3.1074 \(\int \frac{x^2}{(-2-3 x^2) (-1-3 x^2)^{3/4}} \, dx\)

Optimal. Leaf size=61 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{3 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{3 \sqrt{6}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.016959, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {442} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{3 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{3 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 (-2-3 x^2\right ) \left (-1-3 x^2\right )^{3/4}} \, dx &=\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{3 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{3 \sqrt{6}}\\ \end{align*}

Mathematica [C]  time = 0.0417319, size = 52, normalized size = 0.85 \[ -\frac{x^3 \left (3 x^2+1\right )^{3/4} F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-3 x^2,-\frac{3 x^2}{2}\right )}{6 \left (-3 x^2-1\right )^{3/4}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{-3\,{x}^{2}-2} \left ( -3\,{x}^{2}-1 \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)/(-3*x^2-1)^(3/4),x)

[Out]

int(x^2/(-3*x^2-2)/(-3*x^2-1)^(3/4),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{x^{2}}{{\left (3 \, x^{2} + 2\right )}{\left (-3 \, x^{2} - 1\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)/(-3*x^2-1)^(3/4),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [C]  time = 1.66068, size = 343, normalized size = 5.62 \begin{align*} -\frac{1}{36} \, \sqrt{6} \log \left (\frac{\sqrt{6} x + 2 \,{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}}{2 \, x}\right ) + \frac{1}{36} \, \sqrt{6} \log \left (-\frac{\sqrt{6} x - 2 \,{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}}{2 \, x}\right ) - \frac{1}{36} i \, \sqrt{6} \log \left (\frac{i \, \sqrt{6} x + 2 \,{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}}{2 \, x}\right ) + \frac{1}{36} i \, \sqrt{6} \log \left (\frac{-i \, \sqrt{6} x + 2 \,{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}}{2 \, x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-3*x^2-2)/(-3*x^2-1)^(3/4),x, algorithm="fricas")

[Out]

-1/36*sqrt(6)*log(1/2*(sqrt(6)*x + 2*(-3*x^2 - 1)^(1/4))/x) + 1/36*sqrt(6)*log(-1/2*(sqrt(6)*x - 2*(-3*x^2 - 1
)^(1/4))/x) - 1/36*I*sqrt(6)*log(1/2*(I*sqrt(6)*x + 2*(-3*x^2 - 1)^(1/4))/x) + 1/36*I*sqrt(6)*log(1/2*(-I*sqrt
(6)*x + 2*(-3*x^2 - 1)^(1/4))/x)

________________________________________________________________________________________

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

[Out]

-Integral(x**2/(3*x**2*(-3*x**2 - 1)**(3/4) + 2*(-3*x**2 - 1)**(3/4)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x^{2}}{{\left (3 \, x^{2} + 2\right )}{\left (-3 \, x^{2} - 1\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)/(-3*x^2-1)^(3/4),x, algorithm="giac")

[Out]

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