\(\int \frac {A+B x^2}{x^6 \sqrt {a-b x^2} \sqrt {c+d x^2} (e+f x^2)} \, dx\) [14]

Optimal result

Integrand size = 43, antiderivative size = 674 \[ \int \frac {A+B x^2}{x^6 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=-\frac {A \sqrt {a-b x^2} \sqrt {c+d x^2}}{5 a c e x^5}-\frac {(5 a B c e+A (4 b c e-4 a d e-5 a c f)) \sqrt {a-b x^2} \sqrt {c+d x^2}}{15 a^2 c^2 e^2 x^3}-\frac {\left (\frac {8 A b^2 c e}{a}-5 a B (2 d e+3 c f)+b (10 B c e-7 A d e-10 A c f)+a A \left (\frac {8 d^2 e}{c}+10 d f+\frac {15 c f^2}{e}\right )\right ) \sqrt {a-b x^2} \sqrt {c+d x^2}}{15 a^2 c^2 e^2 x}-\frac {\sqrt {b} \left (5 a B c e (2 b c e-2 a d e-3 a c f)+A \left (8 b^2 c^2 e^2-a b c e (7 d e+10 c f)+a^2 \left (8 d^2 e^2+10 c d e f+15 c^2 f^2\right )\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{15 a^{5/2} c^3 e^3 \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {\sqrt {b} \left (5 a B c e (2 b c e-a d e-3 a c f)+A \left (8 b^2 c^2 e^2-a b c e (3 d e+10 c f)+a^2 \left (4 d^2 e^2+5 c d e f+15 c^2 f^2\right )\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{15 a^{5/2} c^2 e^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {a} f^2 (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} e^4 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:

-1/5*A*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/e/x^5-1/15*(5*B*a*c*e+A*(-5*a* 
c*f-4*a*d*e+4*b*c*e))*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a^2/c^2/e^2/x^3-1/1 
5*(8*A*b^2*c*e/a-5*a*B*(3*c*f+2*d*e)+b*(-10*A*c*f-7*A*d*e+10*B*c*e)+a*A*(8 
*d^2*e/c+10*d*f+15*c*f^2/e))*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a^2/c^2/e^2/ 
x-1/15*b^(1/2)*(5*a*B*c*e*(-3*a*c*f-2*a*d*e+2*b*c*e)+A*(8*b^2*c^2*e^2-a*b* 
c*e*(10*c*f+7*d*e)+a^2*(15*c^2*f^2+10*c*d*e*f+8*d^2*e^2)))*(1-b*x^2/a)^(1/ 
2)*(d*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2),(-a*d/b/c)^(1/2))/a^(5/2)/c 
^3/e^3/(-b*x^2+a)^(1/2)/(1+d*x^2/c)^(1/2)+1/15*b^(1/2)*(5*a*B*c*e*(-3*a*c* 
f-a*d*e+2*b*c*e)+A*(8*b^2*c^2*e^2-a*b*c*e*(10*c*f+3*d*e)+a^2*(15*c^2*f^2+5 
*c*d*e*f+4*d^2*e^2)))*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(b^(1/2 
)*x/a^(1/2),(-a*d/b/c)^(1/2))/a^(5/2)/c^2/e^3/(-b*x^2+a)^(1/2)/(d*x^2+c)^( 
1/2)+a^(1/2)*f^2*(-A*f+B*e)*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)*EllipticPi 
(b^(1/2)*x/a^(1/2),-a*f/b/e,(-a*d/b/c)^(1/2))/b^(1/2)/e^4/(-b*x^2+a)^(1/2) 
/(d*x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 15.57 (sec) , antiderivative size = 1196, normalized size of antiderivative = 1.77 \[ \int \frac {A+B x^2}{x^6 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:

Integrate[(A + B*x^2)/(x^6*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)),x]
 

Output:

