\(\int \frac {x^6}{(2-3 x^2)^{3/4} (4-3 x^2)} \, dx\) [1067]

Optimal result

Integrand size = 24, antiderivative size = 182 \[ \int \frac {x^6}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {80}{567} x \sqrt [4]{2-3 x^2}+\frac {2}{63} x^3 \sqrt [4]{2-3 x^2}+\frac {8\ 2^{3/4} \arctan \left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt {3}}-\frac {8\ 2^{3/4} \text {arctanh}\left (\frac {2^{3/4}+\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt {3}}-\frac {160\ 2^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\sqrt {\frac {3}{2}} x\right ),2\right )}{567 \sqrt {3}} \]

[Out]

80/567*x*(-3*x^2+2)^(1/4)+2/63*x^3*(-3*x^2+2)^(1/4)+8/81*2^(3/4)*arctan(1/3*(2^(3/4)-2^(1/4)*(-3*x^2+2)^(1/2))
/x/(-3*x^2+2)^(1/4)*3^(1/2))*3^(1/2)-8/81*2^(3/4)*arctanh(1/3*(2^(3/4)+2^(1/4)*(-3*x^2+2)^(1/2))/x/(-3*x^2+2)^
(1/4)*3^(1/2))*3^(1/2)-160/1701*2^(3/4)*(cos(1/2*arcsin(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)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {454, 238, 327, 409, 452} \[ \int \frac {x^6}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {160\ 2^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\sqrt {\frac {3}{2}} x\right ),2\right )}{567 \sqrt {3}}+\frac {8\ 2^{3/4} \arctan \left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt {3}}-\frac {8\ 2^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {2-3 x^2}+2^{3/4}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt {3}}+\frac {80}{567} \sqrt [4]{2-3 x^2} x+\frac {2}{63} \sqrt [4]{2-3 x^2} x^3 \]

[In]

Int[x^6/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]

[Out]

(80*x*(2 - 3*x^2)^(1/4))/567 + (2*x^3*(2 - 3*x^2)^(1/4))/63 + (8*2^(3/4)*ArcTan[(2^(3/4) - 2^(1/4)*Sqrt[2 - 3*
x^2])/(Sqrt[3]*x*(2 - 3*x^2)^(1/4))])/(27*Sqrt[3]) - (8*2^(3/4)*ArcTanh[(2^(3/4) + 2^(1/4)*Sqrt[2 - 3*x^2])/(S
qrt[3]*x*(2 - 3*x^2)^(1/4))])/(27*Sqrt[3]) - (160*2^(3/4)*EllipticF[ArcSin[Sqrt[3/2]*x]/2, 2])/(567*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]

Rule 409

Int[1/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Dist[1/c, Int[1/(a + b*x^2)^(3/4), x],
x] - Dist[d/c, Int[x^2/((a + b*x^2)^(3/4)*(c + d*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d,
0]

Rule 452

Int[(x_)^2/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Simp[(-b/(a*d*Rt[b^2/a, 4]^3))*Ar
cTan[(b + Rt[b^2/a, 4]^2*Sqrt[a + b*x^2])/(Rt[b^2/a, 4]^3*x*(a + b*x^2)^(1/4))], x] + Simp[(b/(a*d*Rt[b^2/a, 4
]^3))*ArcTanh[(b - Rt[b^2/a, 4]^2*Sqrt[a + b*x^2])/(Rt[b^2/a, 4]^3*x*(a + b*x^2)^(1/4))], x] /; FreeQ[{a, b, c
, d}, x] && EqQ[b*c - 2*a*d, 0] && PosQ[b^2/a]

Rule 454

Int[(x_)^(m_)/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Int[ExpandIntegrand[x^m/((a +
b*x^2)^(3/4)*(c + d*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && IntegerQ[m] && (PosQ[a]
|| IntegerQ[m/2])

