3.170 \(\int \frac{1}{(a+b x^2) \sqrt{4-5 x^4}} \, dx\)

Optimal. Leaf size=40 \[ \frac{\Pi \left (-\frac{2 b}{\sqrt{5} a};\left .\sin ^{-1}\left (\frac{\sqrt [4]{5} x}{\sqrt{2}}\right )\right |-1\right )}{\sqrt{2} \sqrt [4]{5} a} \]

[Out]

EllipticPi[(-2*b)/(Sqrt[5]*a), ArcSin[(5^(1/4)*x)/Sqrt[2]], -1]/(Sqrt[2]*5^(1/4)*a)

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Rubi [A]  time = 0.0630028, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {1213, 537} \[ \frac{\Pi \left (-\frac{2 b}{\sqrt{5} a};\left .\sin ^{-1}\left (\frac{\sqrt [4]{5} x}{\sqrt{2}}\right )\right |-1\right )}{\sqrt{2} \sqrt [4]{5} a} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^2)*Sqrt[4 - 5*x^4]),x]

[Out]

EllipticPi[(-2*b)/(Sqrt[5]*a), ArcSin[(5^(1/4)*x)/Sqrt[2]], -1]/(Sqrt[2]*5^(1/4)*a)

Rule 1213

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[Sqrt[-c],
Int[1/((d + e*x^2)*Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[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{4-5 x^4}} \, dx &=\sqrt{5} \int \frac{1}{\sqrt{2 \sqrt{5}-5 x^2} \sqrt{2 \sqrt{5}+5 x^2} \left (a+b x^2\right )} \, dx\\ &=\frac{\Pi \left (-\frac{2 b}{\sqrt{5} a};\left .\sin ^{-1}\left (\frac{\sqrt [4]{5} x}{\sqrt{2}}\right )\right |-1\right )}{\sqrt{2} \sqrt [4]{5} a}\\ \end{align*}

Mathematica [A]  time = 0.123363, size = 43, normalized size = 1.08 \[ -\frac{\Pi \left (-\frac{2 b}{\sqrt{5} a};\left .-\sin ^{-1}\left (\frac{\sqrt [4]{5} x}{\sqrt{2}}\right )\right |-1\right )}{\sqrt{2} \sqrt [4]{5} a} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^2)*Sqrt[4 - 5*x^4]),x]

[Out]

-(EllipticPi[(-2*b)/(Sqrt[5]*a), -ArcSin[(5^(1/4)*x)/Sqrt[2]], -1]/(Sqrt[2]*5^(1/4)*a))

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Maple [B]  time = 0.191, size = 79, normalized size = 2. \begin{align*}{\frac{\sqrt{2}{5}^{{\frac{3}{4}}}}{5\,a}\sqrt{1-{\frac{{x}^{2}\sqrt{5}}{2}}}\sqrt{1+{\frac{{x}^{2}\sqrt{5}}{2}}}{\it EllipticPi} \left ({\frac{\sqrt [4]{5}x\sqrt{2}}{2}},-{\frac{2\,\sqrt{5}b}{5\,a}},{\frac{\sqrt{-{\frac{\sqrt{5}}{2}}}\sqrt{2}{5}^{{\frac{3}{4}}}}{5}} \right ){\frac{1}{\sqrt{-5\,{x}^{4}+4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2+a)/(-5*x^4+4)^(1/2),x)

[Out]

1/5/a*2^(1/2)*5^(3/4)*(1-1/2*x^2*5^(1/2))^(1/2)*(1+1/2*x^2*5^(1/2))^(1/2)/(-5*x^4+4)^(1/2)*EllipticPi(1/2*5^(1
/4)*x*2^(1/2),-2/5*b/a*5^(1/2),1/5*(-1/2*5^(1/2))^(1/2)*2^(1/2)*5^(3/4))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-5 \, 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)/(-5*x^4+4)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-5*x^4 + 4)*(b*x^2 + a)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-5 \, x^{4} + 4}}{5 \, b x^{6} + 5 \, a x^{4} - 4 \, b x^{2} - 4 \, a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)/(-5*x^4+4)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-5*x^4 + 4)/(5*b*x^6 + 5*a*x^4 - 4*b*x^2 - 4*a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{4 - 5 x^{4}} \left (a + b x^{2}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**2+a)/(-5*x**4+4)**(1/2),x)

[Out]

Integral(1/(sqrt(4 - 5*x**4)*(a + b*x**2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-5 \, 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)/(-5*x^4+4)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-5*x^4 + 4)*(b*x^2 + a)), x)