3.11.78 \(\int \frac {x^2}{(-2 a-3 x^2) (-a-3 x^2)^{3/4}} \, dx\) [1078]

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]

1/18*arctan(1/2*x*6^(1/2)/a^(1/4)/(-3*x^2-a)^(1/4))/a^(1/4)*6^(1/2)-1/18*arctanh(1/2*x*6^(1/2)/a^(1/4)/(-3*x^2
-a)^(1/4))/a^(1/4)*6^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 85, normalized size of antiderivative = 1.00, 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 = {453} \begin {gather*} \frac {\text {ArcTan}\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 {gather*}

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 453

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

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Mathematica [A]
time = 2.06, size = 75, normalized size = 0.88 \begin {gather*} -\frac {-\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )+\tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}} \sqrt [4]{a} \sqrt [4]{-a-3 x^2}}{x}\right )}{3 \sqrt {6} \sqrt [4]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (-3 x^{2}-2 a \right ) \left (-3 x^{2}-a \right )^{\frac {3}{4}}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (59) = 118\).
time = 0.62, size = 145, normalized size = 1.71 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {x^2}{\left (3\,x^2+2\,a\right )\,{\left (-3\,x^2-a\right )}^{3/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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