\(\int \frac {x^6}{(-2+3 x^2) (-1+3 x^2)^{3/4}} \, dx\) [1087]

Optimal result

Integrand size = 24, antiderivative size = 165 \[ \int \frac {x^6}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {40}{567} x \sqrt [4]{-1+3 x^2}+\frac {2}{63} x^3 \sqrt [4]{-1+3 x^2}+\frac {2}{27} \sqrt {\frac {2}{3}} \arctan \left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {2}{27} \sqrt {\frac {2}{3}} \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )+\frac {40 \sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{-1+3 x^2}\right ),\frac {1}{2}\right )}{567 \sqrt {3} x} \]

[Out]

40/567*x*(3*x^2-1)^(1/4)+2/63*x^3*(3*x^2-1)^(1/4)+2/81*arctan(1/2*x*6^(1/2)/(3*x^2-1)^(1/4))*6^(1/2)-2/81*arct
anh(1/2*x*6^(1/2)/(3*x^2-1)^(1/4))*6^(1/2)+40/1701*(cos(2*arctan((3*x^2-1)^(1/4)))^2)^(1/2)/cos(2*arctan((3*x^
2-1)^(1/4)))*EllipticF(sin(2*arctan((3*x^2-1)^(1/4))),1/2*2^(1/2))*(1+(3*x^2-1)^(1/2))*(x^2/(1+(3*x^2-1)^(1/2)
)^2)^(1/2)/x*3^(1/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {454, 240, 226, 327, 409, 453} \[ \int \frac {x^6}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2}{27} \sqrt {\frac {2}{3}} \arctan \left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )+\frac {40 \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-1}+1\right )^2}} \left (\sqrt {3 x^2-1}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{3 x^2-1}\right ),\frac {1}{2}\right )}{567 \sqrt {3} x}-\frac {2}{27} \sqrt {\frac {2}{3}} \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )+\frac {40}{567} \sqrt [4]{3 x^2-1} x+\frac {2}{63} \sqrt [4]{3 x^2-1} x^3 \]

[In]

Int[x^6/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]

[Out]

(40*x*(-1 + 3*x^2)^(1/4))/567 + (2*x^3*(-1 + 3*x^2)^(1/4))/63 + (2*Sqrt[2/3]*ArcTan[(Sqrt[3/2]*x)/(-1 + 3*x^2)
^(1/4)])/27 - (2*Sqrt[2/3]*ArcTanh[(Sqrt[3/2]*x)/(-1 + 3*x^2)^(1/4)])/27 + (40*Sqrt[x^2/(1 + Sqrt[-1 + 3*x^2])
^2]*(1 + Sqrt[-1 + 3*x^2])*EllipticF[2*ArcTan[(-1 + 3*x^2)^(1/4)], 1/2])/(567*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 240

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 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 453

