∫ x2 ____________________(−2−3x2 )3∕4 dx [914]

Optimal. Leaf size=103 \[ -\frac {2}{9} x \sqrt [4]{-2-3 x^2}+\frac {2\ 2^{3/4} \sqrt {-\frac {x^2}{\left (\sqrt {2}+\sqrt {-2-3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2-3 x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-2-3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{9 \sqrt {3} x} \]

-2/9*x*(-3*x^2-2)^(1/4)+2/27*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 = 103, 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 = {327, 240, 226} \begin {gather*} \frac {2\ 2^{3/4} \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 )}{9 \sqrt {3} x}-\frac {2}{9} x \sqrt [4]{-3 x^2-2} \end {gather*}

(-2*x*(-2 - 3*x^2)^(1/4))/9 + (2*2^(3/4)*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])/(9*Sqrt[3]*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^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n

)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p

, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,

\begin {align*} \int \frac {x^2}{\left (-2-3 x^2\right )^{3/4}} \, dx &=-\frac {2}{9} x \sqrt [4]{-2-3 x^2}-\frac {4}{9} \int \frac {1}{\left (-2-3 x^2\right )^{3/4}} \, dx\\ &=-\frac {2}{9} x \sqrt [4]{-2-3 x^2}+\frac {\left (4 \sqrt {\frac {2}{3}} \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 )}{9 x}\\ &=-\frac {2}{9} x \sqrt [4]{-2-3 x^2}+\frac {2\ 2^{3/4} \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 )}{9 \sqrt {3} 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 = 5.60, size = 58, normalized size = 0.56 \begin {gather*} \frac {2 x \left (2+3 x^2-\sqrt [4]{2} \left (2+3 x^2\right )^{3/4} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};-\frac {3 x^2}{2}\right )\right )}{9 \left (-2-3 x^2\right )^{3/4}} \end {gather*}

(2*x*(2 + 3*x^2 - 2^(1/4)*(2 + 3*x^2)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (-3*x^2)/2]))/(9*(-2 - 3*x^2)^(3/

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 4.
time = 0.07, size = 23, normalized size = 0.22


method	result	size

meijerg	\(-\frac {\left (-1\right )^{\frac {1}{4}} 2^{\frac {1}{4}} x^{3} \hypergeom \left (\left [\frac {3}{4}, \frac {3}{2}\right ], \left [\frac {5}{2}\right ], -\frac {3 x^{2}}{2}\right )}{6}\)	\(23\)

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

-1/6*(-1)^(1/4)*2^(1/4)*x^3*hypergeom([3/4,3/2],[5/2],-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(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(x^2/(-3*x^2-2)^(3/4),x, algorithm="fricas")

-2/9*(-3*x^2 - 2)^(1/4)*x + integral(4/9*(-3*x^2 - 2)^(1/4)/(3*x^2 + 2), x)

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Sympy [C] Result contains complex when optimal does not.
time = 0.40, size = 36, normalized size = 0.35 \begin {gather*} \frac {\sqrt [4]{2} x^{3} e^{- \frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{i \pi }}{2}} \right )}}{6} \end {gather*}

2**(1/4)*x**3*exp(-3*I*pi/4)*hyper((3/4, 3/2), (5/2,), 3*x**2*exp_polar(I*pi)/2)/6

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

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2}{{\left (-3\,x^2-2\right )}^{3/4}} \,d x \end {gather*}

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3.10.14 \(\int \frac {x^2}{(-2-3 x^2)^{3/4}} \, dx\) [914]