3.9.98 \(\int \frac {x^6}{\sqrt [4]{-2-3 x^2}} \, dx\) [898]

Optimal. Leaf size=260 \[ -\frac {32 x \left (-2-3 x^2\right )^{3/4}}{1053}+\frac {40 x^3 \left (-2-3 x^2\right )^{3/4}}{1053}-\frac {2}{39} x^5 \left (-2-3 x^2\right )^{3/4}-\frac {128 x \sqrt [4]{-2-3 x^2}}{1053 \left (\sqrt {2}+\sqrt {-2-3 x^2}\right )}-\frac {128 \sqrt [4]{2} \sqrt {-\frac {x^2}{\left (\sqrt {2}+\sqrt {-2-3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2-3 x^2}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-2-3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{1053 \sqrt {3} x}+\frac {64 \sqrt [4]{2} \sqrt {-\frac {x^2}{\left (\sqrt {2}+\sqrt {-2-3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2-3 x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-2-3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{1053 \sqrt {3} x} \]

[Out]

-32/1053*x*(-3*x^2-2)^(3/4)+40/1053*x^3*(-3*x^2-2)^(3/4)-2/39*x^5*(-3*x^2-2)^(3/4)-128/1053*x*(-3*x^2-2)^(1/4)
/(2^(1/2)+(-3*x^2-2)^(1/2))-128/3159*2^(1/4)*(cos(2*arctan(1/2*(-3*x^2-2)^(1/4)*2^(3/4)))^2)^(1/2)/cos(2*arcta
n(1/2*(-3*x^2-2)^(1/4)*2^(3/4)))*EllipticE(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)+64/3159*2^(1/4)*(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 = 0.10, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {327, 236, 311, 226, 1210} \begin {gather*} \frac {64 \sqrt [4]{2} \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 )}{1053 \sqrt {3} x}-\frac {128 \sqrt [4]{2} \sqrt {-\frac {x^2}{\left (\sqrt {-3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {-3 x^2-2}+\sqrt {2}\right ) E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{1053 \sqrt {3} x}-\frac {32 \left (-3 x^2-2\right )^{3/4} x}{1053}-\frac {128 \sqrt [4]{-3 x^2-2} x}{1053 \left (\sqrt {-3 x^2-2}+\sqrt {2}\right )}-\frac {2}{39} \left (-3 x^2-2\right )^{3/4} x^5+\frac {40 \left (-3 x^2-2\right )^{3/4} x^3}{1053} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6/(-2 - 3*x^2)^(1/4),x]

[Out]

(-32*x*(-2 - 3*x^2)^(3/4))/1053 + (40*x^3*(-2 - 3*x^2)^(3/4))/1053 - (2*x^5*(-2 - 3*x^2)^(3/4))/39 - (128*x*(-
2 - 3*x^2)^(1/4))/(1053*(Sqrt[2] + Sqrt[-2 - 3*x^2])) - (128*2^(1/4)*Sqrt[-(x^2/(Sqrt[2] + Sqrt[-2 - 3*x^2])^2
)]*(Sqrt[2] + Sqrt[-2 - 3*x^2])*EllipticE[2*ArcTan[(-2 - 3*x^2)^(1/4)/2^(1/4)], 1/2])/(1053*Sqrt[3]*x) + (64*2
^(1/4)*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])/(1053*Sqrt[3]*x)

Rule 226

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]

Rule 236

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

Rule 311

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

Rule 327

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,
c, n, m, p, x]

Rule 1210

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a +
c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps

