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

Optimal. Leaf size=426 \[ -\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),-\frac{c f}{d e}\right )}{2 a \sqrt{c-d x^2} \sqrt{e+f x^2} (a d+b c)}+\frac{\sqrt{c} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \left (-3 a^2 d f+a b (2 d e-2 c f)+b^2 c e\right ) \Pi \left (-\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a^2 \sqrt{d} \sqrt{c-d x^2} \sqrt{e+f x^2} (a d+b c) (b e-a f)}+\frac{b^2 x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right ) (a d+b c) (b e-a f)}+\frac{b \sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{e+f x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a \sqrt{c-d x^2} \sqrt{\frac{f x^2}{e}+1} (a d+b c) (b e-a f)} \]

[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])

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Rubi [A]  time = 0.373077, antiderivative size = 426, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {549, 524, 427, 426, 424, 421, 419, 538, 537} \[ \frac{\sqrt{c} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \left (-3 a^2 d f+a b (2 d e-2 c f)+b^2 c e\right ) \Pi \left (-\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a^2 \sqrt{d} \sqrt{c-d x^2} \sqrt{e+f x^2} (a d+b c) (b e-a f)}+\frac{b^2 x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right ) (a d+b c) (b e-a f)}-\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a \sqrt{c-d x^2} \sqrt{e+f x^2} (a d+b c)}+\frac{b \sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{e+f x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a \sqrt{c-d x^2} \sqrt{\frac{f x^2}{e}+1} (a d+b c) (b e-a f)} \]

Antiderivative was successfully verified.

[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 549

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] + (-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] + 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]) /; Fre
eQ[{a, b, c, d, e, f}, x]

Rule 524

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 427

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

Rule 426

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

Rule 424

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

Rule 421

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

Rule 419

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

Rule 538

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

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), 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)
])

Rubi steps

\begin{align*} \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{(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]  time = 5.69896, size = 617, normalized size = 1.45 \[ \frac{i c \sqrt{-\frac{d}{c}} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} (b e-a f) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{-\frac{d}{c}}\right ),-\frac{c f}{d e}\right )+\frac{i b^2 c e \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left (-\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )}{a \sqrt{-\frac{d}{c}}}-\frac{b^2 c e x}{a+b x^2}-\frac{b^2 c f x^3}{a+b x^2}+\frac{b^2 d e x^3}{a+b x^2}+\frac{b^2 d f x^5}{a+b x^2}-2 i b c e \sqrt{-\frac{d}{c}} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left (-\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )+3 i a c f \sqrt{-\frac{d}{c}} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left (-\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )+\frac{2 i b d f \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left (-\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )}{\left (-\frac{d}{c}\right )^{3/2}}-i b c e \sqrt{-\frac{d}{c}} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} E\left (i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )}{2 a \sqrt{c-d x^2} \sqrt{e+f x^2} (a d+b c) (a f-b e)} \]

Antiderivative was successfully verified.

[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])

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Maple [B]  time = 0.046, size = 1105, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(-(d/c)^(1/2)*x^5*a*b^2*d*f+(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(d/c)^(1/2),(-c*f/d/e)^(1
/2))*x^2*a^2*b*d*f-(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(d/c)^(1/2),(-c*f/d/e)^(1/2))*x^2*a*b^
2*d*e+(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(d/c)^(1/2),(-c*f/d/e)^(1/2))*x^2*a*b^2*d*e-3*(-(d*
x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(d/c)^(1/2),-b*c/a/d,(-f/e)^(1/2)/(d/c)^(1/2))*x^2*a^2*b*d*f-
2*(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(d/c)^(1/2),-b*c/a/d,(-f/e)^(1/2)/(d/c)^(1/2))*x^2*a*b
^2*c*f+2*(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(d/c)^(1/2),-b*c/a/d,(-f/e)^(1/2)/(d/c)^(1/2))*
x^2*a*b^2*d*e+(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(d/c)^(1/2),-b*c/a/d,(-f/e)^(1/2)/(d/c)^(1
/2))*x^2*b^3*c*e+(d/c)^(1/2)*x^3*a*b^2*c*f-(d/c)^(1/2)*x^3*a*b^2*d*e+(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*
EllipticF(x*(d/c)^(1/2),(-c*f/d/e)^(1/2))*a^3*d*f-(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(d/c)^(
1/2),(-c*f/d/e)^(1/2))*a^2*b*d*e+(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(d/c)^(1/2),(-c*f/d/e)^(
1/2))*a^2*b*d*e-3*(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(d/c)^(1/2),-b*c/a/d,(-f/e)^(1/2)/(d/c
)^(1/2))*a^3*d*f-2*(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(d/c)^(1/2),-b*c/a/d,(-f/e)^(1/2)/(d/
c)^(1/2))*a^2*b*c*f+2*(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(d/c)^(1/2),-b*c/a/d,(-f/e)^(1/2)/
(d/c)^(1/2))*a^2*b*d*e+(-(d*x^2-c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(d/c)^(1/2),-b*c/a/d,(-f/e)^(1/2)
/(d/c)^(1/2))*a*b^2*c*e+(d/c)^(1/2)*x*a*b^2*c*e)*(f*x^2+e)^(1/2)*(-d*x^2+c)^(1/2)/(d/c)^(1/2)/(b*x^2+a)/a^2/(a
*f-b*e)/(a*d+b*c)/(d*f*x^4-c*f*x^2+d*e*x^2-c*e)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{2} \sqrt{-d x^{2} + c} \sqrt{f x^{2} + e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{2} \sqrt{-d x^{2} + c} \sqrt{f x^{2} + e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)