3.96 \(\int \frac{1}{\left (a+b x^2\right ) \sqrt{2+d x^2} \sqrt{3+f x^2}} \, dx\)

Optimal. Leaf size=49 \[ \frac{\Pi \left (\frac{2 b}{a d};\sin ^{-1}\left (\frac{\sqrt{-d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )}{\sqrt{3} a \sqrt{-d}} \]

[Out]

EllipticPi[(2*b)/(a*d), ArcSin[(Sqrt[-d]*x)/Sqrt[2]], (2*f)/(3*d)]/(Sqrt[3]*a*Sq
rt[-d])

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Rubi [A]  time = 0.146693, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.031 \[ \frac{\Pi \left (\frac{2 b}{a d};\sin ^{-1}\left (\frac{\sqrt{-d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )}{\sqrt{3} a \sqrt{-d}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^2)*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2]),x]

[Out]

EllipticPi[(2*b)/(a*d), ArcSin[(Sqrt[-d]*x)/Sqrt[2]], (2*f)/(3*d)]/(Sqrt[3]*a*Sq
rt[-d])

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Rubi in Sympy [A]  time = 41.6285, size = 177, normalized size = 3.61 \[ - \frac{\sqrt{3} \sqrt{f} \sqrt{d x^{2} + 2} F\left (\operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{f} x}{3} \right )}\middle | - \frac{3 d}{2 f} + 1\right )}{2 \sqrt{\frac{3 d x^{2} + 6}{2 f x^{2} + 6}} \left (- a f + 3 b\right ) \sqrt{f x^{2} + 3}} + \frac{3 \sqrt{3} b \sqrt{d x^{2} + 2} \Pi \left (1 - \frac{3 b}{a f}; \operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{f} x}{3} \right )}\middle | - \frac{3 d}{2 f} + 1\right )}{2 a \sqrt{f} \sqrt{\frac{3 d x^{2} + 6}{2 f x^{2} + 6}} \left (- a f + 3 b\right ) \sqrt{f x^{2} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2+a)/(d*x**2+2)**(1/2)/(f*x**2+3)**(1/2),x)

[Out]

-sqrt(3)*sqrt(f)*sqrt(d*x**2 + 2)*elliptic_f(atan(sqrt(3)*sqrt(f)*x/3), -3*d/(2*
f) + 1)/(2*sqrt((3*d*x**2 + 6)/(2*f*x**2 + 6))*(-a*f + 3*b)*sqrt(f*x**2 + 3)) +
3*sqrt(3)*b*sqrt(d*x**2 + 2)*elliptic_pi(1 - 3*b/(a*f), atan(sqrt(3)*sqrt(f)*x/3
), -3*d/(2*f) + 1)/(2*a*sqrt(f)*sqrt((3*d*x**2 + 6)/(2*f*x**2 + 6))*(-a*f + 3*b)
*sqrt(f*x**2 + 3))

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Mathematica [C]  time = 0.0988031, size = 52, normalized size = 1.06 \[ -\frac{i \Pi \left (\frac{2 b}{a d};i \sinh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )}{\sqrt{3} a \sqrt{d}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x^2)*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2]),x]

[Out]

((-I)*EllipticPi[(2*b)/(a*d), I*ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)])/(Sqr
t[3]*a*Sqrt[d])

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Maple [A]  time = 0.032, size = 53, normalized size = 1.1 \[{\frac{\sqrt{2}}{2\,a}{\it EllipticPi} \left ({\frac{x\sqrt{3}}{3}\sqrt{-f}},3\,{\frac{b}{af}},{\frac{\sqrt{3}\sqrt{2}}{2}\sqrt{-d}{\frac{1}{\sqrt{-f}}}} \right ){\frac{1}{\sqrt{-f}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+a)/(d*x^2+2)^(1/2)/(f*x^2+3)^(1/2),x)

[Out]

1/2*2^(1/2)*EllipticPi(1/3*x*3^(1/2)*(-f)^(1/2),3*b/a/f,1/2*2^(1/2)*(-d)^(1/2)*3
^(1/2)/(-f)^(1/2))/(-f)^(1/2)/a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )} \sqrt{d x^{2} + 2} \sqrt{f x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*sqrt(d*x^2 + 2)*sqrt(f*x^2 + 3)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)*sqrt(d*x^2 + 2)*sqrt(f*x^2 + 3)), 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 + 2)*sqrt(f*x^2 + 3)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x^{2}\right ) \sqrt{d x^{2} + 2} \sqrt{f x^{2} + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**2+a)/(d*x**2+2)**(1/2)/(f*x**2+3)**(1/2),x)

[Out]

Integral(1/((a + b*x**2)*sqrt(d*x**2 + 2)*sqrt(f*x**2 + 3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )} \sqrt{d x^{2} + 2} \sqrt{f x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*sqrt(d*x^2 + 2)*sqrt(f*x^2 + 3)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)*sqrt(d*x^2 + 2)*sqrt(f*x^2 + 3)), x)