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

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*Sqrt[-d])

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Rubi [A]  time = 0.0415932, 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, Rules used = {537} \[ \frac{\Pi \left (\frac{2 b}{a d};\sin ^{-1}\left (\frac{\sqrt{-d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )}{\sqrt{3} a \sqrt{-d}} \]

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*Sqrt[-d])

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 ) \sqrt{2+d x^2} \sqrt{3+f x^2}} \, dx &=\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}}\\ \end{align*}

Mathematica [A]  time = 0.01666, size = 49, normalized size = 1. \[ \frac{\Pi \left (\frac{2 b}{a d};\sin ^{-1}\left (\frac{\sqrt{-d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )}{\sqrt{3} a \sqrt{-d}} \]

Antiderivative was successfully verified.

[In]

Integrate[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*Sqrt[-d])

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Maple [A]  time = 0.019, size = 53, normalized size = 1.1 \begin{align*}{\frac{\sqrt{2}}{2\,a}{\it EllipticPi} \left ({\frac{x\sqrt{3}}{3}\sqrt{-f}},3\,{\frac{b}{af}},{\frac{\sqrt{2}\sqrt{3}}{2}\sqrt{-d}{\frac{1}{\sqrt{-f}}}} \right ){\frac{1}{\sqrt{-f}}}} \end{align*}

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. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )} \sqrt{d x^{2} + 2} \sqrt{f x^{2} + 3}}\,{d x} \end{align*}

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, 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. \begin{align*} \text{Timed out} \end{align*}

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, algorithm="fricas")

[Out]

Timed out

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

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

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, algorithm="giac")

[Out]

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