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

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

[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 = 0.0139391, 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} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{2 \sqrt{6} a^{3/4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{2 \sqrt{6} a^{3/4}} \]

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 = 0.146947, size = 157, normalized size = 1.85 \[ \frac{2 a x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right )}{\sqrt [4]{-a-3 x^2} \left (2 a+3 x^2\right ) \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right )\right )-2 a F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right )\right )} \]

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

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Maple [F]  time = 0.04, 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} + 2 \, a\right )}{\left (-3 \, x^{2} - a\right )}^{\frac{1}{4}}}\,{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 + 2*a)*(-3*x^2 - a)^(1/4)), x)

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Fricas [B]  time = 54.896, size = 838, normalized size = 9.86 \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}{2 a \sqrt [4]{- a - 3 x^{2}} + 3 x^{2} \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*(-a - 3*x**2)**(1/4) + 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} + 2 \, a\right )}{\left (-3 \, x^{2} - a\right )}^{\frac{1}{4}}}\,{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 + 2*a)*(-3*x^2 - a)^(1/4)), x)