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

Optimal. Leaf size=100 \[ \frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{-c}}\right )|\frac{c f}{d e}\right )}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

[Out]

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

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Rubi [A]  time = 0.479151, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{-c}}\right )|\frac{c f}{d e}\right )}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 50.4011, size = 173, normalized size = 1.73 \[ \frac{\sqrt{c} f \sqrt{e + f x^{2}} F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{\sqrt{d} e \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}} \left (a f - b e\right )} - \frac{b e^{\frac{3}{2}} \sqrt{c + d x^{2}} \Pi \left (1 - \frac{b e}{a f}; \operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{a c \sqrt{f} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}} \left (a f - b e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c)*f*sqrt(e + f*x**2)*elliptic_f(atan(sqrt(d)*x/sqrt(c)), -c*f/(d*e) + 1)/(
sqrt(d)*e*sqrt(c*(e + f*x**2)/(e*(c + d*x**2)))*sqrt(c + d*x**2)*(a*f - b*e)) -
b*e**(3/2)*sqrt(c + d*x**2)*elliptic_pi(1 - b*e/(a*f), atan(sqrt(f)*x/sqrt(e)),
1 - d*e/(c*f))/(a*c*sqrt(f)*sqrt(e*(c + d*x**2)/(c*(e + f*x**2)))*sqrt(e + f*x**
2)*(a*f - b*e))

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Mathematica [C]  time = 0.150235, size = 101, normalized size = 1.01 \[ -\frac{i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{a \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*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]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [A]  time = 0.03, size = 118, normalized size = 1.2 \[{\frac{1}{a \left ( df{x}^{4}+cf{x}^{2}+de{x}^{2}+ce \right ) }{\it EllipticPi} \left ( x\sqrt{-{\frac{d}{c}}},{\frac{bc}{ad}},{1\sqrt{-{\frac{f}{e}}}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \right ) \sqrt{{\frac{f{x}^{2}+e}{e}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{f{x}^{2}+e}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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