3.9.74 \(\int \frac {1}{x^2 \sqrt [4]{2-3 x^2}} \, dx\) [874]

Optimal. Leaf size=47 \[ -\frac {\left (2-3 x^2\right )^{3/4}}{2 x}-\frac {\sqrt {3} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{2^{3/4}} \]

[Out]

-1/2*(-3*x^2+2)^(3/4)/x-1/2*2^(1/4)*(cos(1/2*arcsin(1/2*x*6^(1/2)))^2)^(1/2)/cos(1/2*arcsin(1/2*x*6^(1/2)))*El
lipticE(sin(1/2*arcsin(1/2*x*6^(1/2))),2^(1/2))*3^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {331, 234} \begin {gather*} -\frac {\sqrt {3} E\left (\left .\frac {1}{2} \text {ArcSin}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{2^{3/4}}-\frac {\left (2-3 x^2\right )^{3/4}}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(2 - 3*x^2)^(1/4)),x]

[Out]

-1/2*(2 - 3*x^2)^(3/4)/x - (Sqrt[3]*EllipticE[ArcSin[Sqrt[3/2]*x]/2, 2])/2^(3/4)

Rule 234

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(2/(a^(1/4)*Rt[-b/a, 2]))*EllipticE[(1/2)*ArcSin[Rt[-b/a,
2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \sqrt [4]{2-3 x^2}} \, dx &=-\frac {\left (2-3 x^2\right )^{3/4}}{2 x}-\frac {3}{4} \int \frac {1}{\sqrt [4]{2-3 x^2}} \, dx\\ &=-\frac {\left (2-3 x^2\right )^{3/4}}{2 x}-\frac {\sqrt {3} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{2^{3/4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 5.87, size = 27, normalized size = 0.57 \begin {gather*} -\frac {\, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {1}{2};\frac {3 x^2}{2}\right )}{\sqrt [4]{2} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 4.
time = 0.08, size = 20, normalized size = 0.43

method	result	size
meijerg	\(-\frac {2^{\frac {3}{4}} \hypergeom \left (\left [-\frac {1}{2}, \frac {1}{4}\right ], \left [\frac {1}{2}\right ], \frac {3 x^{2}}{2}\right )}{2 x}\)	\(20\)
risch	\(\frac {3 x^{2}-2}{2 x \left (-3 x^{2}+2\right )^{\frac {1}{4}}}-\frac {3 \,2^{\frac {3}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], \frac {3 x^{2}}{2}\right )}{8}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(-3*x^2+2)^(1/4),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.40, size = 31, normalized size = 0.66 \begin {gather*} - \frac {2^{\frac {3}{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{2 i \pi }}{2}} \right )}}{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 5.05, size = 36, normalized size = 0.77 \begin {gather*} -\frac {2\,3^{3/4}\,{\left (3-\frac {2}{x^2}\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {3}{4};\ \frac {7}{4};\ \frac {2}{3\,x^2}\right )}{9\,x\,{\left (2-3\,x^2\right )}^{1/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(2 - 3*x^2)^(1/4)),x)

[Out]

-(2*3^(3/4)*(3 - 2/x^2)^(1/4)*hypergeom([1/4, 3/4], 7/4, 2/(3*x^2)))/(9*x*(2 - 3*x^2)^(1/4))

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