3.316 \(\int \frac{1}{(-2 a+3 x^2) \sqrt [4]{-a+3 x^2}} \, dx\)

Optimal. Leaf size=85 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{a} \sqrt [4]{3 x^2-a}}\right )}{2 \sqrt{6} a^{3/4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{a} \sqrt [4]{3 x^2-a}}\right )}{2 \sqrt{6} a^{3/4}} \]

[Out]

-ArcTan[(Sqrt[3/2]*x)/(a^(1/4)*(-a + 3*x^2)^(1/4))]/(2*Sqrt[6]*a^(3/4)) - ArcTanh[(Sqrt[3/2]*x)/(a^(1/4)*(-a +
3*x^2)^(1/4))]/(2*Sqrt[6]*a^(3/4))

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Rubi [A]  time = 0.0178728, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {398} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{a} \sqrt [4]{3 x^2-a}}\right )}{2 \sqrt{6} a^{3/4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{a} \sqrt [4]{3 x^2-a}}\right )}{2 \sqrt{6} a^{3/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTan[(Sqrt[3/2]*x)/(a^(1/4)*(-a + 3*x^2)^(1/4))]/(2*Sqrt[6]*a^(3/4)) - ArcTanh[(Sqrt[3/2]*x)/(a^(1/4)*(-a +
3*x^2)^(1/4))]/(2*Sqrt[6]*a^(3/4))

Rule 398

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

Rubi steps

\begin{align*} \int \frac{1}{\left (-2 a+3 x^2\right ) \sqrt [4]{-a+3 x^2}} \, dx &=-\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a+3 x^2}}\right )}{2 \sqrt{6} a^{3/4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a+3 x^2}}\right )}{2 \sqrt{6} a^{3/4}}\\ \end{align*}

Mathematica [C]  time = 0.160353, size = 157, normalized size = 1.85 \[ \frac{2 a x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right )}{\left (3 x^2-2 a\right ) \sqrt [4]{3 x^2-a} \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right )\right )+2 a F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*a*x*AppellF1[1/2, 1/4, 1, 3/2, (3*x^2)/a, (3*x^2)/(2*a)])/((-2*a + 3*x^2)*(-a + 3*x^2)^(1/4)*(2*a*AppellF1[
1/2, 1/4, 1, 3/2, (3*x^2)/a, (3*x^2)/(2*a)] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, (3*x^2)/a, (3*x^2)/(2*a)] + Ap
pellF1[3/2, 5/4, 1, 5/2, (3*x^2)/a, (3*x^2)/(2*a)])))

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{3\,{x}^{2}-2\,a}{\frac{1}{\sqrt [4]{3\,{x}^{2}-a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 64.0755, size = 826, normalized size = 9.72 \begin{align*} -\left (\frac{1}{36}\right )^{\frac{1}{4}} \frac{1}{a^{3}}^{\frac{1}{4}} \arctan \left (\frac{2 \,{\left (\sqrt{\frac{1}{2}}{\left (6 \, \left (\frac{1}{36}\right )^{\frac{3}{4}} a^{3} \frac{1}{a^{3}}^{\frac{3}{4}} + \left (\frac{1}{36}\right )^{\frac{1}{4}} \sqrt{3 \, x^{2} - a} a \frac{1}{a^{3}}^{\frac{1}{4}}\right )} \sqrt{a \sqrt{\frac{1}{a^{3}}}} - \left (\frac{1}{36}\right )^{\frac{1}{4}}{\left (3 \, x^{2} - a\right )}^{\frac{1}{4}} a \frac{1}{a^{3}}^{\frac{1}{4}}\right )}}{x}\right ) - \frac{1}{4} \, \left (\frac{1}{36}\right )^{\frac{1}{4}} \frac{1}{a^{3}}^{\frac{1}{4}} \log \left (\frac{18 \, \left (\frac{1}{36}\right )^{\frac{3}{4}} \sqrt{3 \, x^{2} - a} a^{2} \frac{1}{a^{3}}^{\frac{3}{4}} x +{\left (3 \, x^{2} - a\right )}^{\frac{1}{4}} a^{2} \sqrt{\frac{1}{a^{3}}} + 3 \, \left (\frac{1}{36}\right )^{\frac{1}{4}} a \frac{1}{a^{3}}^{\frac{1}{4}} x +{\left (3 \, x^{2} - a\right )}^{\frac{3}{4}}}{3 \, x^{2} - 2 \, a}\right ) + \frac{1}{4} \, \left (\frac{1}{36}\right )^{\frac{1}{4}} \frac{1}{a^{3}}^{\frac{1}{4}} \log \left (-\frac{18 \, \left (\frac{1}{36}\right )^{\frac{3}{4}} \sqrt{3 \, x^{2} - a} a^{2} \frac{1}{a^{3}}^{\frac{3}{4}} x -{\left (3 \, x^{2} - a\right )}^{\frac{1}{4}} a^{2} \sqrt{\frac{1}{a^{3}}} + 3 \, \left (\frac{1}{36}\right )^{\frac{1}{4}} a \frac{1}{a^{3}}^{\frac{1}{4}} x -{\left (3 \, x^{2} - a\right )}^{\frac{3}{4}}}{3 \, x^{2} - 2 \, a}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1/36)^(1/4)*(a^(-3))^(1/4)*arctan(2*(sqrt(1/2)*(6*(1/36)^(3/4)*a^3*(a^(-3))^(3/4) + (1/36)^(1/4)*sqrt(3*x^2
- a)*a*(a^(-3))^(1/4))*sqrt(a*sqrt(a^(-3))) - (1/36)^(1/4)*(3*x^2 - a)^(1/4)*a*(a^(-3))^(1/4))/x) - 1/4*(1/36)
^(1/4)*(a^(-3))^(1/4)*log((18*(1/36)^(3/4)*sqrt(3*x^2 - a)*a^2*(a^(-3))^(3/4)*x + (3*x^2 - a)^(1/4)*a^2*sqrt(a
^(-3)) + 3*(1/36)^(1/4)*a*(a^(-3))^(1/4)*x + (3*x^2 - a)^(3/4))/(3*x^2 - 2*a)) + 1/4*(1/36)^(1/4)*(a^(-3))^(1/
4)*log(-(18*(1/36)^(3/4)*sqrt(3*x^2 - a)*a^2*(a^(-3))^(3/4)*x - (3*x^2 - a)^(1/4)*a^2*sqrt(a^(-3)) + 3*(1/36)^
(1/4)*a*(a^(-3))^(1/4)*x - (3*x^2 - a)^(3/4))/(3*x^2 - 2*a))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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