3.9.88 \(\int \frac {1}{x^2 (2-3 x^2)^{3/4}} \, dx\) [888]

Optimal. Leaf size=49 \[ -\frac {\sqrt [4]{2-3 x^2}}{2 x}+\frac {\sqrt {3} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{2 \sqrt [4]{2}} \]

[Out]

-1/2*(-3*x^2+2)^(1/4)/x+1/4*2^(3/4)*(cos(1/2*arcsin(1/2*x*6^(1/2)))^2)^(1/2)/cos(1/2*arcsin(1/2*x*6^(1/2)))*El
lipticF(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 = 49, 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, 238} \begin {gather*} \frac {\sqrt {3} F\left (\left .\frac {1}{2} \text {ArcSin}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{2 \sqrt [4]{2}}-\frac {\sqrt [4]{2-3 x^2}}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(2 - 3*x^2)^(3/4)),x]

[Out]

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

Rule 238

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[-b/a, 2]))*EllipticF[(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 \left (2-3 x^2\right )^{3/4}} \, dx &=-\frac {\sqrt [4]{2-3 x^2}}{2 x}+\frac {3}{4} \int \frac {1}{\left (2-3 x^2\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{2-3 x^2}}{2 x}+\frac {\sqrt {3} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{2 \sqrt [4]{2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 5.99, size = 27, normalized size = 0.55 \begin {gather*} -\frac {\, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {1}{2};\frac {3 x^2}{2}\right )}{2^{3/4} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Hypergeometric2F1[-1/2, 3/4, 1/2, (3*x^2)/2]/(2^(3/4)*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 = 0.41

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.47, size = 31, normalized size = 0.63 \begin {gather*} - \frac {\sqrt [4]{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{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)**(3/4),x)

[Out]

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

[Out]

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

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Mupad [B]
time = 5.03, size = 36, normalized size = 0.73 \begin {gather*} -\frac {2\,3^{1/4}\,{\left (3-\frac {2}{x^2}\right )}^{3/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{4},\frac {5}{4};\ \frac {9}{4};\ \frac {2}{3\,x^2}\right )}{15\,x\,{\left (2-3\,x^2\right )}^{3/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(2 - 3*x^2)^(3/4)),x)

[Out]

-(2*3^(1/4)*(3 - 2/x^2)^(3/4)*hypergeom([3/4, 5/4], 9/4, 2/(3*x^2)))/(15*x*(2 - 3*x^2)^(3/4))

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