-1/15*(3*a^3*A*b*c^3*e^3 + a^2*A*b^2*c^3*e^3*x^2 + 5*a^3*b*B*c^3*e^3*x^2 - 
 a^3*A*b*c^2*d*e^3*x^2 - 5*a^3*A*b*c^3*e^2*f*x^2 + 4*a*A*b^3*c^3*e^3*x^4 + 
 5*a^2*b^2*B*c^3*e^3*x^4 - 2*a^2*A*b^2*c^2*d*e^3*x^4 - 5*a^3*b*B*c^2*d*e^3 
*x^4 + 4*a^3*A*b*c*d^2*e^3*x^4 - 5*a^2*A*b^2*c^3*e^2*f*x^4 - 15*a^3*b*B*c^ 
3*e^2*f*x^4 + 5*a^3*A*b*c^2*d*e^2*f*x^4 + 15*a^3*A*b*c^3*e*f^2*x^4 - 8*A*b 
^4*c^3*e^3*x^6 - 10*a*b^3*B*c^3*e^3*x^6 + 11*a*A*b^3*c^2*d*e^3*x^6 + 15*a^ 
2*b^2*B*c^2*d*e^3*x^6 - 11*a^2*A*b^2*c*d^2*e^3*x^6 - 10*a^3*b*B*c*d^2*e^3* 
x^6 + 8*a^3*A*b*d^3*e^3*x^6 + 10*a*A*b^3*c^3*e^2*f*x^6 + 15*a^2*b^2*B*c^3* 
e^2*f*x^6 - 15*a^2*A*b^2*c^2*d*e^2*f*x^6 - 15*a^3*b*B*c^2*d*e^2*f*x^6 + 10 
*a^3*A*b*c*d^2*e^2*f*x^6 - 15*a^2*A*b^2*c^3*e*f^2*x^6 + 15*a^3*A*b*c^2*d*e 
*f^2*x^6 - 8*A*b^4*c^2*d*e^3*x^8 - 10*a*b^3*B*c^2*d*e^3*x^8 + 7*a*A*b^3*c* 
d^2*e^3*x^8 + 10*a^2*b^2*B*c*d^2*e^3*x^8 - 8*a^2*A*b^2*d^3*e^3*x^8 + 10*a* 
A*b^3*c^2*d*e^2*f*x^8 + 15*a^2*b^2*B*c^2*d*e^2*f*x^8 - 10*a^2*A*b^2*c*d^2* 
e^2*f*x^8 - 15*a^2*A*b^2*c^2*d*e*f^2*x^8 - I*a*b*Sqrt[-(b/a)]*c*e*(-8*A*b^ 
2*c^2*e^2 + a*A*b*c*e*(7*d*e + 10*c*f) + 5*a*B*c*e*(-2*b*c*e + 2*a*d*e + 3 
*a*c*f) - a^2*A*(8*d^2*e^2 + 10*c*d*e*f + 15*c^2*f^2))*x^5*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*b*Sqrt[-(b/a)]*c*e*(-8*A*b^2*c^2*e^2 + a*A*b*c*e*(3*d*e + 10*c*f 
) + 5*a*B*c*e*(-2*b*c*e + a*d*e + 3*a*c*f) - a^2*A*(4*d^2*e^2 + 5*c*d*e*f 
+ 15*c^2*f^2))*x^5*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*...
 

Rubi [A] (verified)

Time = 3.20 (sec) , antiderivative size = 1077, normalized size of antiderivative = 1.60, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, 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.

\(\displaystyle \int \frac {A+B x^2}{x^6 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (\frac {f^2 (B e-A f)}{e^3 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )}-\frac {f (B e-A f)}{e^3 x^2 \sqrt {a-b x^2} \sqrt {c+d x^2}}+\frac {B e-A f}{e^2 x^4 \sqrt {a-b x^2} \sqrt {c+d x^2}}+\frac {A}{e x^6 \sqrt {a-b x^2} \sqrt {c+d x^2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {a} (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right ) f^2}{\sqrt {b} e^4 \sqrt {a-b x^2} \sqrt {d x^2+c}}+\frac {\sqrt {b} (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right ) f}{\sqrt {a} c e^3 \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {b} (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right ) f}{\sqrt {a} e^3 \sqrt {a-b x^2} \sqrt {d x^2+c}}+\frac {(B e-A f) \sqrt {a-b x^2} \sqrt {d x^2+c} f}{a c e^3 x}-\frac {2 \sqrt {b} (b c-a d) (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{3 a^{3/2} c^2 e^2 \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {A \sqrt {b} \left (8 b^2 c^2-7 a b d c+8 a^2 d^2\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{15 a^{5/2} c^3 e \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}+\frac {\sqrt {b} (2 b c-a d) (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{3 a^{3/2} c e^2 \sqrt {a-b x^2} \sqrt {d x^2+c}}+\frac {A \sqrt {b} \left (8 b^2 c^2-3 a b d c+4 a^2 d^2\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{15 a^{5/2} c^2 e \sqrt {a-b x^2} \sqrt {d x^2+c}}-\frac {2 (b c-a d) (B e-A f) \sqrt {a-b x^2} \sqrt {d x^2+c}}{3 a^2 c^2 e^2 x}-\frac {A \left (8 b^2 c^2-7 a b d c+8 a^2 d^2\right ) \sqrt {a-b x^2} \sqrt {d x^2+c}}{15 a^3 c^3 e x}-\frac {(B e-A f) \sqrt {a-b x^2} \sqrt {d x^2+c}}{3 a c e^2 x^3}-\frac {4 A (b c-a d) \sqrt {a-b x^2} \sqrt {d x^2+c}}{15 a^2 c^2 e x^3}-\frac {A \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 a c e x^5}\)