Int[(x_)^2/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Simp[(-b/(Sqrt[2]*a*d*Rt[-b^2/a,
4]^3))*ArcTan[(Rt[-b^2/a, 4]*x)/(Sqrt[2]*(a + b*x^2)^(1/4))], x] + Simp[(b/(Sqrt[2]*a*d*Rt[-b^2/a, 4]^3))*ArcT
anh[(Rt[-b^2/a, 4]*x)/(Sqrt[2]*(a + b*x^2)^(1/4))], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && Neg
Q[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 {4}{27 \left (-1+3 x^2\right )^{3/4}}+\frac {2 x^2}{9 \left (-1+3 x^2\right )^{3/4}}+\frac {x^4}{3 \left (-1+3 x^2\right )^{3/4}}+\frac {8}{27 \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}}\right ) \, dx \\ & = \frac {4}{27} \int \frac {1}{\left (-1+3 x^2\right )^{3/4}} \, dx+\frac {2}{9} \int \frac {x^2}{\left (-1+3 x^2\right )^{3/4}} \, dx+\frac {8}{27} \int \frac {1}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx+\frac {1}{3} \int \frac {x^4}{\left (-1+3 x^2\right )^{3/4}} \, dx \\ & = \frac {4}{81} x \sqrt [4]{-1+3 x^2}+\frac {2}{63} x^3 \sqrt [4]{-1+3 x^2}+\frac {4}{81} \int \frac {1}{\left (-1+3 x^2\right )^{3/4}} \, dx+\frac {2}{21} \int \frac {x^2}{\left (-1+3 x^2\right )^{3/4}} \, dx-\frac {4}{27} \int \frac {1}{\left (-1+3 x^2\right )^{3/4}} \, dx+\frac {4}{9} \int \frac {x^2}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx+\frac {\left (8 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{27 \sqrt {3} x} \\ & = \frac {40}{567} x \sqrt [4]{-1+3 x^2}+\frac {2}{63} x^3 \sqrt [4]{-1+3 x^2}+\frac {2}{27} \sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {2}{27} \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )+\frac {4 \sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )|\frac {1}{2}\right )}{27 \sqrt {3} x}+\frac {4}{189} \int \frac {1}{\left (-1+3 x^2\right )^{3/4}} \, dx+\frac {\left (8 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{81 \sqrt {3} x}-\frac {\left (8 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{27 \sqrt {3} x} \\ & = \frac {40}{567} x \sqrt [4]{-1+3 x^2}+\frac {2}{63} x^3 \sqrt [4]{-1+3 x^2}+\frac {2}{27} \sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {2}{27} \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )+\frac {4 \sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )|\frac {1}{2}\right )}{81 \sqrt {3} x}+\frac {\left (8 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{189 \sqrt {3} x} \\ & = \frac {40}{567} x \sqrt [4]{-1+3 x^2}+\frac {2}{63} x^3 \sqrt [4]{-1+3 x^2}+\frac {2}{27} \sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {2}{27} \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )+\frac {40 \sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )|\frac {1}{2}\right )}{567 \sqrt {3} x} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

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

Time = 6.07 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.12 \[ \int \frac {x^6}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2 x \left (-20+51 x^2+27 x^4-31 x^2 \left (1-3 x^2\right )^{3/4} \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},1,\frac {5}{2},3 x^2,\frac {3 x^2}{2}\right )-\frac {80 \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},3 x^2,\frac {3 x^2}{2}\right )}{\left (-2+3 x^2\right ) \left (2 \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},3 x^2,\frac {3 x^2}{2}\right )+x^2 \left (2 \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},3 x^2,\frac {3 x^2}{2}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},3 x^2,\frac {3 x^2}{2}\right )\right )\right )}\right )}{567 \left (-1+3 x^2\right )^{3/4}} \]

[In]

Integrate[x^6/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4)),x]

[Out]

(2*x*(-20 + 51*x^2 + 27*x^4 - 31*x^2*(1 - 3*x^2)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, 3*x^2, (3*x^2)/2] - (80*Appe
llF1[1/2, 3/4, 1, 3/2, 3*x^2, (3*x^2)/2])/((-2 + 3*x^2)*(2*AppellF1[1/2, 3/4, 1, 3/2, 3*x^2, (3*x^2)/2] + x^2*
(2*AppellF1[3/2, 3/4, 2, 5/2, 3*x^2, (3*x^2)/2] + 3*AppellF1[3/2, 7/4, 1, 5/2, 3*x^2, (3*x^2)/2])))))/(567*(-1
+ 3*x^2)^(3/4))

Maple [F]

\[\int \frac {x^{6}}{\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )^{\frac {3}{4}}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {x^6}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\int { \frac {x^{6}}{{\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} {\left (3 \, x^{2} - 2\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x^6}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\int \frac {x^{6}}{\left (3 x^{2} - 2\right ) \left (3 x^{2} - 1\right )^{\frac {3}{4}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {x^6}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\int { \frac {x^{6}}{{\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} {\left (3 \, x^{2} - 2\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {x^6}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\int { \frac {x^{6}}{{\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} {\left (3 \, x^{2} - 2\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\int \frac {x^6}{{\left (3\,x^2-1\right )}^{3/4}\,\left (3\,x^2-2\right )} \,d x \]

[In]

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

[Out]

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