d^2*e*f^2-15*B*b^2*c*d*e*f^2+ 
15*B*b^2*d^2*e^2*f-C*a^2*c*d*f^3-C*a*b*c^2*f^3-10*C*a*b*c*d*e*f^2+15*C*a*b 
*d^2*e^2*f+15*C*b^2*c*d*e^2*f-15*C*b^2*d^2*e^3)/f^2/(-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))-1/f*(15*A*b^2*d^2*f^2+5*B*a*b*d^2* 
f^2+5*B*b^2*c*d*f^2-15*B*b^2*d^2*e*f-2*C*a^2*d^2*f^2+2*C*a*b*c*d*f^2-5*C*a 
*b*d^2*e*f-2*C*b^2*c^2*f^2-5*C*b^2*c*d*e*f+15*C*b^2*d^2*e^2)*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)))+15*(A*a*c*f^4-A*a*d*e*f^3-A*b*c*e*f^3+A*b* 
d*e^2*f^2-B*a*c*e*f^3+B*a*d*e^2*f^2+B*b*c*e^2*f^2-B*b*d*e^3*f+C*a*c*e^2*f^ 
2-C*a*d*e^3*f-C*b*c*e^3*f+C*b*d*e^4)*b*d/f^2/e/(-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)*EllipticPi(x*(-b 
/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2)))*((b*x^2+a)*(d*x^2+c))^(1/2 
)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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