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

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 = 0.0235324, 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, Rules used = {442} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{3 \sqrt{6} \sqrt [4]{a}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{3 \sqrt{6} \sqrt [4]{a}} \]

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))

Rule 442

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

Rubi steps

\begin{align*} \int \frac{x^2}{\left (-2 a-3 x^2\right ) \left (-a-3 x^2\right )^{3/4}} \, dx &=\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}}\\ \end{align*}

Mathematica [C]  time = 0.0560152, size = 67, normalized size = 0.79 \[ -\frac{x^3 \left (\frac{a+3 x^2}{a}\right )^{3/4} F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right )}{6 a \left (-a-3 x^2\right )^{3/4}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(x^3*((a + 3*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, (-3*x^2)/a, (-3*x^2)/(2*a)])/(6*a*(-a - 3*x^2)^(3/4))

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{-3\,{x}^{2}-2\,a} \left ( -3\,{x}^{2}-a \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}

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. \begin{align*} -\int \frac{x^{2}}{{\left (3 \, x^{2} + 2 \, a\right )}{\left (-3 \, x^{2} - a\right )}^{\frac{3}{4}}}\,{d x} \end{align*}

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 [B]  time = 1.64697, size = 443, normalized size = 5.21 \begin{align*} \frac{2 \, \left (\frac{1}{36}\right )^{\frac{1}{4}} \arctan \left (\frac{12 \,{\left (\sqrt{\frac{1}{2}} \left (\frac{1}{36}\right )^{\frac{3}{4}} a^{\frac{1}{4}} x \sqrt{\frac{\frac{3 \, x^{2}}{\sqrt{a}} + 2 \, \sqrt{-3 \, x^{2} - a}}{x^{2}}} - \left (\frac{1}{36}\right )^{\frac{3}{4}}{\left (-3 \, x^{2} - a\right )}^{\frac{1}{4}} a^{\frac{1}{4}}\right )}}{x}\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}}} \end{align*}

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(12*(sqrt(1/2)*(1/36)^(3/4)*a^(1/4)*x*sqrt((3*x^2/sqrt(a) + 2*sqrt(-3*x^2 - a))/x^2) -
(1/36)^(3/4)*(-3*x^2 - a)^(1/4)*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) + 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. \begin{align*} - \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 \end{align*}

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

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)