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

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)) - ArcTan
h[(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 = 0.0538701, 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 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{a} \sqrt [4]{3 x^2-a}}\right )}{2 \sqrt{6} a^{3/4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{a} \sqrt [4]{3 x^2-a}}\right )}{2 \sqrt{6} a^{3/4}} \]

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

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Rubi in Sympy [A]  time = 90.765, size = 231, normalized size = 2.72 \[ \frac{\sqrt{6} \sqrt{- \frac{1}{\sqrt{a}}} \sqrt{\frac{x^{2}}{a}} \operatorname{atan}{\left (\frac{\sqrt{6} \sqrt{- \frac{1}{\sqrt{a}}} \sqrt [4]{- a + 3 x^{2}}}{3 \sqrt{\frac{x^{2}}{a}}} \right )}}{12 x} + \frac{x \sqrt [4]{- a} \Pi \left (\frac{\sqrt{a}}{\sqrt{- a}}; \operatorname{asin}{\left (\frac{\sqrt [4]{- a + 3 x^{2}}}{\sqrt [4]{- a}} \right )}\middle | -1\right )}{a^{\frac{3}{2}} \sqrt{1 - \frac{\sqrt{- a + 3 x^{2}}}{\sqrt{- a}}} \sqrt{1 + \frac{\sqrt{- a + 3 x^{2}}}{\sqrt{- a}}}} - \frac{\sqrt{3} \sqrt{\frac{x^{2}}{\left (\sqrt{a} + \sqrt{- a + 3 x^{2}}\right )^{2}}} \left (\sqrt{a} + \sqrt{- a + 3 x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{- a + 3 x^{2}}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{12 a^{\frac{3}{4}} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3*x**2-2*a)/(3*x**2-a)**(1/4),x)

[Out]

sqrt(6)*sqrt(-1/sqrt(a))*sqrt(x**2/a)*atan(sqrt(6)*sqrt(-1/sqrt(a))*(-a + 3*x**2
)**(1/4)/(3*sqrt(x**2/a)))/(12*x) + x*(-a)**(1/4)*elliptic_pi(sqrt(a)/sqrt(-a),
asin((-a + 3*x**2)**(1/4)/(-a)**(1/4)), -1)/(a**(3/2)*sqrt(1 - sqrt(-a + 3*x**2)
/sqrt(-a))*sqrt(1 + sqrt(-a + 3*x**2)/sqrt(-a))) - sqrt(3)*sqrt(x**2/(sqrt(a) +
sqrt(-a + 3*x**2))**2)*(sqrt(a) + sqrt(-a + 3*x**2))*elliptic_f(2*atan((-a + 3*x
**2)**(1/4)/a**(1/4)), 1/2)/(12*a**(3/4)*x)

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Mathematica [C]  time = 0.269638, 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)] + AppellF1[3/2, 5/4, 1,
5/2, (3*x^2)/a, (3*x^2)/(2*a)])))

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \[ \int{\frac{1}{3\,{x}^{2}-2\,a}{\frac{1}{\sqrt [4]{3\,{x}^{2}-a}}}}\, dx \]

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

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 - a)^(1/4)*(3*x^2 - 2*a)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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

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

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 - a)^(1/4)*(3*x^2 - 2*a)), x)