\(\int \frac {A+B x^2}{\sqrt {a+b x^2} (c-d x^2)^{3/2} (e+f x^2)} \, dx\) [23]

Optimal result

Integrand size = 40, antiderivative size = 390 \[ \int \frac {A+B x^2}{\sqrt {a+b x^2} \left (c-d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {d (B c+A d) x \sqrt {a+b x^2}}{c (b c+a d) (d e+c f) \sqrt {c-d x^2}}-\frac {a \sqrt {d} (B c+A d) \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{c^{3/2} (b c+a d) (d e+c f) \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}}}+\frac {(B c+A d) \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{c^{3/2} \sqrt {d} (d e+c f) \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}}}-\frac {\sqrt {c} (B e-A f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {c f}{d e},\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {d} e (d e+c f) \sqrt {a+b x^2} \sqrt {c-d x^2}} \] Output:

d*(A*d+B*c)*x*(b*x^2+a)^(1/2)/c/(a*d+b*c)/(c*f+d*e)/(-d*x^2+c)^(1/2)-a*d^( 
1/2)*(A*d+B*c)*(1+b*x^2/a)^(1/2)*(-d*x^2+c)^(1/2)*EllipticE(d^(1/2)*x/c^(1 
/2),(-b*c/a/d)^(1/2))/c^(3/2)/(a*d+b*c)/(c*f+d*e)/(b*x^2+a)^(1/2)/(1-d*x^2 
/c)^(1/2)+(A*d+B*c)*(1+b*x^2/a)^(1/2)*(-d*x^2+c)^(1/2)*EllipticF(d^(1/2)*x 
/c^(1/2),(-b*c/a/d)^(1/2))/c^(3/2)/d^(1/2)/(c*f+d*e)/(b*x^2+a)^(1/2)/(1-d* 
x^2/c)^(1/2)-c^(1/2)*(-A*f+B*e)*(1+b*x^2/a)^(1/2)*(1-d*x^2/c)^(1/2)*Ellipt 
icPi(d^(1/2)*x/c^(1/2),-c*f/d/e,(-b*c/a/d)^(1/2))/d^(1/2)/e/(c*f+d*e)/(b*x 
^2+a)^(1/2)/(-d*x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.11 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.62 \[ \int \frac {A+B x^2}{\sqrt {a+b x^2} \left (c-d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {\sqrt {\frac {b}{a}} d (B c+A d) e x \left (a+b x^2\right )-i b c (B c+A d) e \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+i c (b c+a d) (B e-A f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} c (b c+a d) e (d e+c f) \sqrt {a+b x^2} \sqrt {c-d x^2}} \] Input:

Integrate[(A + B*x^2)/(Sqrt[a + b*x^2]*(c - d*x^2)^(3/2)*(e + f*x^2)),x]
 

Output:

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

Rubi [A] (verified)

Time = 1.85 (sec) , antiderivative size = 750, normalized size of antiderivative = 1.92, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

Int[(A + B*x^2)/(Sqrt[a + b*x^2]*(c - d*x^2)^(3/2)*(e + f*x^2)),x]
 

Output:

(B*d*x*Sqrt[a + b*x^2])/(c*(b*c + a*d)*f*Sqrt[c - d*x^2]) - (d^2*(B*e - A* 
f)*x*Sqrt[a + b*x^2])/(c*(b*c + a*d)*f*(d*e + c*f)*Sqrt[c - d*x^2]) - (B*S 
qrt[d]*Sqrt[a + b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sq 
rt[c]], -((b*c)/(a*d))])/(Sqrt[c]*(b*c + a*d)*f*Sqrt[1 + (b*x^2)/a]*Sqrt[c 
 - d*x^2]) + (d^(3/2)*(B*e - A*f)*Sqrt[a + b*x^2]*Sqrt[1 - (d*x^2)/c]*Elli 
pticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[c]*(b*c + a*d)*f 
*(d*e + c*f)*Sqrt[1 + (b*x^2)/a]*Sqrt[c - d*x^2]) + (B*Sqrt[1 + (b*x^2)/a] 
*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d)) 
])/(Sqrt[c]*Sqrt[d]*f*Sqrt[a + b*x^2]*Sqrt[c - d*x^2]) + (Sqrt[c]*Sqrt[d]* 
(B*e - A*f)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqrt 
[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/((d*e + c*f)^2*Sqrt[a + b*x^2]*Sqrt[c - 
d*x^2]) - (Sqrt[d]*(B*e - A*f)*(d*e + 2*c*f)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - 
(d*x^2)/c]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[c 
]*f*(d*e + c*f)^2*Sqrt[a + b*x^2]*Sqrt[c - d*x^2]) - (Sqrt[c]*(B*e - A*f)* 
Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((c*f)/(d*e)), ArcSin[ 
(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[d]*e*(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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(751\) vs. \(2(341)=682\).

