3.915 \(\int \frac{1}{(-2-3 x^2)^{3/4}} \, dx\)

Optimal. Leaf size=84 \[ -\frac{\sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right ),\frac{1}{2}\right )}{\sqrt [4]{2} \sqrt{3} x} \]

[Out]

-((Sqrt[-(x^2/(Sqrt[2] + Sqrt[-2 - 3*x^2])^2)]*(Sqrt[2] + Sqrt[-2 - 3*x^2])*EllipticF[2*ArcTan[(-2 - 3*x^2)^(1
/4)/2^(1/4)], 1/2])/(2^(1/4)*Sqrt[3]*x))

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Rubi [A]  time = 0.0270031, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {234, 220} \[ -\frac{\sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{\sqrt [4]{2} \sqrt{3} x} \]

Antiderivative was successfully verified.

[In]

Int[(-2 - 3*x^2)^(-3/4),x]

[Out]

-((Sqrt[-(x^2/(Sqrt[2] + Sqrt[-2 - 3*x^2])^2)]*(Sqrt[2] + Sqrt[-2 - 3*x^2])*EllipticF[2*ArcTan[(-2 - 3*x^2)^(1
/4)/2^(1/4)], 1/2])/(2^(1/4)*Sqrt[3]*x))

Rule 234

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Dist[(2*Sqrt[-((b*x^2)/a)])/(b*x), Subst[Int[1/Sqrt[1 - x^4/a],
x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rubi steps

\begin{align*} \int \frac{1}{\left (-2-3 x^2\right )^{3/4}} \, dx &=-\frac{\left (\sqrt{\frac{2}{3}} \sqrt{-x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^4}{2}}} \, dx,x,\sqrt [4]{-2-3 x^2}\right )}{x}\\ &=-\frac{\sqrt{-\frac{x^2}{\left (\sqrt{2}+\sqrt{-2-3 x^2}\right )^2}} \left (\sqrt{2}+\sqrt{-2-3 x^2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-2-3 x^2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{\sqrt [4]{2} \sqrt{3} x}\\ \end{align*}

Mathematica [C]  time = 0.0067792, size = 43, normalized size = 0.51 \[ \frac{x \left (\frac{3 x^2}{2}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{3 x^2}{2}\right )}{\left (-3 x^2-2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-2 - 3*x^2)^(-3/4),x]

[Out]

(x*(1 + (3*x^2)/2)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (-3*x^2)/2])/(-2 - 3*x^2)^(3/4)

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Maple [C]  time = 0.011, size = 21, normalized size = 0.3 \begin{align*} -{\frac{\sqrt [4]{-1}\sqrt [4]{2}x}{2}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{3}{4}};\,{\frac{3}{2}};\,-{\frac{3\,{x}^{2}}{2}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-3*x^2-2)^(3/4),x)

[Out]

-1/2*(-1)^(1/4)*2^(1/4)*x*hypergeom([1/2,3/4],[3/2],-3/2*x^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-3 \, x^{2} - 2\right )}^{\frac{3}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2-2)^(3/4),x, algorithm="maxima")

[Out]

integrate((-3*x^2 - 2)^(-3/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (-3 \, x^{2} - 2\right )}^{\frac{1}{4}}}{3 \, x^{2} + 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2-2)^(3/4),x, algorithm="fricas")

[Out]

integral(-(-3*x^2 - 2)^(1/4)/(3*x^2 + 2), x)

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Sympy [C]  time = 0.598736, size = 34, normalized size = 0.4 \begin{align*} \frac{\sqrt [4]{2} x e^{- \frac{3 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{3}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{i \pi }}{2}} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x**2-2)**(3/4),x)

[Out]

2**(1/4)*x*exp(-3*I*pi/4)*hyper((1/2, 3/4), (3/2,), 3*x**2*exp_polar(I*pi)/2)/2

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-3 \, x^{2} - 2\right )}^{\frac{3}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2-2)^(3/4),x, algorithm="giac")

[Out]

integrate((-3*x^2 - 2)^(-3/4), x)