\(\int \frac {1}{(a+b x^2)^2 \sqrt {c-d x^2} \sqrt {e+f x^2}} \, dx\) [101]

Optimal result

Integrand size = 33, antiderivative size = 426 \[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c-d x^2} \sqrt {e+f x^2}} \, dx=\frac {b^2 x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a (b c+a d) (b e-a f) \left (a+b x^2\right )}+\frac {b \sqrt {c} \sqrt {d} \sqrt {1-\frac {d x^2}{c}} \sqrt {e+f x^2} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{2 a (b c+a d) (b e-a f) \sqrt {c-d x^2} \sqrt {1+\frac {f x^2}{e}}}-\frac {\sqrt {c} \sqrt {d} \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {c f}{d e}\right )}{2 a (b c+a d) \sqrt {c-d x^2} \sqrt {e+f x^2}}+\frac {\sqrt {c} \left (b^2 c e-3 a^2 d f+a b (2 d e-2 c f)\right ) \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (-\frac {b c}{a d},\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {c f}{d e}\right )}{2 a^2 \sqrt {d} (b c+a d) (b e-a f) \sqrt {c-d x^2} \sqrt {e+f x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {563, 552, 551, 538, 438, 437, 435, 432, 430} \[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c-d x^2} \sqrt {e+f x^2}} \, dx=\frac {\sqrt {c} \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} \left (-3 a^2 d f+a b (2 d e-2 c f)+b^2 c e\right ) \operatorname {EllipticPi}\left (-\frac {b c}{a d},\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {c f}{d e}\right )}{2 a^2 \sqrt {d} \sqrt {c-d x^2} \sqrt {e+f x^2} (a d+b c) (b e-a f)}-\frac {\sqrt {c} \sqrt {d} \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {c f}{d e}\right )}{2 a \sqrt {c-d x^2} \sqrt {e+f x^2} (a d+b c)}+\frac {b \sqrt {c} \sqrt {d} \sqrt {1-\frac {d x^2}{c}} \sqrt {e+f x^2} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{2 a \sqrt {c-d x^2} \sqrt {\frac {f x^2}{e}+1} (a d+b c) (b e-a f)}+\frac {b^2 x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right ) (a d+b c) (b e-a f)} \]

[In]

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

[Out]

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

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 432

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

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 437

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

Rule 438

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

Rule 538

Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c]))))))

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[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] && SimplerSqrtQ[-f/e, -d/c])

Rule 552

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[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 563

