∫ 1_____ x2(−2−3x2)3∕4 dx [916]

Optimal. Leaf size=105 \[ \frac {\sqrt [4]{-2-3 x^2}}{2 x}+\frac {\sqrt {3} \sqrt {-\frac {x^2}{\left (\sqrt {2}+\sqrt {-2-3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2-3 x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-2-3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} x} \]

1/2*(-3*x^2-2)^(1/4)/x+1/8*2^(3/4)*(cos(2*arctan(1/2*(-3*x^2-2)^(1/4)*2^(3/4)))^2)^(1/2)/cos(2*arctan(1/2*(-3*

x^2-2)^(1/4)*2^(3/4)))*EllipticF(sin(2*arctan(1/2*(-3*x^2-2)^(1/4)*2^(3/4))),1/2*2^(1/2))*(2^(1/2)+(-3*x^2-2)^

(1/2))*(-x^2/(2^(1/2)+(-3*x^2-2)^(1/2))^2)^(1/2)/x*3^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {331, 240, 226} \begin {gather*} \frac {\sqrt {3} \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 \text {ArcTan}\left (\frac {\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} x}+\frac {\sqrt [4]{-3 x^2-2}}{2 x} \end {gather*}

(-2 - 3*x^2)^(1/4)/(2*x) + (Sqrt[3]*Sqrt[-(x^2/(Sqrt[2] + Sqrt[-2 - 3*x^2])^2)]*(Sqrt[2] + Sqrt[-2 - 3*x^2])*E

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

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)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

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]

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]

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 5.89, size = 46, normalized size = 0.44 \begin {gather*} -\frac {\left (1+\frac {3 x^2}{2}\right )^{3/4} \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {1}{2};-\frac {3 x^2}{2}\right )}{x \left (-2-3 x^2\right )^{3/4}} \end {gather*}

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

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 4.
time = 0.08, size = 23, normalized size = 0.22


method	result	size

meijerg	\(\frac {\left (-1\right )^{\frac {1}{4}} 2^{\frac {1}{4}} \hypergeom \left (\left [-\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {1}{2}\right ], -\frac {3 x^{2}}{2}\right )}{2 x}\)	\(23\)

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

1/2*(-1)^(1/4)*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*}

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

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

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

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.44, size = 34, normalized size = 0.32 \begin {gather*} \frac {\sqrt [4]{2} e^{\frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{i \pi }}{2}} \right )}}{2 x} \end {gather*}

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

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

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Mupad [B]
time = 4.73, size = 36, normalized size = 0.34 \begin {gather*} -\frac {2\,3^{1/4}\,{\left (\frac {2}{x^2}+3\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 (-3\,x^2-2\right )}^{3/4}} \end {gather*}

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

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3.10.16 \(\int \frac {1}{x^2 (-2-3 x^2)^{3/4}} \, dx\) [916]