3.1078 \(\int \frac{x^2}{\left (-2 a-3 x^2\right ) \left (-a-3 x^2\right )^{3/4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0956848, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{3 \sqrt{6} \sqrt [4]{a}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{3 \sqrt{6} \sqrt [4]{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 34.3596, size = 54, normalized size = 0.64 \[ \frac{x^{3} \sqrt [4]{- a - 3 x^{2}} \operatorname{appellf_{1}}{\left (\frac{3}{2},\frac{3}{4},1,\frac{5}{2},- \frac{3 x^{2}}{a},- \frac{3 x^{2}}{2 a} \right )}}{6 a^{2} \sqrt [4]{1 + \frac{3 x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*(-a - 3*x**2)**(1/4)*appellf1(3/2, 3/4, 1, 5/2, -3*x**2/a, -3*x**2/(2*a))/(
6*a**2*(1 + 3*x**2/a)**(1/4))

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Mathematica [C]  time = 0.276683, size = 164, normalized size = 1.93 \[ \frac{10 a x^3 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right )}{3 \left (-a-3 x^2\right )^{3/4} \left (2 a+3 x^2\right ) \left (3 x^2 \left (2 F_1\left (\frac{5}{2};\frac{3}{4},2;\frac{7}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right )+3 F_1\left (\frac{5}{2};\frac{7}{4},1;\frac{7}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right )\right )-10 a F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(10*a*x^3*AppellF1[3/2, 3/4, 1, 5/2, (-3*x^2)/a, (-3*x^2)/(2*a)])/(3*(-a - 3*x^2
)^(3/4)*(2*a + 3*x^2)*(-10*a*AppellF1[3/2, 3/4, 1, 5/2, (-3*x^2)/a, (-3*x^2)/(2*
a)] + 3*x^2*(2*AppellF1[5/2, 3/4, 2, 7/2, (-3*x^2)/a, (-3*x^2)/(2*a)] + 3*Appell
F1[5/2, 7/4, 1, 7/2, (-3*x^2)/a, (-3*x^2)/(2*a)])))

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Maple [F]  time = 0.075, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2}}{-3\,{x}^{2}-2\,a} \left ( -3\,{x}^{2}-a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-3*x^2-2*a)/(-3*x^2-a)^(3/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2/((3*x^2 + 2*a)*(-3*x^2 - a)^(3/4)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.235559, size = 185, normalized size = 2.18 \[ \frac{2 \, \left (\frac{1}{36}\right )^{\frac{1}{4}} \arctan \left (\frac{3 \, \left (\frac{1}{36}\right )^{\frac{1}{4}} x}{{\left (\sqrt{\frac{1}{2}} x \sqrt{\frac{\frac{3 \, x^{2}}{\sqrt{a}} + 2 \, \sqrt{-3 \, x^{2} - a}}{x^{2}}} +{\left (-3 \, x^{2} - a\right )}^{\frac{1}{4}}\right )} a^{\frac{1}{4}}}\right )}{3 \, a^{\frac{1}{4}}} - \frac{\left (\frac{1}{36}\right )^{\frac{1}{4}} \log \left (\frac{\frac{3 \, \left (\frac{1}{36}\right )^{\frac{1}{4}} x}{a^{\frac{1}{4}}} +{\left (-3 \, x^{2} - a\right )}^{\frac{1}{4}}}{x}\right )}{6 \, a^{\frac{1}{4}}} + \frac{\left (\frac{1}{36}\right )^{\frac{1}{4}} \log \left (-\frac{\frac{3 \, \left (\frac{1}{36}\right )^{\frac{1}{4}} x}{a^{\frac{1}{4}}} -{\left (-3 \, x^{2} - a\right )}^{\frac{1}{4}}}{x}\right )}{6 \, a^{\frac{1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2/((3*x^2 + 2*a)*(-3*x^2 - a)^(3/4)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2/((3*x^2 + 2*a)*(-3*x^2 - a)^(3/4)),x, algorithm="giac")

[Out]

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