3.905 \(\int \frac {x^6}{(-2+3 x^2)^{3/4}} \, dx\)

Optimal. Leaf size=138 \[ \frac {160\ 2^{3/4} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {3 x^2-2}+\sqrt {2}\right ) \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{2079 \sqrt {3} x}+\frac {160 \sqrt [4]{3 x^2-2} x}{2079}+\frac {2}{33} \sqrt [4]{3 x^2-2} x^5+\frac {40}{693} \sqrt [4]{3 x^2-2} x^3 \]

[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)+160/6237*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 = 0.06, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {321, 234, 220} \[ \frac {2}{33} \sqrt [4]{3 x^2-2} x^5+\frac {40}{693} \sqrt [4]{3 x^2-2} x^3+\frac {160 \sqrt [4]{3 x^2-2} x}{2079}+\frac {160\ 2^{3/4} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {3 x^2-2}+\sqrt {2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2079 \sqrt {3} x} \]

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 + (160*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])/(2079*Sqrt[3]*x)

Rule 220

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

Rule 234

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]

Rule 321

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 {\left (320 \sqrt {\frac {2}{3}} \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{2}}} \, dx,x,\sqrt [4]{-2+3 x^2}\right )}{2079 x}\\ &=\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 {160\ 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 )}{2079 \sqrt {3} x}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 68, normalized size = 0.49 \[ \frac {2 x \left (80 \sqrt [4]{2} \left (2-3 x^2\right )^{3/4} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};\frac {3 x^2}{2}\right )+189 x^6+54 x^4+120 x^2-160\right )}{2079 \left (3 x^2-2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(-2 + 3*x^2)^(3/4),x]

[Out]

(2*x*(-160 + 120*x^2 + 54*x^4 + 189*x^6 + 80*2^(1/4)*(2 - 3*x^2)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (3*x^2
)/2]))/(2079*(-2 + 3*x^2)^(3/4))

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fricas [F]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{6}}{{\left (3 \, x^{2} - 2\right )}^{\frac {3}{4}}}, x\right ) \]

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(x^6/(3*x^2 - 2)^(3/4), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{{\left (3 \, x^{2} - 2\right )}^{\frac {3}{4}}}\,{d x} \]

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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maple [C]  time = 0.30, size = 65, normalized size = 0.47 \[ \frac {160 \,2^{\frac {1}{4}} \left (-\mathrm {signum}\left (\frac {3 x^{2}}{2}-1\right )\right )^{\frac {3}{4}} x \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {3}{2}\right ], \frac {3 x^{2}}{2}\right )}{2079 \mathrm {signum}\left (\frac {3 x^{2}}{2}-1\right )^{\frac {3}{4}}}+\frac {2 \left (63 x^{4}+60 x^{2}+80\right ) \left (3 x^{2}-2\right )^{\frac {1}{4}} x}{2079} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(3*x^2-2)^(3/4),x)

[Out]

2/2079*x*(63*x^4+60*x^2+80)*(3*x^2-2)^(1/4)+160/2079*2^(1/4)/signum(3/2*x^2-1)^(3/4)*(-signum(3/2*x^2-1))^(3/4
)*x*hypergeom([1/2,3/4],[3/2],3/2*x^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{{\left (3 \, x^{2} - 2\right )}^{\frac {3}{4}}}\,{d x} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(3*x^2 - 2)^(3/4),x)

[Out]

int(x^6/(3*x^2 - 2)^(3/4), x)

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sympy [C]  time = 0.86, size = 31, normalized size = 0.22 \[ \frac {\sqrt [4]{2} x^{7} e^{- \frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {3 x^{2}}{2}} \right )}}{14} \]

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

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