Input:

Int[(A + B*x^2)/(x^6*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)),x]
 

Output:

-1/5*(A*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])/(a*c*e*x^5) - (4*A*(b*c - a*d)*Sq 
rt[a - b*x^2]*Sqrt[c + d*x^2])/(15*a^2*c^2*e*x^3) - ((B*e - A*f)*Sqrt[a - 
b*x^2]*Sqrt[c + d*x^2])/(3*a*c*e^2*x^3) - (A*(8*b^2*c^2 - 7*a*b*c*d + 8*a^ 
2*d^2)*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])/(15*a^3*c^3*e*x) - (2*(b*c - a*d)* 
(B*e - A*f)*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])/(3*a^2*c^2*e^2*x) + (f*(B*e - 
 A*f)*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])/(a*c*e^3*x) - (A*Sqrt[b]*(8*b^2*c^2 
 - 7*a*b*c*d + 8*a^2*d^2)*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[Ar 
cSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(15*a^(5/2)*c^3*e*Sqrt[a - b*x 
^2]*Sqrt[1 + (d*x^2)/c]) - (2*Sqrt[b]*(b*c - a*d)*(B*e - A*f)*Sqrt[1 - (b* 
x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b* 
c))])/(3*a^(3/2)*c^2*e^2*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c]) + (Sqrt[b]*f 
*(B*e - A*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b] 
*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[a]*c*e^3*Sqrt[a - b*x^2]*Sqrt[1 + (d* 
x^2)/c]) + (A*Sqrt[b]*(8*b^2*c^2 - 3*a*b*c*d + 4*a^2*d^2)*Sqrt[1 - (b*x^2) 
/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b* 
c))])/(15*a^(5/2)*c^2*e*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]) + (Sqrt[b]*(2*b*c 
 - a*d)*(B*e - A*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcS 
in[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(3*a^(3/2)*c*e^2*Sqrt[a - b*x^2] 
*Sqrt[c + d*x^2]) - (Sqrt[b]*f*(B*e - A*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d 
*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[...
 

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 = 20.97 (sec) , antiderivative size = 1028, normalized size of antiderivative = 1.53

method	result	size
risch	\(\text {Expression too large to display}\)	\(1028\)
default	\(\text {Expression too large to display}\)	\(2277\)
elliptic	\(\text {Expression too large to display}\)	\(2317\)

Input:

int((B*x^2+A)/x^6/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x,method=_RET 
URNVERBOSE)
 

Output:

-1/15*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(15*A*a^2*c^2*f^2*x^4+10*A*a^2*c*d* 
e*f*x^4+8*A*a^2*d^2*e^2*x^4-10*A*a*b*c^2*e*f*x^4-7*A*a*b*c*d*e^2*x^4+8*A*b 
^2*c^2*e^2*x^4-15*B*a^2*c^2*e*f*x^4-10*B*a^2*c*d*e^2*x^4+10*B*a*b*c^2*e^2* 
x^4-5*A*a^2*c^2*e*f*x^2-4*A*a^2*c*d*e^2*x^2+4*A*a*b*c^2*e^2*x^2+5*B*a^2*c^ 
2*e^2*x^2+3*A*a^2*c^2*e^2)/a^3/c^3/e^3/x^5-1/15/c^3/a^3/e^3*(-b*(15*A*a^2* 
c^2*f^2+10*A*a^2*c*d*e*f+8*A*a^2*d^2*e^2-10*A*a*b*c^2*e*f-7*A*a*b*c*d*e^2+ 
8*A*b^2*c^2*e^2-15*B*a^2*c^2*e*f-10*B*a^2*c*d*e^2+10*B*a*b*c^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)*(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^3*c^3*f^2*(A*f-B*e)/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)*E 
llipticPi(x*(b/a)^(1/2),-a*f/b/e,(-1/c*d)^(1/2)/(b/a)^(1/2))-4*a*b^2*c^2*d 
*e^2*A/(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))+4*a^2* 
b*c*d^2*e^2*A/(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)) 
-5*a^2*b*c^2*d*e^2*B/(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))+5*b*c^2*e*a^2*d*A*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)*EllipticF(x*(b/a)^(1/2),(-1-(a*d...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x^2}{x^6 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((B*x^2+A)/x^6/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algo 
rithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate((B*x**2+A)/x**6/(-b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e), 
x)
 

