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

Optimal result

Integrand size = 44, antiderivative size = 545 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=-\frac {\left (C e^2-B e f+A f^2\right ) x \sqrt {a+b x^2}}{2 e f (b e-a f) \sqrt {c+d x^2} \left (e+f x^2\right )}+\frac {\sqrt {c} \sqrt {d} \left (C e^2-B e f+A f^2\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{2 e f (b e-a f) (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\sqrt {c} \left (2 c^2 C e f+2 A d^2 e f-c d \left (C e^2+f (B e+A f)\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{2 a \sqrt {d} e f (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} \left (a f \left (C e^2 (2 d e-3 c f)+f^2 (B c e-2 A d e+A c f)\right )-b \left (C e^3 (d e-2 c f)+e f \left (B d e^2-A f (3 d e-2 c f)\right )\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{2 a \sqrt {d} e^2 f (b e-a f) (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

-1/2*(A*f^2-B*e*f+C*e^2)*x*(b*x^2+a)^(1/2)/e/f/(-a*f+b*e)/(d*x^2+c)^(1/2)/ 
(f*x^2+e)+1/2*c^(1/2)*d^(1/2)*(A*f^2-B*e*f+C*e^2)*(b*x^2+a)^(1/2)*Elliptic 
E(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))/e/f/(-a*f+b*e)/(- 
c*f+d*e)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+1/2*c^(1/2)*(2*c^ 
2*C*e*f+2*A*d^2*e*f-c*d*(C*e^2+f*(A*f+B*e)))*(b*x^2+a)^(1/2)*InverseJacobi 
AM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/a/d^(1/2)/e/f/(-c*f+d*e)^2 
/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/2*c^(3/2)*(a*f*(C*e^2*( 
-3*c*f+2*d*e)+f^2*(A*c*f-2*A*d*e+B*c*e))-b*(C*e^3*(-2*c*f+d*e)+e*f*(B*d*e^ 
2-A*f*(-2*c*f+3*d*e))))*(b*x^2+a)^(1/2)*EllipticPi(d^(1/2)*x/c^(1/2)/(1+d* 
x^2/c)^(1/2),1-c*f/d/e,(1-b*c/a/d)^(1/2))/a/d^(1/2)/e^2/f/(-a*f+b*e)/(-c*f 
+d*e)^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 14.67 (sec) , antiderivative size = 2038, normalized size of antiderivative = 3.74 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.71 (sec) , antiderivative size = 777, normalized size of antiderivative = 1.43, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {7293, 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/2*(d*(C*e^2 - B*e*f + A*f^2)*x*Sqrt[a + b*x^2])/(e*f*(b*e - a*f)*(d*e - 
 c*f)*Sqrt[c + d*x^2]) + ((C*e^2 - B*e*f + A*f^2)*x*Sqrt[a + b*x^2]*Sqrt[c 
 + d*x^2])/(2*e*(b*e - a*f)*(d*e - c*f)*(e + f*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)])/(2*e*f*(b*e - a*f)*(d*e - c*f)*Sqrt[(c*(a + b*x^2))/( 
a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqrt[c]*C*Sqrt[a + b*x^2]*EllipticF[Ar 
cTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*f^2*Sqrt[(c*(a + b 
*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (b*Sqrt[c]*Sqrt[d]*(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)])/(2*a*f^2*(b*e - a*f)*(d*e - c*f)*Sqrt[(c*(a + b*x^2))/(a*(c + d 
*x^2))]*Sqrt[c + d*x^2]) - (Sqrt[-a]*(2*C*e - B*f)*Sqrt[1 + (b*x^2)/a]*Sqr 
t[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), ArcSin[(Sqrt[b]*x)/Sqrt[-a]], (a* 
d)/(b*c)])/(Sqrt[b]*e*f^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]) + (Sqrt[-a]*(C* 
e^2 - B*e*f + A*f^2)*(b*e*(3*d*e - 2*c*f) - a*f*(2*d*e - c*f))*Sqrt[1 + (b 
*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), ArcSin[(Sqrt[b]*x)/Sq 
rt[-a]], (a*d)/(b*c)])/(2*Sqrt[b]*e^2*f^2*(b*e - a*f)*(d*e - c*f)*Sqrt[a + 
 b*x^2]*Sqrt[c + d*x^2])
 

Defintions of rubi rules used

rule 2009

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

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2572\) vs. \(2(519)=1038\).