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

Rubi steps \begin{align*} \text {integral}& = \frac {b^2 x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a (b c+a d) (b e-a f) \left (a+b x^2\right )}+\frac {(d f) \int \frac {a+b x^2}{\sqrt {c-d x^2} \sqrt {e+f x^2}} \, dx}{2 a (b c+a d) (b e-a f)}+\frac {\left (b^2 c e-3 a^2 d f-2 a b (-d e+c f)\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c-d x^2} \sqrt {e+f x^2}} \, 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 (b c+a d) (b e-a f) \left (a+b x^2\right )}-\frac {d \int \frac {1}{\sqrt {c-d x^2} \sqrt {e+f x^2}} \, dx}{2 a (b c+a d)}+\frac {(b d) \int \frac {\sqrt {e+f x^2}}{\sqrt {c-d x^2}} \, dx}{2 a (b c+a d) (b e-a f)}+\frac {\left (\left (b^2 c e-3 a^2 d f-2 a b (-d e+c f)\right ) \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \sqrt {e+f x^2}} \, dx}{2 a (b c+a d) (b e-a f) \sqrt {c-d x^2}} \\ & = \frac {b^2 x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a (b c+a d) (b e-a f) \left (a+b x^2\right )}+\frac {\left (b d \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {\sqrt {e+f x^2}}{\sqrt {1-\frac {d x^2}{c}}} \, dx}{2 a (b c+a d) (b e-a f) \sqrt {c-d x^2}}-\frac {\left (d \sqrt {1+\frac {f x^2}{e}}\right ) \int \frac {1}{\sqrt {c-d x^2} \sqrt {1+\frac {f x^2}{e}}} \, dx}{2 a (b c+a d) \sqrt {e+f x^2}}+\frac {\left (\left (b^2 c e-3 a^2 d f-2 a b (-d e+c f)\right ) \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}}\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}}} \, dx}{2 a (b c+a d) (b e-a f) \sqrt {c-d x^2} \sqrt {e+f x^2}} \\ & = \frac {b^2 x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a (b c+a d) (b e-a f) \left (a+b x^2\right )}+\frac {\sqrt {c} \left (b^2 c e-3 a^2 d f+a b (2 d e-2 c f)\right ) \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (-\frac {b c}{a d};\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{2 a^2 \sqrt {d} (b c+a d) (b e-a f) \sqrt {c-d x^2} \sqrt {e+f x^2}}+\frac {\left (b d \sqrt {1-\frac {d x^2}{c}} \sqrt {e+f x^2}\right ) \int \frac {\sqrt {1+\frac {f x^2}{e}}}{\sqrt {1-\frac {d x^2}{c}}} \, dx}{2 a (b c+a d) (b e-a f) \sqrt {c-d x^2} \sqrt {1+\frac {f x^2}{e}}}-\frac {\left (d \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}}\right ) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}}} \, dx}{2 a (b c+a d) \sqrt {c-d x^2} \sqrt {e+f x^2}} \\ & = \frac {b^2 x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a (b c+a d) (b e-a f) \left (a+b x^2\right )}+\frac {b \sqrt {c} \sqrt {d} \sqrt {1-\frac {d x^2}{c}} \sqrt {e+f x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{2 a (b c+a d) (b e-a f) \sqrt {c-d x^2} \sqrt {1+\frac {f x^2}{e}}}-\frac {\sqrt {c} \sqrt {d} \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{2 a (b c+a d) \sqrt {c-d x^2} \sqrt {e+f x^2}}+\frac {\sqrt {c} \left (b^2 c e-3 a^2 d f+a b (2 d e-2 c f)\right ) \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (-\frac {b c}{a d};\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{2 a^2 \sqrt {d} (b c+a d) (b e-a f) \sqrt {c-d x^2} \sqrt {e+f x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.21 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.45 \[ \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c-d x^2} \sqrt {e+f x^2}} \, dx=\frac {-\frac {b^2 c e x}{a+b x^2}+\frac {b^2 d e x^3}{a+b x^2}-\frac {b^2 c f x^3}{a+b x^2}+\frac {b^2 d f x^5}{a+b x^2}-i b c \sqrt {-\frac {d}{c}} e \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {d}{c}} x\right )|-\frac {c f}{d e}\right )+i c \sqrt {-\frac {d}{c}} (b e-a f) \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {d}{c}} x\right ),-\frac {c f}{d e}\right )+\frac {i b^2 c e \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (-\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {-\frac {d}{c}} x\right ),-\frac {c f}{d e}\right )}{a \sqrt {-\frac {d}{c}}}-2 i b c \sqrt {-\frac {d}{c}} e \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (-\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {-\frac {d}{c}} x\right ),-\frac {c f}{d e}\right )+\frac {2 i b d f \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (-\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {-\frac {d}{c}} x\right ),-\frac {c f}{d e}\right )}{\left (-\frac {d}{c}\right )^{3/2}}+3 i a c \sqrt {-\frac {d}{c}} f \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (-\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {-\frac {d}{c}} x\right ),-\frac {c f}{d e}\right )}{2 a (b c+a d) (-b e+a f) \sqrt {c-d x^2} \sqrt {e+f x^2}} \]

[In]

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

[Out]

(-((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*ArcSin
h[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*d*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))])/(-(d/c))^(3/2)
+ (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])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(988\) vs. \(2(369)=738\).

Time = 6.06 (sec) , antiderivative size = 989, normalized size of antiderivative = 2.32

method	result	size
elliptic	\(\frac {\sqrt {\left (-d \,x^{2}+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 \left (a^{2} d f +a c f b -a b d e -b^{2} c e \right ) a \left (b \,x^{2}+a \right )}-\frac {d f \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {c f -d e}{e d}}\right )}{2 \left (a^{2} d f +a c f b -a b d e -b^{2} c e \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}}\, F\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {c f -d e}{e d}}\right )}{2 \left (a^{2} d f +a c f b -a b d e -b^{2} c e \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}}\, E\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {c f -d e}{e d}}\right )}{2 \left (a^{2} d f +a c f b -a b d e -b^{2} c e \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}}\, \Pi \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 c f b -a b d e -b^{2} c e \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}}\, \Pi \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 c f b -a b d e -b^{2} c e \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}}\, \Pi \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 c f b -a b d e -b^{2} c e \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}}\, \Pi \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 c f b -a b d e -b^{2} c e \right ) a^{2} \sqrt {\frac {d}{c}}\, \sqrt {-d f \,x^{4}+c f \,x^{2}-d e \,x^{2}+c e}}\right )}{\sqrt {-d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(989\)
default	\(\text {Expression too large to display}\)	\(1077\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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