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {16}{27 \left (2-3 x^2\right )^{3/4}}-\frac {4 x^2}{9 \left (2-3 x^2\right )^{3/4}}-\frac {x^4}{3 \left (2-3 x^2\right )^{3/4}}+\frac {64}{27 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )}\right ) \, dx \\ & = -\left (\frac {1}{3} \int \frac {x^4}{\left (2-3 x^2\right )^{3/4}} \, dx\right )-\frac {4}{9} \int \frac {x^2}{\left (2-3 x^2\right )^{3/4}} \, dx-\frac {16}{27} \int \frac {1}{\left (2-3 x^2\right )^{3/4}} \, dx+\frac {64}{27} \int \frac {1}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx \\ & = \frac {8}{81} x \sqrt [4]{2-3 x^2}+\frac {2}{63} x^3 \sqrt [4]{2-3 x^2}-\frac {16\ 2^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{27 \sqrt {3}}-\frac {4}{21} \int \frac {x^2}{\left (2-3 x^2\right )^{3/4}} \, dx-\frac {16}{81} \int \frac {1}{\left (2-3 x^2\right )^{3/4}} \, dx+\frac {16}{27} \int \frac {1}{\left (2-3 x^2\right )^{3/4}} \, dx+\frac {16}{9} \int \frac {x^2}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx \\ & = \frac {80}{567} x \sqrt [4]{2-3 x^2}+\frac {2}{63} x^3 \sqrt [4]{2-3 x^2}+\frac {8\ 2^{3/4} \tan ^{-1}\left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt {3}}-\frac {8\ 2^{3/4} \tanh ^{-1}\left (\frac {2^{3/4}+\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt {3}}-\frac {16\ 2^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{81 \sqrt {3}}-\frac {16}{189} \int \frac {1}{\left (2-3 x^2\right )^{3/4}} \, dx \\ & = \frac {80}{567} x \sqrt [4]{2-3 x^2}+\frac {2}{63} x^3 \sqrt [4]{2-3 x^2}+\frac {8\ 2^{3/4} \tan ^{-1}\left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt {3}}-\frac {8\ 2^{3/4} \tanh ^{-1}\left (\frac {2^{3/4}+\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt {3}}-\frac {160\ 2^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{567 \sqrt {3}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 6.39 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.04 \[ \int \frac {x^6}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {2}{567} x \left (31 \sqrt [4]{2} x^2 \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},1,\frac {5}{2},\frac {3 x^2}{2},\frac {3 x^2}{4}\right )+\frac {80-102 x^2-27 x^4+\frac {1280 \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},\frac {3 x^2}{2},\frac {3 x^2}{4}\right )}{\left (-4+3 x^2\right ) \left (4 \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},\frac {3 x^2}{2},\frac {3 x^2}{4}\right )+x^2 \left (2 \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},\frac {3 x^2}{2},\frac {3 x^2}{4}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},\frac {3 x^2}{2},\frac {3 x^2}{4}\right )\right )\right )}}{\left (2-3 x^2\right )^{3/4}}\right ) \]

[In]

Integrate[x^6/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]

[Out]

(2*x*(31*2^(1/4)*x^2*AppellF1[3/2, 3/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4] + (80 - 102*x^2 - 27*x^4 + (1280*AppellF
1[1/2, 3/4, 1, 3/2, (3*x^2)/2, (3*x^2)/4])/((-4 + 3*x^2)*(4*AppellF1[1/2, 3/4, 1, 3/2, (3*x^2)/2, (3*x^2)/4] +
x^2*(2*AppellF1[3/2, 3/4, 2, 5/2, (3*x^2)/2, (3*x^2)/4] + 3*AppellF1[3/2, 7/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4])
)))/(2 - 3*x^2)^(3/4)))/567

Maple [F]

\[\int \frac {x^{6}}{\left (-3 x^{2}+2\right )^{\frac {3}{4}} \left (-3 x^{2}+4\right )}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {x^6}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\int { -\frac {x^{6}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}} \,d x } \]

[In]

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

[Out]

integral((-3*x^2 + 2)^(1/4)*x^6/(9*x^4 - 18*x^2 + 8), x)

Sympy [F]

\[ \int \frac {x^6}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=- \int \frac {x^{6}}{3 x^{2} \left (2 - 3 x^{2}\right )^{\frac {3}{4}} - 4 \left (2 - 3 x^{2}\right )^{\frac {3}{4}}}\, dx \]

[In]

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

[Out]

-Integral(x**6/(3*x**2*(2 - 3*x**2)**(3/4) - 4*(2 - 3*x**2)**(3/4)), x)

Maxima [F]

\[ \int \frac {x^6}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\int { -\frac {x^{6}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {x^6}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\int { -\frac {x^{6}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\int \frac {x^6}{{\left (2-3\,x^2\right )}^{3/4}\,\left (3\,x^2-4\right )} \,d x \]

[In]

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

[Out]

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