3.1088 \(\int \frac{x^4}{(-2+3 x^2) (-1+3 x^2)^{3/4}} \, dx\)

Optimal. Leaf size=147 \[ \frac{2 \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right ),\frac{1}{2}\right )}{27 \sqrt{3} x}+\frac{2}{27} \sqrt [4]{3 x^2-1} x+\frac{1}{9} \sqrt{\frac{2}{3}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac{1}{9} \sqrt{\frac{2}{3}} \tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right ) \]

[Out]

(2*x*(-1 + 3*x^2)^(1/4))/27 + (Sqrt[2/3]*ArcTan[(Sqrt[3/2]*x)/(-1 + 3*x^2)^(1/4)])/9 - (Sqrt[2/3]*ArcTanh[(Sqr
t[3/2]*x)/(-1 + 3*x^2)^(1/4)])/9 + (2*Sqrt[x^2/(1 + Sqrt[-1 + 3*x^2])^2]*(1 + Sqrt[-1 + 3*x^2])*EllipticF[2*Ar
cTan[(-1 + 3*x^2)^(1/4)], 1/2])/(27*Sqrt[3]*x)

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Rubi [A]  time = 0.13704, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {443, 234, 220, 321, 400, 442} \[ \frac{2}{27} \sqrt [4]{3 x^2-1} x+\frac{1}{9} \sqrt{\frac{2}{3}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac{1}{9} \sqrt{\frac{2}{3}} \tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )+\frac{2 \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{27 \sqrt{3} x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(-1 + 3*x^2)^(1/4))/27 + (Sqrt[2/3]*ArcTan[(Sqrt[3/2]*x)/(-1 + 3*x^2)^(1/4)])/9 - (Sqrt[2/3]*ArcTanh[(Sqr
t[3/2]*x)/(-1 + 3*x^2)^(1/4)])/9 + (2*Sqrt[x^2/(1 + Sqrt[-1 + 3*x^2])^2]*(1 + Sqrt[-1 + 3*x^2])*EllipticF[2*Ar
cTan[(-1 + 3*x^2)^(1/4)], 1/2])/(27*Sqrt[3]*x)

Rule 443

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])

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 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 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]

Rule 400

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 442

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

Rubi steps

\begin{align*} \int \frac{x^4}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx &=\int \left (\frac{2}{9 \left (-1+3 x^2\right )^{3/4}}+\frac{x^2}{3 \left (-1+3 x^2\right )^{3/4}}+\frac{4}{9 \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}}\right ) \, dx\\ &=\frac{2}{9} \int \frac{1}{\left (-1+3 x^2\right )^{3/4}} \, dx+\frac{1}{3} \int \frac{x^2}{\left (-1+3 x^2\right )^{3/4}} \, dx+\frac{4}{9} \int \frac{1}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx\\ &=\frac{2}{27} x \sqrt [4]{-1+3 x^2}+\frac{2}{27} \int \frac{1}{\left (-1+3 x^2\right )^{3/4}} \, dx-\frac{2}{9} \int \frac{1}{\left (-1+3 x^2\right )^{3/4}} \, dx+\frac{2}{3} \int \frac{x^2}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx+\frac{\left (4 \sqrt{x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{9 \sqrt{3} x}\\ &=\frac{2}{27} x \sqrt [4]{-1+3 x^2}+\frac{1}{9} \sqrt{\frac{2}{3}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac{1}{9} \sqrt{\frac{2}{3}} \tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )+\frac{2 \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 )}{9 \sqrt{3} x}+\frac{\left (4 \sqrt{x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{27 \sqrt{3} x}-\frac{\left (4 \sqrt{x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{9 \sqrt{3} x}\\ &=\frac{2}{27} x \sqrt [4]{-1+3 x^2}+\frac{1}{9} \sqrt{\frac{2}{3}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac{1}{9} \sqrt{\frac{2}{3}} \tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )+\frac{2 \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}\\ \end{align*}

Mathematica [C]  time = 0.0820678, size = 179, normalized size = 1.22 \[ \frac{2 x \left (-2 \left (1-3 x^2\right )^{3/4} x^2 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )-\frac{4 F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right )}{\left (3 x^2-2\right ) \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{3}{4},2;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )+3 F_1\left (\frac{3}{2};\frac{7}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )\right )+2 F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right )\right )}+3 x^2-1\right )}{27 \left (3 x^2-1\right )^{3/4}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*x*(-1 + 3*x^2 - 2*x^2*(1 - 3*x^2)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, 3*x^2, (3*x^2)/2] - (4*AppellF1[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])))))/(27*(-1 + 3*x^2)^(3/4
))

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{4}}{3\,{x}^{2}-2} \left ( 3\,{x}^{2}-1 \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}}{\left (3 \, x^{2} - 2\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} x^{4}}{9 \, x^{4} - 9 \, x^{2} + 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\left (3 x^{2} - 2\right ) \left (3 x^{2} - 1\right )^{\frac{3}{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}}{\left (3 \, x^{2} - 2\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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