\(\int \frac {1}{(a+b x^2)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\) [171]

Optimal result

Integrand size = 32, antiderivative size = 424 \[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\frac {b x \sqrt {c+d x^2}}{2 a (b c-a d) \left (a+b x^2\right ) \sqrt {e+f x^2}}+\frac {b \sqrt {e} \sqrt {f} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{2 a (b c-a d) (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {f} (b e-2 a f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{2 a c (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {b e^{3/2} \left (b^2 c e+3 a^2 d f-2 a b (d e+c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {b e}{a f},\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{2 a^2 c (b c-a d) \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/2*b*x*(d*x^2+c)^(1/2)/a/(-a*d+b*c)/(b*x^2+a)/(f*x^2+e)^(1/2)+1/2*b*e^(1/ 
2)*f^(1/2)*(d*x^2+c)^(1/2)*EllipticE(f^(1/2)*x/e^(1/2)/(1+f*x^2/e)^(1/2),( 
1-d*e/c/f)^(1/2))/a/(-a*d+b*c)/(-a*f+b*e)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/ 
(f*x^2+e)^(1/2)-1/2*e^(1/2)*f^(1/2)*(-2*a*f+b*e)*(d*x^2+c)^(1/2)*InverseJa 
cobiAM(arctan(f^(1/2)*x/e^(1/2)),(1-d*e/c/f)^(1/2))/a/c/(-a*f+b*e)^2/(e*(d 
*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+1/2*b*e^(3/2)*(b^2*c*e+3*a^2*d* 
f-2*a*b*(c*f+d*e))*(d*x^2+c)^(1/2)*EllipticPi(f^(1/2)*x/e^(1/2)/(1+f*x^2/e 
)^(1/2),1-b*e/a/f,(1-d*e/c/f)^(1/2))/a^2/c/(-a*d+b*c)/f^(1/2)/(-a*f+b*e)^2 
/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.73 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\frac {\frac {b^2 c e x}{a+b x^2}+\frac {b^2 d e x^3}{a+b x^2}+\frac {b^2 c f x^3}{a+b x^2}+\frac {b^2 d f x^5}{a+b x^2}+i b c \sqrt {\frac {d}{c}} e \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i c \sqrt {\frac {d}{c}} (b e-a f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )-\frac {i b^2 c e \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )}{a \sqrt {\frac {d}{c}}}+2 i b c \sqrt {\frac {d}{c}} e \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )+\frac {2 i b c f \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )}{\sqrt {\frac {d}{c}}}-3 i a c \sqrt {\frac {d}{c}} f \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )}{2 a (-b c+a d) (-b e+a f) \sqrt {c+d x^2} \sqrt {e+f x^2}} \] Input:

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

Output:

((b^2*c*e*x)/(a + b*x^2) + (b^2*d*e*x^3)/(a + b*x^2) + (b^2*c*f*x^3)/(a + 
b*x^2) + (b^2*d*f*x^5)/(a + b*x^2) + I*b*c*Sqrt[d/c]*e*Sqrt[1 + (d*x^2)/c] 
*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*c* 
Sqrt[d/c]*(b*e - a*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I* 
ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (I*b^2*c*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 
 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] 
)/(a*Sqrt[d/c]) + (2*I)*b*c*Sqrt[d/c]*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^ 
2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + ((2*I 
)*b*c*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I* 
ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/Sqrt[d/c] - (3*I)*a*c*Sqrt[d/c]*f*Sqrt 
[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt 
[d/c]*x], (c*f)/(d*e)])/(2*a*(-(b*c) + a*d)*(-(b*e) + a*f)*Sqrt[c + d*x^2] 
*Sqrt[e + f*x^2])
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 447, 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.

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

\(\Big \downarrow \) 424

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 412

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

Input:

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

Output:

(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)) - (d*f*(b*((x*Sqrt[c + d*x^2])/(d*Sqrt[e + f*x^2]) - (Sqrt[e]*Sqrt 
[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(d*Sq 
rt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])) + (a*Sqrt[e] 
*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/ 
(c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])))/(2*a*( 
b*c - a*d)*(b*e - a*f)) + (Sqrt[-c]*(b^2*c*e + 3*a^2*d*f - 2*a*b*(d*e + c* 
f))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), ArcSin 
[(Sqrt[d]*x)/Sqrt[-c]], (c*f)/(d*e)])/(2*a^2*Sqrt[d]*(b*c - a*d)*(b*e - a* 
f)*Sqrt[c + d*x^2]*Sqrt[e + f*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. \(2(398)=796\).

