3.9.84 \(\int \frac {x^6}{(2-3 x^2)^{3/4}} \, dx\) [884]

Optimal. Leaf size=83 \[ -\frac {160 x \sqrt [4]{2-3 x^2}}{2079}-\frac {40}{693} x^3 \sqrt [4]{2-3 x^2}-\frac {2}{33} x^5 \sqrt [4]{2-3 x^2}+\frac {320\ 2^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{2079 \sqrt {3}} \]

[Out]

-160/2079*x*(-3*x^2+2)^(1/4)-40/693*x^3*(-3*x^2+2)^(1/4)-2/33*x^5*(-3*x^2+2)^(1/4)+320/6237*2^(3/4)*(cos(1/2*a
rcsin(1/2*x*6^(1/2)))^2)^(1/2)/cos(1/2*arcsin(1/2*x*6^(1/2)))*EllipticF(sin(1/2*arcsin(1/2*x*6^(1/2))),2^(1/2)
)*3^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {327, 238} \begin {gather*} \frac {320\ 2^{3/4} F\left (\left .\frac {1}{2} \text {ArcSin}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{2079 \sqrt {3}}-\frac {160 \sqrt [4]{2-3 x^2} x}{2079}-\frac {2}{33} \sqrt [4]{2-3 x^2} x^5-\frac {40}{693} \sqrt [4]{2-3 x^2} x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6/(2 - 3*x^2)^(3/4),x]

[Out]

(-160*x*(2 - 3*x^2)^(1/4))/2079 - (40*x^3*(2 - 3*x^2)^(1/4))/693 - (2*x^5*(2 - 3*x^2)^(1/4))/33 + (320*2^(3/4)
*EllipticF[ArcSin[Sqrt[3/2]*x]/2, 2])/(2079*Sqrt[3])

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 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]

Rubi steps

\begin {align*} \int \frac {x^6}{\left (2-3 x^2\right )^{3/4}} \, dx &=-\frac {2}{33} x^5 \sqrt [4]{2-3 x^2}+\frac {20}{33} \int \frac {x^4}{\left (2-3 x^2\right )^{3/4}} \, dx\\ &=-\frac {40}{693} x^3 \sqrt [4]{2-3 x^2}-\frac {2}{33} x^5 \sqrt [4]{2-3 x^2}+\frac {80}{231} \int \frac {x^2}{\left (2-3 x^2\right )^{3/4}} \, dx\\ &=-\frac {160 x \sqrt [4]{2-3 x^2}}{2079}-\frac {40}{693} x^3 \sqrt [4]{2-3 x^2}-\frac {2}{33} x^5 \sqrt [4]{2-3 x^2}+\frac {320 \int \frac {1}{\left (2-3 x^2\right )^{3/4}} \, dx}{2079}\\ &=-\frac {160 x \sqrt [4]{2-3 x^2}}{2079}-\frac {40}{693} x^3 \sqrt [4]{2-3 x^2}-\frac {2}{33} x^5 \sqrt [4]{2-3 x^2}+\frac {320\ 2^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{2079 \sqrt {3}}\\ \end {align*}

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Mathematica [A]
time = 6.26, size = 59, normalized size = 0.71 \begin {gather*} \frac {-6 x \sqrt [4]{2-3 x^2} \left (80+60 x^2+63 x^4\right )+320\ 2^{3/4} \sqrt {3} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{6237} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6/(2 - 3*x^2)^(3/4),x]

[Out]

(-6*x*(2 - 3*x^2)^(1/4)*(80 + 60*x^2 + 63*x^4) + 320*2^(3/4)*Sqrt[3]*EllipticF[ArcSin[Sqrt[3/2]*x]/2, 2])/6237

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 4.
time = 0.10, size = 20, normalized size = 0.24

method	result	size
meijerg	\(\frac {2^{\frac {1}{4}} x^{7} \hypergeom \left (\left [\frac {3}{4}, \frac {7}{2}\right ], \left [\frac {9}{2}\right ], \frac {3 x^{2}}{2}\right )}{14}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

integral(-(-3*x^2 + 2)^(1/4)*x^6/(3*x^2 - 2), x)

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Sympy [C] Result contains complex when optimal does not.
time = 0.46, size = 29, normalized size = 0.35 \begin {gather*} \frac {\sqrt [4]{2} x^{7} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{2 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)**(3/4),x)

[Out]

2**(1/4)*x**7*hyper((3/4, 7/2), (9/2,), 3*x**2*exp_polar(2*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)^(3/4),x, algorithm="giac")

[Out]

integrate(x^6/(-3*x^2 + 2)^(3/4), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^6}{{\left (2-3\,x^2\right )}^{3/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(2 - 3*x^2)^(3/4),x)

[Out]

int(x^6/(2 - 3*x^2)^(3/4), x)

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