Time = 8.47 (sec) , antiderivative size = 2573, normalized size of antiderivative = 4.72

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2573\)
default	\(\text {Expression too large to display}\)	\(3121\)

Input:

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

Output:

((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/2/e/(a*c*f^ 
2-a*d*e*f-b*c*e*f+b*d*e^2)*x*(A*f^2-B*e*f+C*e^2)*(b*d*x^4+a*d*x^2+b*c*x^2+ 
a*c)^(1/2)/(f*x^2+e)+1/2/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)*f/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*a*c-1/2/( 
a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)*e/f/(-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*b*d+1/(a*c*f^2-a*d*e*f-b*c*e*f+b*d* 
e^2)*e/f/(-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))*C*a*d+1/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)*e/f/(-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)*Elliptic 
Pi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*C*b*c-1/2/(a*c*f^2- 
a*d*e*f-b*c*e*f+b*d*e^2)*e^2/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)*EllipticPi(x*(-b/a)^(1/2),a*f/ 
b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*C*b*d+1/2/(-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))*d*b/f/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)*B*e 
-1/2/(-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))*d*b/...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b d \,f^{2} x^{8}+a d \,f^{2} x^{6}+b c \,f^{2} x^{6}+2 b d e f \,x^{6}+a c \,f^{2} x^{4}+2 a d e f \,x^{4}+2 b c e f \,x^{4}+b d \,e^{2} x^{4}+2 a c e f \,x^{2}+a d \,e^{2} x^{2}+b c \,e^{2} x^{2}+a c \,e^{2}}d x \right ) c +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,f^{2} x^{8}+a d \,f^{2} x^{6}+b c \,f^{2} x^{6}+2 b d e f \,x^{6}+a c \,f^{2} x^{4}+2 a d e f \,x^{4}+2 b c e f \,x^{4}+b d \,e^{2} x^{4}+2 a c e f \,x^{2}+a d \,e^{2} x^{2}+b c \,e^{2} x^{2}+a c \,e^{2}}d x \right ) b +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,f^{2} x^{8}+a d \,f^{2} x^{6}+b c \,f^{2} x^{6}+2 b d e f \,x^{6}+a c \,f^{2} x^{4}+2 a d e f \,x^{4}+2 b c e f \,x^{4}+b d \,e^{2} x^{4}+2 a c e f \,x^{2}+a d \,e^{2} x^{2}+b c \,e^{2} x^{2}+a c \,e^{2}}d x \right ) a \] Input:

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

Output:

int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e**2 + 2*a*c*e*f*x**2 + 
a*c*f**2*x**4 + a*d*e**2*x**2 + 2*a*d*e*f*x**4 + a*d*f**2*x**6 + b*c*e**2* 
x**2 + 2*b*c*e*f*x**4 + b*c*f**2*x**6 + b*d*e**2*x**4 + 2*b*d*e*f*x**6 + b 
*d*f**2*x**8),x)*c + int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c*e** 
2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 + a*d*e**2*x**2 + 2*a*d*e*f*x**4 + a*d* 
f**2*x**6 + b*c*e**2*x**2 + 2*b*c*e*f*x**4 + b*c*f**2*x**6 + b*d*e**2*x**4 
 + 2*b*d*e*f*x**6 + b*d*f**2*x**8),x)*b + int((sqrt(c + d*x**2)*sqrt(a + b 
*x**2))/(a*c*e**2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 + a*d*e**2*x**2 + 2*a*d 
*e*f*x**4 + a*d*f**2*x**6 + b*c*e**2*x**2 + 2*b*c*e*f*x**4 + b*c*f**2*x**6 
 + b*d*e**2*x**4 + 2*b*d*e*f*x**6 + b*d*f**2*x**8),x)*a