3.165 \(\int \frac{1}{(d+e x^2) \sqrt{-a+c x^4}} \, dx\)

Optimal. Leaf size=73 \[ \frac{\sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d \sqrt{c x^4-a}} \]

[Out]

(a^(1/4)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)
*d*Sqrt[-a + c*x^4])

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Rubi [A]  time = 0.041897, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {1219, 1218} \[ \frac{\sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{c} d \sqrt{c x^4-a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^(1/4)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)
*d*Sqrt[-a + c*x^4])

Rule 1219

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[Sqrt[1 + (c*x^4)/a]/Sqrt[a + c*x^4]
, Int[1/((d + e*x^2)*Sqrt[1 + (c*x^4)/a]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 0]

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

Mathematica [C]  time = 0.149835, size = 92, normalized size = 1.26 \[ -\frac{i \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{d \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} \sqrt{c x^4-a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(
Sqrt[-(Sqrt[c]/Sqrt[a])]*d*Sqrt[-a + c*x^4])

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Maple [A]  time = 0.18, size = 99, normalized size = 1.4 \begin{align*}{\frac{1}{d}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticPi} \left ( x\sqrt{-{\sqrt{c}{\frac{1}{\sqrt{a}}}}},{\frac{e}{d}\sqrt{a}{\frac{1}{\sqrt{c}}}},{\sqrt{{\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\frac{1}{\sqrt{-{\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}} \right ){\frac{1}{\sqrt{-{\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}-a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)/(c*x^4-a)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^4 - a)/(c*e*x^6 + c*d*x^4 - a*e*x^2 - a*d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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