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

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 = 0.120808, 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, Rules used = {538, 537} \[ \frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{-c}}\right )|\frac{c f}{d e}\right )}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

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])

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

Mathematica [C]  time = 0.220757, 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 = 0.021, size = 118, normalized size = 1.2 \begin{align*}{\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}},{\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}}}}}} \end{align*}

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

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

[Out]

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