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

Optimal. Leaf size=120 \[ \frac{\tan ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a-3 x^2}}{\sqrt{a}}\right )}{\sqrt{3} x \sqrt [4]{a-3 x^2}}\right )}{2 \sqrt{3} a^{3/4}}+\frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (\frac{\sqrt{a-3 x^2}}{\sqrt{a}}+1\right )}{\sqrt{3} x \sqrt [4]{a-3 x^2}}\right )}{2 \sqrt{3} a^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.0171325, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {397} \[ \frac{\tan ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a-3 x^2}}{\sqrt{a}}\right )}{\sqrt{3} x \sqrt [4]{a-3 x^2}}\right )}{2 \sqrt{3} a^{3/4}}+\frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (\frac{\sqrt{a-3 x^2}}{\sqrt{a}}+1\right )}{\sqrt{3} x \sqrt [4]{a-3 x^2}}\right )}{2 \sqrt{3} a^{3/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 397

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

Rubi steps

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

Mathematica [C]  time = 0.153428, size = 155, normalized size = 1.29 \[ -\frac{2 a x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right )}{\sqrt [4]{a-3 x^2} \left (3 x^2-2 a\right ) \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right )\right )+2 a F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a - 3*x^2)^(1/4)*(2*a - 3*x^2)),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*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 = 0.044, 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+a)^(1/4)/(-3*x^2+2*a),x)

[Out]

int(1/(-3*x^2+a)^(1/4)/(-3*x^2+2*a),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+a)^(1/4)/(-3*x^2+2*a),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 = 62.8448, size = 836, normalized size = 6.97 \begin{align*} \left (\frac{1}{36}\right )^{\frac{1}{4}} \left (-\frac{1}{a^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{2 \,{\left (\sqrt{\frac{1}{2}}{\left (6 \, \left (\frac{1}{36}\right )^{\frac{3}{4}} a^{3} \left (-\frac{1}{a^{3}}\right )^{\frac{3}{4}} - \left (\frac{1}{36}\right )^{\frac{1}{4}} \sqrt{-3 \, x^{2} + a} a \left (-\frac{1}{a^{3}}\right )^{\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 \left (-\frac{1}{a^{3}}\right )^{\frac{1}{4}}\right )}}{x}\right ) + \frac{1}{4} \, \left (\frac{1}{36}\right )^{\frac{1}{4}} \left (-\frac{1}{a^{3}}\right )^{\frac{1}{4}} \log \left (-\frac{18 \, \left (\frac{1}{36}\right )^{\frac{3}{4}} \sqrt{-3 \, x^{2} + a} a^{2} x \left (-\frac{1}{a^{3}}\right )^{\frac{3}{4}} +{\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 x \left (-\frac{1}{a^{3}}\right )^{\frac{1}{4}} -{\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}} \left (-\frac{1}{a^{3}}\right )^{\frac{1}{4}} \log \left (\frac{18 \, \left (\frac{1}{36}\right )^{\frac{3}{4}} \sqrt{-3 \, x^{2} + a} a^{2} x \left (-\frac{1}{a^{3}}\right )^{\frac{3}{4}} -{\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 x \left (-\frac{1}{a^{3}}\right )^{\frac{1}{4}} +{\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+a)^(1/4)/(-3*x^2+2*a),x, algorithm="fricas")

[Out]

(1/36)^(1/4)*(-1/a^3)^(1/4)*arctan(2*(sqrt(1/2)*(6*(1/36)^(3/4)*a^3*(-1/a^3)^(3/4) - (1/36)^(1/4)*sqrt(-3*x^2
+ a)*a*(-1/a^3)^(1/4))*sqrt(a*sqrt(-1/a^3)) - (1/36)^(1/4)*(-3*x^2 + a)^(1/4)*a*(-1/a^3)^(1/4))/x) + 1/4*(1/36
)^(1/4)*(-1/a^3)^(1/4)*log(-(18*(1/36)^(3/4)*sqrt(-3*x^2 + a)*a^2*x*(-1/a^3)^(3/4) + (-3*x^2 + a)^(1/4)*a^2*sq
rt(-1/a^3) + 3*(1/36)^(1/4)*a*x*(-1/a^3)^(1/4) - (-3*x^2 + a)^(3/4))/(3*x^2 - 2*a)) - 1/4*(1/36)^(1/4)*(-1/a^3
)^(1/4)*log((18*(1/36)^(3/4)*sqrt(-3*x^2 + a)*a^2*x*(-1/a^3)^(3/4) - (-3*x^2 + a)^(1/4)*a^2*sqrt(-1/a^3) + 3*(
1/36)^(1/4)*a*x*(-1/a^3)^(1/4) + (-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+a)**(1/4)/(-3*x**2+2*a),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+a)^(1/4)/(-3*x^2+2*a),x, algorithm="giac")

[Out]

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