Time = 8.44 (sec) , antiderivative size = 752, normalized size of antiderivative = 1.93

method	result	size
elliptic	\(\frac {\sqrt {\left (-x^{2} d +c \right ) \left (b \,x^{2}+a \right )}\, \left (-\frac {\left (-b d \,x^{2}-a d \right ) x \left (A d +B c \right )}{c \left (a d +b c \right ) \left (c f +d e \right ) \sqrt {\left (x^{2}-\frac {c}{d}\right ) \left (-b d \,x^{2}-a d \right )}}+\frac {\sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right ) A d}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}\, c \left (c f +d e \right )}+\frac {\sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right ) B}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}\, \left (c f +d e \right )}-\frac {d^{2} a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right ) A}{c \left (a d +b c \right ) \left (c f +d e \right ) \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}-\frac {d a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right ) B}{\left (a d +b c \right ) \left (c f +d e \right ) \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}+\frac {f \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {d}{c}}, -\frac {c f}{d e}, \frac {\sqrt {-\frac {b}{a}}}{\sqrt {\frac {d}{c}}}\right ) A}{\left (c f +d e \right ) e \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}-\frac {\sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {d}{c}}, -\frac {c f}{d e}, \frac {\sqrt {-\frac {b}{a}}}{\sqrt {\frac {d}{c}}}\right ) B}{\left (c f +d e \right ) \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {-x^{2} d +c}\, \sqrt {b \,x^{2}+a}}\)	\(752\)
default	\(\frac {\left (A \sqrt {\frac {d}{c}}\, b \,d^{2} e \,x^{3}+B \sqrt {\frac {d}{c}}\, b c d e \,x^{3}+A \sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a \,d^{2} e +A \sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) b c d e -A \sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a \,d^{2} e +A \sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {d}{c}}, -\frac {c f}{d e}, \frac {\sqrt {-\frac {b}{a}}}{\sqrt {\frac {d}{c}}}\right ) a c d f +A \sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {d}{c}}, -\frac {c f}{d e}, \frac {\sqrt {-\frac {b}{a}}}{\sqrt {\frac {d}{c}}}\right ) b \,c^{2} f +B \sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a c d e +B \sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) b \,c^{2} e -B \sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a c d e -B \sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {d}{c}}, -\frac {c f}{d e}, \frac {\sqrt {-\frac {b}{a}}}{\sqrt {\frac {d}{c}}}\right ) a c d e -B \sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {d}{c}}, -\frac {c f}{d e}, \frac {\sqrt {-\frac {b}{a}}}{\sqrt {\frac {d}{c}}}\right ) b \,c^{2} e +A \sqrt {\frac {d}{c}}\, a \,d^{2} e x +B \sqrt {\frac {d}{c}}\, a c d e x \right ) \sqrt {-x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{e \sqrt {\frac {d}{c}}\, c \left (c f +d e \right ) \left (a d +b c \right ) \left (-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c \right )}\)	\(756\)

Input:

int((B*x^2+A)/(b*x^2+a)^(1/2)/(-d*x^2+c)^(3/2)/(f*x^2+e),x,method=_RETURNV 
ERBOSE)
 

Output:

((-d*x^2+c)*(b*x^2+a))^(1/2)/(-d*x^2+c)^(1/2)/(b*x^2+a)^(1/2)*(-(-b*d*x^2- 
a*d)/c/(a*d+b*c)*x*(A*d+B*c)/(c*f+d*e)/((x^2-c/d)*(-b*d*x^2-a*d))^(1/2)+1/ 
(1/c*d)^(1/2)*(1-d*x^2/c)^(1/2)*(1+b*x^2/a)^(1/2)/(-b*d*x^4-a*d*x^2+b*c*x^ 
2+a*c)^(1/2)*EllipticF(x*(1/c*d)^(1/2),(-1-(-a*d+b*c)/a/d)^(1/2))/c/(c*f+d 
*e)*A*d+1/(1/c*d)^(1/2)*(1-d*x^2/c)^(1/2)*(1+b*x^2/a)^(1/2)/(-b*d*x^4-a*d* 
x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(1/c*d)^(1/2),(-1-(-a*d+b*c)/a/d)^(1/2) 
)/(c*f+d*e)*B-d^2/c/(a*d+b*c)/(c*f+d*e)*a/(1/c*d)^(1/2)*(1-d*x^2/c)^(1/2)* 
(1+b*x^2/a)^(1/2)/(-b*d*x^4-a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticE(x*(1/c*d) 
^(1/2),(-1-(-a*d+b*c)/a/d)^(1/2))*A-d/(a*d+b*c)/(c*f+d*e)*a/(1/c*d)^(1/2)* 
(1-d*x^2/c)^(1/2)*(1+b*x^2/a)^(1/2)/(-b*d*x^4-a*d*x^2+b*c*x^2+a*c)^(1/2)*E 
llipticE(x*(1/c*d)^(1/2),(-1-(-a*d+b*c)/a/d)^(1/2))*B+1/(c*f+d*e)/e*f/(1/c 
*d)^(1/2)*(1-d*x^2/c)^(1/2)*(1+b*x^2/a)^(1/2)/(-b*d*x^4-a*d*x^2+b*c*x^2+a* 
c)^(1/2)*EllipticPi(x*(1/c*d)^(1/2),-c*f/d/e,(-b/a)^(1/2)/(1/c*d)^(1/2))*A 
-1/(c*f+d*e)/(1/c*d)^(1/2)*(1-d*x^2/c)^(1/2)*(1+b*x^2/a)^(1/2)/(-b*d*x^4-a 
*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticPi(x*(1/c*d)^(1/2),-c*f/d/e,(-b/a)^(1/2) 
/(1/c*d)^(1/2))*B)
 

Fricas [F(-1)]

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

integrate((B*x^2+A)/(b*x^2+a)^(1/2)/(-d*x^2+c)^(3/2)/(f*x^2+e),x, algorith 
m="fricas")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral((A + B*x**2)/(sqrt(a + b*x**2)*(c - d*x**2)**(3/2)*(e + f*x**2)), 
 x)
 

Maxima [F]

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

integrate((B*x^2+A)/(b*x^2+a)^(1/2)/(-d*x^2+c)^(3/2)/(f*x^2+e),x, algorith 
m="maxima")
 

Output:

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

Giac [F]

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

integrate((B*x^2+A)/(b*x^2+a)^(1/2)/(-d*x^2+c)^(3/2)/(f*x^2+e),x, algorith 
m="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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