\(\int \frac {A+B x^2}{x^4 \sqrt {a-b x^2} \sqrt {c+d x^2} (e+f x^2)} \, dx\) [13]

Optimal result

Integrand size = 43, antiderivative size = 457 \[ \int \frac {A+B x^2}{x^4 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=-\frac {A \sqrt {a-b x^2} \sqrt {c+d x^2}}{3 a c e x^3}-\frac {(3 a B c e+A (2 b c e-2 a d e-3 a c f)) \sqrt {a-b x^2} \sqrt {c+d x^2}}{3 a^2 c^2 e^2 x}-\frac {\sqrt {b} (3 a B c e+A (2 b c e-2 a d e-3 a c f)) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{3 a^{3/2} c^2 e^2 \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {\sqrt {b} (2 A b c e+3 a B c e-a A (d e+3 c f)) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{3 a^{3/2} c e^2 \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} f (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} e^3 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:

-1/3*A*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/e/x^3-1/3*(3*B*a*c*e+A*(-3*a*c 
*f-2*a*d*e+2*b*c*e))*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a^2/c^2/e^2/x-1/3*b^ 
(1/2)*(3*B*a*c*e+A*(-3*a*c*f-2*a*d*e+2*b*c*e))*(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^(3/2)/c^2/e^2/(-b*x 
^2+a)^(1/2)/(1+d*x^2/c)^(1/2)+1/3*b^(1/2)*(2*A*b*c*e+3*B*a*c*e-a*A*(3*c*f+ 
d*e))*(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^(3/2)/c/e^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)-a^(1/2)*f*(-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^3/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 14.66 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.43 \[ \int \frac {A+B x^2}{x^4 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=-\frac {a A b c^2 e^2+A b^2 c^2 e^2 x^2+3 a b B c^2 e^2 x^2-a A b c d e^2 x^2-3 a A b c^2 e f x^2-\frac {2 A b^3 c^2 e^2 x^4}{a}-3 b^2 B c^2 e^2 x^4+3 A b^2 c d e^2 x^4+3 a b B c d e^2 x^4-2 a A b d^2 e^2 x^4+3 A b^2 c^2 e f x^4-3 a A b c d e f x^4-\frac {2 A b^3 c d e^2 x^6}{a}-3 b^2 B c d e^2 x^6+2 A b^2 d^2 e^2 x^6+3 A b^2 c d e f x^6+i b \sqrt {-\frac {b}{a}} c e (3 a B c e+A (2 b c e-2 a d e-3 a c f)) x^3 \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+i b \sqrt {-\frac {b}{a}} c e (-3 a B c e+A (-2 b c e+a d e+3 a c f)) x^3 \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )+3 i a^2 \sqrt {-\frac {b}{a}} B c^2 e f x^3 \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )+\frac {3 i a A b c^2 f^2 x^3 \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}}}}{3 a b c^2 e^3 x^3 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

-1/3*(a*A*b*c^2*e^2 + A*b^2*c^2*e^2*x^2 + 3*a*b*B*c^2*e^2*x^2 - a*A*b*c*d* 
e^2*x^2 - 3*a*A*b*c^2*e*f*x^2 - (2*A*b^3*c^2*e^2*x^4)/a - 3*b^2*B*c^2*e^2* 
x^4 + 3*A*b^2*c*d*e^2*x^4 + 3*a*b*B*c*d*e^2*x^4 - 2*a*A*b*d^2*e^2*x^4 + 3* 
A*b^2*c^2*e*f*x^4 - 3*a*A*b*c*d*e*f*x^4 - (2*A*b^3*c*d*e^2*x^6)/a - 3*b^2* 
B*c*d*e^2*x^6 + 2*A*b^2*d^2*e^2*x^6 + 3*A*b^2*c*d*e*f*x^6 + I*b*Sqrt[-(b/a 
)]*c*e*(3*a*B*c*e + A*(2*b*c*e - 2*a*d*e - 3*a*c*f))*x^3*Sqrt[1 - (b*x^2)/ 
a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c)) 
] + I*b*Sqrt[-(b/a)]*c*e*(-3*a*B*c*e + A*(-2*b*c*e + a*d*e + 3*a*c*f))*x^3 
*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[-(b/a)]* 
x], -((a*d)/(b*c))] + (3*I)*a^2*Sqrt[-(b/a)]*B*c^2*e*f*x^3*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*A*b*c^2*f^2*x^3*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))])/Sqrt[-(b/a)])/(a*b*c^2*e^3*x^3*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 2.64 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.43, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/3*(A*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])/(a*c*e*x^3) - (2*A*(b*c - a*d)*Sq 
rt[a - b*x^2]*Sqrt[c + d*x^2])/(3*a^2*c^2*e*x) - ((B*e - A*f)*Sqrt[a - b*x 
^2]*Sqrt[c + d*x^2])/(a*c*e^2*x) - (2*A*Sqrt[b]*(b*c - a*d)*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*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c]) - (Sqrt[b]*(B*e 
- A*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sq 
rt[a]], -((a*d)/(b*c))])/(Sqrt[a]*c*e^2*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c 
]) + (A*Sqrt[b]*(2*b*c - a*d)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Elli 
pticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(3*a^(3/2)*c*e*Sqrt[a 
- b*x^2]*Sqrt[c + d*x^2]) + (Sqrt[b]*(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))])/(Sq 
rt[a]*e^2*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]) - (Sqrt[a]*f*(B*e - A*f)*Sqrt[1 
 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[-((a*f)/(b*e)), ArcSin[(Sqrt[ 
b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*e^3*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 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 = 10.46 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.06