Output:

Integral((A + B*x**2)/(x**6*sqrt(a - b*x**2)*sqrt(c + d*x**2)*(e + f*x**2) 
), x)
 

Maxima [F]

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

integrate((B*x^2+A)/x^6/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algo 
rithm="maxima")
 

Output:

integrate((B*x^2 + A)/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)*x^6), 
x)
 

Giac [F]

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

integrate((B*x^2+A)/x^6/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algo 
rithm="giac")
 

Output:

integrate((B*x^2 + A)/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)*x^6), 
x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int((A + B*x^2)/(x^6*(a - b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)),x)
 

Output:

int((A + B*x^2)/(x^6*(a - b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)), x)
 

Reduce [F]

\[ \int \frac {A+B x^2}{x^6 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:

int((B*x^2+A)/x^6/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x)
 

Output:

( - 3*sqrt(c + d*x**2)*sqrt(a - b*x**2)*a - 5*sqrt(c + d*x**2)*sqrt(a - b* 
x**2)*b*x**2 + 5*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**2)/(a*c*e + a*c 
*f*x**2 + a*d*e*x**2 + a*d*f*x**4 - b*c*e*x**2 - b*c*f*x**4 - b*d*e*x**4 - 
 b*d*f*x**6),x)*b**2*d*f*x**5 - 15*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)) 
/(a*c*e*x**4 + a*c*f*x**6 + a*d*e*x**6 + a*d*f*x**8 - b*c*e*x**6 - b*c*f*x 
**8 - b*d*e*x**8 - b*d*f*x**10),x)*a**2*c*f*x**5 - 12*int((sqrt(c + d*x**2 
)*sqrt(a - b*x**2))/(a*c*e*x**4 + a*c*f*x**6 + a*d*e*x**6 + a*d*f*x**8 - b 
*c*e*x**6 - b*c*f*x**8 - b*d*e*x**8 - b*d*f*x**10),x)*a**2*d*e*x**5 + 12*i 
nt((sqrt(c + d*x**2)*sqrt(a - b*x**2))/(a*c*e*x**4 + a*c*f*x**6 + a*d*e*x* 
*6 + a*d*f*x**8 - b*c*e*x**6 - b*c*f*x**8 - b*d*e*x**8 - b*d*f*x**10),x)*a 
*b*c*e*x**5 - 12*int((sqrt(c + d*x**2)*sqrt(a - b*x**2))/(a*c*e*x**2 + a*c 
*f*x**4 + a*d*e*x**4 + a*d*f*x**6 - b*c*e*x**4 - b*c*f*x**6 - b*d*e*x**6 - 
 b*d*f*x**8),x)*a**2*d*f*x**5 - 3*int((sqrt(c + d*x**2)*sqrt(a - b*x**2))/ 
(a*c*e*x**2 + a*c*f*x**4 + a*d*e*x**4 + a*d*f*x**6 - b*c*e*x**4 - b*c*f*x* 
*6 - b*d*e*x**6 - b*d*f*x**8),x)*a*b*c*f*x**5 - int((sqrt(c + d*x**2)*sqrt 
(a - b*x**2))/(a*c*e*x**2 + a*c*f*x**4 + a*d*e*x**4 + a*d*f*x**6 - b*c*e*x 
**4 - b*c*f*x**6 - b*d*e*x**6 - b*d*f*x**8),x)*a*b*d*e*x**5 + 10*int((sqrt 
(c + d*x**2)*sqrt(a - b*x**2))/(a*c*e*x**2 + a*c*f*x**4 + a*d*e*x**4 + a*d 
*f*x**6 - b*c*e*x**4 - b*c*f*x**6 - b*d*e*x**6 - b*d*f*x**8),x)*b**2*c*e*x 
**5 - int((sqrt(c + d*x**2)*sqrt(a - b*x**2))/(a*c*e + a*c*f*x**2 + a*d...