3.172 \(\int \frac{1}{(a+b x^2) \sqrt{4-d x^4}} \, dx\)

Optimal. Leaf size=40 \[ \frac{\Pi \left (-\frac{2 b}{a \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt{2}}\right )\right |-1\right )}{\sqrt{2} a \sqrt [4]{d}} \]

[Out]

EllipticPi[(-2*b)/(a*Sqrt[d]), ArcSin[(d^(1/4)*x)/Sqrt[2]], -1]/(Sqrt[2]*a*d^(1/4))

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Rubi [A]  time = 0.0179991, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {1218} \[ \frac{\Pi \left (-\frac{2 b}{a \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt{2}}\right )\right |-1\right )}{\sqrt{2} a \sqrt [4]{d}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^2)*Sqrt[4 - d*x^4]),x]

[Out]

EllipticPi[(-2*b)/(a*Sqrt[d]), ArcSin[(d^(1/4)*x)/Sqrt[2]], -1]/(Sqrt[2]*a*d^(1/4))

Rule 1218

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-(c/a), 4]}, Simp[(1*Ellipt
icPi[-(e/(d*q^2)), ArcSin[q*x], -1])/(d*Sqrt[a]*q), x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b x^2\right ) \sqrt{4-d x^4}} \, dx &=\frac{\Pi \left (-\frac{2 b}{a \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt{2}}\right )\right |-1\right )}{\sqrt{2} a \sqrt [4]{d}}\\ \end{align*}

Mathematica [C]  time = 0.125321, size = 59, normalized size = 1.48 \[ -\frac{i \Pi \left (-\frac{2 b}{a \sqrt{d}};\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{d}} x}{\sqrt{2}}\right )\right |-1\right )}{\sqrt{2} a \sqrt{-\sqrt{d}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^2)*Sqrt[4 - d*x^4]),x]

[Out]

((-I)*EllipticPi[(-2*b)/(a*Sqrt[d]), I*ArcSinh[(Sqrt[-Sqrt[d]]*x)/Sqrt[2]], -1])/(Sqrt[2]*a*Sqrt[-Sqrt[d]])

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Maple [B]  time = 0.095, size = 78, normalized size = 2. \begin{align*}{\frac{\sqrt{2}}{a}\sqrt{1-{\frac{{x}^{2}}{2}\sqrt{d}}}\sqrt{1+{\frac{{x}^{2}}{2}\sqrt{d}}}{\it EllipticPi} \left ({\frac{x\sqrt{2}}{2}\sqrt [4]{d}},-2\,{\frac{b}{a\sqrt{d}}},{\sqrt{2}\sqrt{-{\frac{1}{2}\sqrt{d}}}{\frac{1}{\sqrt [4]{d}}}} \right ){\frac{1}{\sqrt [4]{d}}}{\frac{1}{\sqrt{-d{x}^{4}+4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2+a)/(-d*x^4+4)^(1/2),x)

[Out]

1/a*2^(1/2)/d^(1/4)*(1-1/2*x^2*d^(1/2))^(1/2)*(1+1/2*x^2*d^(1/2))^(1/2)/(-d*x^4+4)^(1/2)*EllipticPi(1/2*d^(1/4
)*x*2^(1/2),-2*b/a/d^(1/2),(-1/2*d^(1/2))^(1/2)*2^(1/2)/d^(1/4))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-d x^{4} + 4}{\left (b x^{2} + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)/(-d*x^4+4)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-d*x^4 + 4)*(b*x^2 + a)), 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^4+4)^(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^{4} + 4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**2+a)/(-d*x**4+4)**(1/2),x)

[Out]

Integral(1/((a + b*x**2)*sqrt(-d*x**4 + 4)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-d x^{4} + 4}{\left (b x^{2} + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)/(-d*x^4+4)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-d*x^4 + 4)*(b*x^2 + a)), x)