method	result	size
risch	\(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (-3 A a c f \,x^{2}-2 A a d e \,x^{2}+2 A b c e \,x^{2}+3 B a c e \,x^{2}+A a c e \right )}{3 a^{2} c^{2} e^{2} x^{3}}+\frac {\left (-\frac {b \left (3 A a c f +2 A a d e -2 A b c e -3 B a c e \right ) c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {a b c d e A \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {3 a^{2} c^{2} f \left (A f -B e \right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right )}{e \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 a^{2} c^{2} e^{2} \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(485\)
default	\(\text {Expression too large to display}\)	\(1053\)
elliptic	\(\text {Expression too large to display}\)	\(1145\)

Input:

int((B*x^2+A)/x^4/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x,method=_RET 
URNVERBOSE)
 

Output:

-1/3*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(-3*A*a*c*f*x^2-2*A*a*d*e*x^2+2*A*b* 
c*e*x^2+3*B*a*c*e*x^2+A*a*c*e)/a^2/c^2/e^2/x^3+1/3/a^2/c^2/e^2*(-b*(3*A*a* 
c*f+2*A*a*d*e-2*A*b*c*e-3*B*a*c*e)*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)) 
)+a*b*c*d*e*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)) 
+3*a^2*c^2*f*(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)))*((-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. \[ \int \frac {A+B x^2}{x^4 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((B*x^2+A)/x^4/(-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]

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

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

Output:

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

Maxima [F]

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

integrate((B*x^2+A)/x^4/(-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^4), 
x)
 

Giac [F]

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

integrate((B*x^2+A)/x^4/(-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^4), 
x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x^2}{x^4 \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {-\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}+\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d f \,x^{6}+a d f \,x^{4}-b c f \,x^{4}-b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}-b c e \,x^{2}+a c e}d x \right ) b d f \,x^{3}-3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{-b d f \,x^{8}+a d f \,x^{6}-b c f \,x^{6}-b d e \,x^{6}+a c f \,x^{4}+a d e \,x^{4}-b c e \,x^{4}+a c e \,x^{2}}d x \right ) a c f \,x^{3}-2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{-b d f \,x^{8}+a d f \,x^{6}-b c f \,x^{6}-b d e \,x^{6}+a c f \,x^{4}+a d e \,x^{4}-b c e \,x^{4}+a c e \,x^{2}}d x \right ) a d e \,x^{3}+5 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{-b d f \,x^{8}+a d f \,x^{6}-b c f \,x^{6}-b d e \,x^{6}+a c f \,x^{4}+a d e \,x^{4}-b c e \,x^{4}+a c e \,x^{2}}d x \right ) b c e \,x^{3}-2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{-b d f \,x^{6}+a d f \,x^{4}-b c f \,x^{4}-b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}-b c e \,x^{2}+a c e}d x \right ) a d f \,x^{3}+2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{-b d f \,x^{6}+a d f \,x^{4}-b c f \,x^{4}-b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}-b c e \,x^{2}+a c e}d x \right ) b c f \,x^{3}+\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{-b d f \,x^{6}+a d f \,x^{4}-b c f \,x^{4}-b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}-b c e \,x^{2}+a c e}d x \right ) b d e \,x^{3}}{3 c e \,x^{3}} \] Input:

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

Output:

( - sqrt(c + d*x**2)*sqrt(a - b*x**2) + 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*d*f*x**3 - 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*c*f*x**3 - 2*int((s 
qrt(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*d*e*x 
**3 + 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)*b*c*e*x**3 - 2*int((sqrt(c + d*x**2)*sqrt(a - b*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)*a*d*f*x**3 + 2*int((sqrt(c + d*x**2)*sqrt(a - b*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*c*f*x**3 + int((sqrt(c + d*x**2)*sqrt(a - b*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*d*e*x**3)/(3*c*e*x**3)