Time = 8.59 (sec) , antiderivative size = 973, normalized size of antiderivative = 2.29

method	result	size
elliptic	\(\frac {\sqrt {\left (x^{2} d +c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {b^{2} x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{2 a \left (a^{2} d f -a b c f -a b d e +c e \,b^{2}\right ) \left (b \,x^{2}+a \right )}-\frac {d f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{2 \left (a^{2} d f -a b c f -a b d e +c e \,b^{2}\right ) \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {b d e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{2 \left (a^{2} d f -a b c f -a b d e +c e \,b^{2}\right ) a \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {b d e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{2 \left (a^{2} d f -a b c f -a b d e +c e \,b^{2}\right ) a \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {3 \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) d f}{2 \left (a^{2} d f -a b c f -a b d e +c e \,b^{2}\right ) \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {b \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) c f}{\left (a^{2} d f -a b c f -a b d e +c e \,b^{2}\right ) a \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {b \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) d e}{\left (a^{2} d f -a b c f -a b d e +c e \,b^{2}\right ) a \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {b^{2} \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) c e}{2 \left (a^{2} d f -a b c f -a b d e +c e \,b^{2}\right ) a^{2} \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{\sqrt {x^{2} d +c}\, \sqrt {f \,x^{2}+e}}\)	\(973\)
default	\(\text {Expression too large to display}\)	\(1078\)

Input:

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

Output:

((d*x^2+c)*(f*x^2+e))^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)*(1/2*b^2/a/(a^ 
2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)*x*(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)/(b*x^ 
2+a)-1/2/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)*d*f/(-1/c*d)^(1/2)*(1+d*x^2/c)^ 
(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*EllipticF(x*(- 
1/c*d)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))+1/2*b*d/(a^2*d*f-a*b*c*f-a*b*d*e+b^ 
2*c*e)/a*e/(-1/c*d)^(1/2)*(1+d*x^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f 
*x^2+d*e*x^2+c*e)^(1/2)*EllipticF(x*(-1/c*d)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2 
))-1/2*b*d/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/a*e/(-1/c*d)^(1/2)*(1+d*x^2/c 
)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*EllipticE(x* 
(-1/c*d)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))+3/2/(a^2*d*f-a*b*c*f-a*b*d*e+b^2* 
c*e)/(-1/c*d)^(1/2)*(1+d*x^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d 
*e*x^2+c*e)^(1/2)*EllipticPi(x*(-1/c*d)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-1/c*d 
)^(1/2))*d*f-1/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/a*b/(-1/c*d)^(1/2)*(1+d*x 
^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*Elliptic 
Pi(x*(-1/c*d)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-1/c*d)^(1/2))*c*f-1/(a^2*d*f-a* 
b*c*f-a*b*d*e+b^2*c*e)/a*b/(-1/c*d)^(1/2)*(1+d*x^2/c)^(1/2)*(1+f*x^2/e)^(1 
/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*EllipticPi(x*(-1/c*d)^(1/2),b*c/a/ 
d,(-f/e)^(1/2)/(-1/c*d)^(1/2))*d*e+1/2/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/a 
^2*b^2/(-1/c*d)^(1/2)*(1+d*x^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2 
+d*e*x^2+c*e)^(1/2)*EllipticPi(x*(-1/c*d)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((sqrt(e + f*x**2)*sqrt(c + d*x**2))/(a**2*c*e + a**2*c*f*x**2 + a**2*d 
*e*x**2 + a**2*d*f*x**4 + 2*a*b*c*e*x**2 + 2*a*b*c*f*x**4 + 2*a*b*d*e*x**4 
 + 2*a*b*d*f*x**6 + b**2*c*e*x**4 + b**2*c*f*x**6 + b**2*d*e*x**6 + b**2*d 
*f*x**8),x)