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

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

[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[(Sqr
t[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)), 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])

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Rubi [A]  time = 1.33234, 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 \[ \frac{\sqrt{c} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \left (-3 a^2 d f+2 a b (d e-c f)+b^2 c e\right ) \Pi \left (-\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a^2 \sqrt{d} \sqrt{c-d x^2} \sqrt{e+f x^2} (a d+b c) (b e-a f)}+\frac{b^2 x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right ) (a d+b c) (b e-a f)}-\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a \sqrt{c-d x^2} \sqrt{e+f x^2} (a d+b c)}+\frac{b \sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{e+f x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a \sqrt{c-d x^2} \sqrt{\frac{f x^2}{e}+1} (a d+b c) (b e-a f)} \]

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[(Sqr
t[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 + 2*a*b*(d*e - c*f))*S
qrt[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])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2+a)**2/(-d*x**2+c)**(1/2)/(f*x**2+e)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 6.64912, size = 773, normalized size = 1.81 \[ -\frac{b^2 x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right ) (a d+b c) (a f-b e)}+\frac{\sqrt{\left (c-d x^2\right ) \left (e+f x^2\right )} \left (\frac{i b^2 c e \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left (-\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )}{a \sqrt{-\frac{d}{c}} \sqrt{\left (c-d x^2\right ) \left (e+f x^2\right )}}+\frac{2 i b d e \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left (-\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )}{\sqrt{-\frac{d}{c}} \sqrt{\left (c-d x^2\right ) \left (e+f x^2\right )}}-\frac{2 i b c f \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left (-\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )}{\sqrt{-\frac{d}{c}} \sqrt{\left (c-d x^2\right ) \left (e+f x^2\right )}}-\frac{3 i a d f \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left (-\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )}{\sqrt{-\frac{d}{c}} \sqrt{\left (c-d x^2\right ) \left (e+f x^2\right )}}+\frac{i a d f \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )}{\sqrt{-\frac{d}{c}} \sqrt{\left (c-d x^2\right ) \left (e+f x^2\right )}}+\frac{i b d e \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \left (E\left (i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )-F\left (i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )\right )}{\sqrt{-\frac{d}{c}} \sqrt{\left (c-d x^2\right ) \left (e+f x^2\right )}}\right )}{2 a \sqrt{c-d x^2} \sqrt{e+f x^2} (a d+b c) (a f-b e)} \]

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*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)) + (Sqrt[(c - d*x^2)*(e + f*x^2)]*((I*b*d*e*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f
*x^2)/e]*(EllipticE[I*ArcSinh[Sqrt[-(d/c)]*x], -((c*f)/(d*e))] - EllipticF[I*Arc
Sinh[Sqrt[-(d/c)]*x], -((c*f)/(d*e))]))/(Sqrt[-(d/c)]*Sqrt[(c - d*x^2)*(e + f*x^
2)]) + (I*a*d*f*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt
[-(d/c)]*x], -((c*f)/(d*e))])/(Sqrt[-(d/c)]*Sqrt[(c - d*x^2)*(e + f*x^2)]) + (I*
b^2*c*e*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[-((b*c)/(a*d)), I*Arc
Sinh[Sqrt[-(d/c)]*x], -((c*f)/(d*e))])/(a*Sqrt[-(d/c)]*Sqrt[(c - d*x^2)*(e + f*x
^2)]) + ((2*I)*b*d*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))])/(Sqrt[-(d/c)]*Sqrt[(c - d*x^
2)*(e + f*x^2)]) - ((2*I)*b*c*f*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*Elliptic
Pi[-((b*c)/(a*d)), I*ArcSinh[Sqrt[-(d/c)]*x], -((c*f)/(d*e))])/(Sqrt[-(d/c)]*Sqr
t[(c - d*x^2)*(e + f*x^2)]) - ((3*I)*a*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))])/(Sqrt[
-(d/c)]*Sqrt[(c - d*x^2)*(e + f*x^2)])))/(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 = 0.073, size = 1105, normalized size = 2.6 \[ \text{result too large to display} \]

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)*Ellipti
cF(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)*Ellip
ticPi(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. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{2} \sqrt{-d x^{2} + c} \sqrt{f x^{2} + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*sqrt(-d*x^2 + c)*sqrt(f*x^2 + e)),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. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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