\(\int \frac {A+B x^2}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2} (e+f x^2)} \, dx\) [7]

Optimal result

Integrand size = 42, antiderivative size = 706 \[ \int \frac {A+B x^2}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {\left (5 a B c e (2 b c e+2 a d e+3 a c f)-A \left (8 b^2 c^2 e^2+a b c e (7 d e+10 c f)+a^2 \left (8 d^2 e^2+10 c d e f+15 c^2 f^2\right )\right )\right ) \sqrt {a+b x^2}}{15 a^3 c^2 e^3 x \sqrt {c+d x^2}}-\frac {A \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 a c e x^5}-\frac {(5 a B c e-A (4 b c e+4 a d e+5 a c f)) \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 a^2 c^2 e^2 x^3}+\frac {\sqrt {d} \left (5 a B c e (2 b c e+2 a d e+3 a c f)-A \left (8 b^2 c^2 e^2+a b c e (7 d e+10 c f)+a^2 \left (8 d^2 e^2+10 c d e f+15 c^2 f^2\right )\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 a^3 c^{5/2} e^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\sqrt {d} \left (5 a B c e \left (3 a c f^2-b e (d e-c f)\right )-A \left (15 a^2 c^2 f^3-4 b^2 c e^2 (d e-c f)-a b e \left (4 d^2 e^2+c d e f-5 c^2 f^2\right )\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 a^3 c^{3/2} e^3 (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} f^3 (B e-A f) \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e^4 (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

1/15*(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)))*(b*x^2+a)^(1/2)/a^3/c^2/e^ 
3/x/(d*x^2+c)^(1/2)-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+1/15*d^(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)))*(b*x^ 
2+a)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2) 
)/a^3/c^(5/2)/e^3/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+1/15*d^( 
1/2)*(5*a*B*c*e*(3*a*c*f^2-b*e*(-c*f+d*e))-A*(15*a^2*c^2*f^3-4*b^2*c*e^2*( 
-c*f+d*e)-a*b*e*(-5*c^2*f^2+c*d*e*f+4*d^2*e^2)))*(b*x^2+a)^(1/2)*InverseJa 
cobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/a^3/c^(3/2)/e^3/(-c*f+ 
d*e)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-c^(3/2)*f^3*(-A*f+B*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^4/(-c*f+d*e)/(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 = 12.39 (sec) , antiderivative size = 1419, normalized size of antiderivative = 2.01 \[ \int \frac {A+B x^2}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 2.76 (sec) , antiderivative size = 1087, normalized size of antiderivative = 1.54, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, 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 {\frac {b x^2}{a}+1} \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 {b x^2+a} \sqrt {d x^2+c}}+\frac {\sqrt {d} (B e-A f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right ) f}{a \sqrt {c} e^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {(B e-A f) \sqrt {b x^2+a} \sqrt {d x^2+c} f}{a c e^3 x}-\frac {d (B e-A f) x \sqrt {b x^2+a} f}{a c e^3 \sqrt {d x^2+c}}+\frac {2 \sqrt {d} (b c+a d) (B e-A 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 a^2 c^{3/2} e^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {A \sqrt {d} \left (8 b^2 c^2+7 a b d c+8 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 a^3 c^{5/2} e \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \sqrt {d} (B e-A 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 a^2 \sqrt {c} e^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {4 A b \sqrt {d} (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 a^3 c^{3/2} e \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 (b c+a d) (B e-A f) \sqrt {b x^2+a} \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 {b x^2+a} \sqrt {d x^2+c}}{15 a^3 c^3 e x}-\frac {(B e-A f) \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 a c e^2 x^3}+\frac {4 A (b c+a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}{15 a^2 c^2 e x^3}-\frac {A \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 a c e x^5}-\frac {2 d (b c+a d) (B e-A f) x \sqrt {b x^2+a}}{3 a^2 c^2 e^2 \sqrt {d x^2+c}}+\frac {A d \left (8 b^2 c^2+7 a b d c+8 a^2 d^2\right ) x \sqrt {b x^2+a}}{15 a^3 c^3 e \sqrt {d x^2+c}}\)

Input:

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

Output:

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

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 = 20.56 (sec) , antiderivative size = 1008, normalized size of antiderivative = 1.43

method	result	size
risch	\(\text {Expression too large to display}\)	\(1008\)
elliptic	\(\text {Expression too large to display}\)	\(2253\)
default	\(\text {Expression too large to display}\)	\(2295\)

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=_RETU 
RNVERBOSE)
 

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)* 
EllipticPi(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),(-...
 

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

\[ \int \frac {A+B x^2}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{d f \,x^{10}+c f \,x^{8}+d e \,x^{8}+c e \,x^{6}}d x \] 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:

int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(c*e*x**6 + c*f*x**8 + d*e*x**8 + 
d*f*x**10),x)