\begin {align*} \int \frac {x^6}{\sqrt [4]{-2-3 x^2}} \, dx &=-\frac {2}{39} x^5 \left (-2-3 x^2\right )^{3/4}-\frac {20}{39} \int \frac {x^4}{\sqrt [4]{-2-3 x^2}} \, dx\\ &=\frac {40 x^3 \left (-2-3 x^2\right )^{3/4}}{1053}-\frac {2}{39} x^5 \left (-2-3 x^2\right )^{3/4}+\frac {80}{351} \int \frac {x^2}{\sqrt [4]{-2-3 x^2}} \, dx\\ &=-\frac {32 x \left (-2-3 x^2\right )^{3/4}}{1053}+\frac {40 x^3 \left (-2-3 x^2\right )^{3/4}}{1053}-\frac {2}{39} x^5 \left (-2-3 x^2\right )^{3/4}-\frac {64 \int \frac {1}{\sqrt [4]{-2-3 x^2}} \, dx}{1053}\\ &=-\frac {32 x \left (-2-3 x^2\right )^{3/4}}{1053}+\frac {40 x^3 \left (-2-3 x^2\right )^{3/4}}{1053}-\frac {2}{39} x^5 \left (-2-3 x^2\right )^{3/4}+\frac {\left (64 \sqrt {\frac {2}{3}} \sqrt {-x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^4}{2}}} \, dx,x,\sqrt [4]{-2-3 x^2}\right )}{1053 x}\\ &=-\frac {32 x \left (-2-3 x^2\right )^{3/4}}{1053}+\frac {40 x^3 \left (-2-3 x^2\right )^{3/4}}{1053}-\frac {2}{39} x^5 \left (-2-3 x^2\right )^{3/4}+\frac {\left (128 \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 )}{1053 \sqrt {3} x}-\frac {\left (128 \sqrt {-x^2}\right ) \text {Subst}\left (\int \frac {1-\frac {x^2}{\sqrt {2}}}{\sqrt {1+\frac {x^4}{2}}} \, dx,x,\sqrt [4]{-2-3 x^2}\right )}{1053 \sqrt {3} x}\\ &=-\frac {32 x \left (-2-3 x^2\right )^{3/4}}{1053}+\frac {40 x^3 \left (-2-3 x^2\right )^{3/4}}{1053}-\frac {2}{39} x^5 \left (-2-3 x^2\right )^{3/4}-\frac {128 x \sqrt [4]{-2-3 x^2}}{1053 \left (\sqrt {2}+\sqrt {-2-3 x^2}\right )}-\frac {128 \sqrt [4]{2} \sqrt {-\frac {x^2}{\left (\sqrt {2}+\sqrt {-2-3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2-3 x^2}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-2-3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{1053 \sqrt {3} x}+\frac {64 \sqrt [4]{2} \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 )}{1053 \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 = 5.90, size = 68, normalized size = 0.26 \begin {gather*} \frac {2 x \left (32+8 x^2-6 x^4+81 x^6-16\ 2^{3/4} \sqrt [4]{2+3 x^2} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-\frac {3 x^2}{2}\right )\right )}{1053 \sqrt [4]{-2-3 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6/(-2 - 3*x^2)^(1/4),x]

[Out]

(2*x*(32 + 8*x^2 - 6*x^4 + 81*x^6 - 16*2^(3/4)*(2 + 3*x^2)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (-3*x^2)/2])
)/(1053*(-2 - 3*x^2)^(1/4))

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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 = 0.09

method	result	size
meijerg	\(-\frac {\left (-1\right )^{\frac {3}{4}} 2^{\frac {3}{4}} x^{7} \hypergeom \left (\left [\frac {1}{4}, \frac {7}{2}\right ], \left [\frac {9}{2}\right ], -\frac {3 x^{2}}{2}\right )}{14}\)	\(23\)
risch	\(\frac {2 x \left (27 x^{4}-20 x^{2}+16\right ) \left (3 x^{2}+2\right )}{1053 \left (-3 x^{2}-2\right )^{\frac {1}{4}}}+\frac {32 \left (-1\right )^{\frac {3}{4}} 2^{\frac {3}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -\frac {3 x^{2}}{2}\right )}{1053}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(-3*x^2-2)^(1/4),x,method=_RETURNVERBOSE)

[Out]

-1/14*(-1)^(3/4)*2^(3/4)*x^7*hypergeom([1/4,7/2],[9/2],-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(x^6/(-3*x^2-2)^(1/4),x, algorithm="maxima")

[Out]

integrate(x^6/(-3*x^2 - 2)^(1/4), 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(x^6/(-3*x^2-2)^(1/4),x, algorithm="fricas")

[Out]

1/3159*(3159*x*integral(256/3159*(-3*x^2 - 2)^(3/4)/(3*x^4 + 2*x^2), x) - 2*(81*x^6 - 60*x^4 + 48*x^2 - 64)*(-
3*x^2 - 2)^(3/4))/x

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Sympy [C] Result contains complex when optimal does not.
time = 0.45, size = 34, normalized size = 0.13 \begin {gather*} \frac {2^{\frac {3}{4}} x^{7} e^{- \frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{i \pi }}{2}} \right )}}{14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(-3*x**2-2)**(1/4),x)

[Out]

2**(3/4)*x**7*exp(-I*pi/4)*hyper((1/4, 7/2), (9/2,), 3*x**2*exp_polar(I*pi)/2)/14

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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(x^6/(-3*x^2-2)^(1/4),x, algorithm="giac")

[Out]

integrate(x^6/(-3*x^2 - 2)^(1/4), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^6}{{\left (-3\,x^2-2\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(- 3*x^2 - 2)^(1/4),x)

[Out]

int(x^6/(- 3*x^2 - 2)^(1/4), x)

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