\(\int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2} (e+f x^2)^2} \, dx\) [193]

Optimal result

Integrand size = 32, antiderivative size = 426 \[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=-\frac {f x \sqrt {a+b x^2}}{2 e (b e-a f) \sqrt {c+d x^2} \left (e+f x^2\right )}+\frac {\sqrt {c} \sqrt {d} f \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{2 e (b e-a f) (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\sqrt {c} \sqrt {d} (2 d e-c f) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{2 a e (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} f (b e (3 d e-2 c f)-a f (2 d e-c f)) \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{2 a \sqrt {d} e^2 (b e-a f) (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

-1/2*f*x*(b*x^2+a)^(1/2)/e/(-a*f+b*e)/(d*x^2+c)^(1/2)/(f*x^2+e)+1/2*c^(1/2 
)*d^(1/2)*f*(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))/e/(-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)*d^(1/2)*(-c*f+2*d*e)*(b*x^2+a)^(1/2)*InverseJ 
acobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/a/e/(-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)*f*(b*e*(-2*c*f+3*d 
*e)-a*f*(-c*f+2*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/(-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 = 4.21 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\frac {\frac {a c f^2 x}{e+f x^2}+\frac {b c f^2 x^3}{e+f x^2}+\frac {a d f^2 x^3}{e+f x^2}+\frac {b d f^2 x^5}{e+f x^2}+i a \sqrt {\frac {b}{a}} c f \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i a \sqrt {\frac {b}{a}} (-d e+c f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-3 i a \sqrt {\frac {b}{a}} d e \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+2 i a \sqrt {\frac {b}{a}} c f \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+\frac {2 i a d f \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}}}-\frac {i a c f^2 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} e}}{2 e (b e-a f) (d e-c f) \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^2),x]
 

Output:

((a*c*f^2*x)/(e + f*x^2) + (b*c*f^2*x^3)/(e + f*x^2) + (a*d*f^2*x^3)/(e + 
f*x^2) + (b*d*f^2*x^5)/(e + f*x^2) + I*a*Sqrt[b/a]*c*f*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* 
Sqrt[b/a]*(-(d*e) + c*f)*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*Sqrt[b/a]*d*e*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*Sqrt[b/a]*c*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*d*f*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), I*A 
rcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/Sqrt[b/a] - (I*a*c*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)])/(Sqrt[b/a]*e))/(2*e*(b*e - a*f)*(d*e - c*f)*Sqrt[a + b*x^2]*Sq 
rt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {424, 406, 320, 388, 313, 413, 413, 412}

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.

\(\displaystyle \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx\)

\(\Big \downarrow \) 424

\(\displaystyle \frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}-\frac {b d \int \frac {f x^2+e}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{2 e (b e-a f) (d e-c f)}+\frac {f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right ) (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 406

\(\displaystyle -\frac {b d \left (e \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+f \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}+\frac {f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right ) (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 320

\(\displaystyle -\frac {b d \left (f \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {\sqrt {c} e \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}+\frac {f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right ) (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 388

\(\displaystyle -\frac {b d \left (f \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {\sqrt {c} e \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}+\frac {f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right ) (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 313

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 412

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

Input:

Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^2),x]
 

Output:

(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)) - (b*d*(f*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt 
[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sq 
rt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (Sqrt[c]*e 
*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/ 
(a*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])))/(2*e*( 
b*e - a*f)*(d*e - c*f)) + (Sqrt[-a]*(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)/Sqrt[-a]], (a*d)/(b*c)])/(2*Sqrt[b]*e^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 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 406

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[e   Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim 
p[f   Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, 
f, p, q}, x]
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 413

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/((a + 
 b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] &&  !GtQ[c, 0]
 

rule 424

Int[1/(((a_) + (b_.)*(x_)^2)^2*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)* 
(x_)^2]), x_Symbol] :> Simp[b^2*x*Sqrt[c + d*x^2]*(Sqrt[e + f*x^2]/(2*a*(b* 
c - a*d)*(b*e - a*f)*(a + b*x^2))), x] + (Simp[(b^2*c*e + 3*a^2*d*f - 2*a*b 
*(d*e + c*f))/(2*a*(b*c - a*d)*(b*e - a*f))   Int[1/((a + b*x^2)*Sqrt[c + d 
*x^2]*Sqrt[e + f*x^2]), x], x] - Simp[d*(f/(2*a*(b*c - a*d)*(b*e - a*f))) 
 Int[(a + b*x^2)/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x]) /; FreeQ[{a, b, 
 c, d, e, f}, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(972\) vs. \(2(400)=800\).

Time = 8.60 (sec) , antiderivative size = 973, normalized size of antiderivative = 2.28

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {f^{2} x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{2 \left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) e \left (f \,x^{2}+e \right )}-\frac {d b \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 )}{2 \left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {f b c \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 )}{2 \left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {f b c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{2 \left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {f^{2} \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 ) a c}{2 \left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) e^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {f \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 ) a d}{\left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {f \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 ) b c}{\left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {3 \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 ) b d}{2 \left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(973\)
default	\(\text {Expression too large to display}\)	\(1074\)

Input:

int(1/(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*f^2/(a*c* 
f^2-a*d*e*f-b*c*e*f+b*d*e^2)/e*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(f*x^ 
2+e)-1/2*d*b/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^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))+1/2*f*b/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^ 
2)/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))-1/2*f 
*b/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/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)*EllipticE(x*(-b/a)^(1/ 
2),(-1+(a*d+b*c)/c/b)^(1/2))+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)) 
*a*c-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))*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)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2) 
/(-b/a)^(1/2))*b*c+3/2/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^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)*Ellipt 
icPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*b*d)
 

Fricas [F(-1)]

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

integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="fric 
as")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {1}{\sqrt {a + b x^{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{2}}\, dx \] Input:

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

Output:

Integral(1/(sqrt(a + b*x**2)*sqrt(c + d*x**2)*(e + f*x**2)**2), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="maxi 
ma")
 

Output:

integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^2), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="giac 
")
 

Output:

integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {1}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,f^{2} x^{8}+a d \,f^{2} x^{6}+b c \,f^{2} x^{6}+2 b d e f \,x^{6}+a c \,f^{2} x^{4}+2 a d e f \,x^{4}+2 b c e f \,x^{4}+b d \,e^{2} x^{4}+2 a c e f \,x^{2}+a d \,e^{2} x^{2}+b c \,e^{2} x^{2}+a c \,e^{2}}d x \] Input:

int(1/(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))/(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)