3.1054 \(\int \frac{x^2}{(2-3 x^2)^{3/4} (4-3 x^2)} \, dx\)

Optimal. Leaf size=120 \[ \frac{\tan ^{-1}\left (\frac{2-\sqrt{2} \sqrt{2-3 x^2}}{\sqrt [4]{2} \sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt [4]{2} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{2-3 x^2}+2}{\sqrt [4]{2} \sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt [4]{2} \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0226413, antiderivative size = 120, 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 = {441} \[ \frac{\tan ^{-1}\left (\frac{2-\sqrt{2} \sqrt{2-3 x^2}}{\sqrt [4]{2} \sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt [4]{2} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{2-3 x^2}+2}{\sqrt [4]{2} \sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt [4]{2} \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 441

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

Rubi steps

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

Mathematica [C]  time = 0.0365984, size = 37, normalized size = 0.31 \[ \frac{x^3 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )}{12\ 2^{3/4}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x^3*AppellF1[3/2, 3/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4])/(12*2^(3/4))

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{-3\,{x}^{2}+4} \left ( -3\,{x}^{2}+2 \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)^(3/4)/(-3*x^2+4),x)

[Out]

int(x^2/(-3*x^2+2)^(3/4)/(-3*x^2+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} - 4\right )}{\left (-3 \, x^{2} + 2\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)^(3/4)/(-3*x^2+4),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.00185, size = 871, normalized size = 7.26 \begin{align*} \frac{1}{216} \cdot 72^{\frac{3}{4}} \sqrt{2} \arctan \left (\frac{72^{\frac{1}{4}} \sqrt{6} \sqrt{2} x \sqrt{\frac{72^{\frac{3}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} x + 18 \, \sqrt{2} x^{2} + 24 \, \sqrt{-3 \, x^{2} + 2}}{x^{2}}} - 12 \cdot 72^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} - 36 \, x}{36 \, x}\right ) + \frac{1}{216} \cdot 72^{\frac{3}{4}} \sqrt{2} \arctan \left (\frac{72^{\frac{1}{4}} \sqrt{6} \sqrt{2} x \sqrt{-\frac{72^{\frac{3}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} x - 18 \, \sqrt{2} x^{2} - 24 \, \sqrt{-3 \, x^{2} + 2}}{x^{2}}} - 12 \cdot 72^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 36 \, x}{36 \, x}\right ) - \frac{1}{864} \cdot 72^{\frac{3}{4}} \sqrt{2} \log \left (\frac{96 \,{\left (72^{\frac{3}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} x + 18 \, \sqrt{2} x^{2} + 24 \, \sqrt{-3 \, x^{2} + 2}\right )}}{x^{2}}\right ) + \frac{1}{864} \cdot 72^{\frac{3}{4}} \sqrt{2} \log \left (-\frac{96 \,{\left (72^{\frac{3}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} x - 18 \, \sqrt{2} x^{2} - 24 \, \sqrt{-3 \, x^{2} + 2}\right )}}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/216*72^(3/4)*sqrt(2)*arctan(1/36*(72^(1/4)*sqrt(6)*sqrt(2)*x*sqrt((72^(3/4)*sqrt(2)*(-3*x^2 + 2)^(1/4)*x + 1
8*sqrt(2)*x^2 + 24*sqrt(-3*x^2 + 2))/x^2) - 12*72^(1/4)*sqrt(2)*(-3*x^2 + 2)^(1/4) - 36*x)/x) + 1/216*72^(3/4)
*sqrt(2)*arctan(1/36*(72^(1/4)*sqrt(6)*sqrt(2)*x*sqrt(-(72^(3/4)*sqrt(2)*(-3*x^2 + 2)^(1/4)*x - 18*sqrt(2)*x^2
- 24*sqrt(-3*x^2 + 2))/x^2) - 12*72^(1/4)*sqrt(2)*(-3*x^2 + 2)^(1/4) + 36*x)/x) - 1/864*72^(3/4)*sqrt(2)*log(
96*(72^(3/4)*sqrt(2)*(-3*x^2 + 2)^(1/4)*x + 18*sqrt(2)*x^2 + 24*sqrt(-3*x^2 + 2))/x^2) + 1/864*72^(3/4)*sqrt(2
)*log(-96*(72^(3/4)*sqrt(2)*(-3*x^2 + 2)^(1/4)*x - 18*sqrt(2)*x^2 - 24*sqrt(-3*x^2 + 2))/x^2)

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

[Out]

-Integral(x**2/(3*x**2*(2 - 3*x**2)**(3/4) - 4*(2 - 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} - 4\right )}{\left (-3 \, x^{2} + 2\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)^(3/4)/(-3*x^2+4),x, algorithm="giac")

[Out]

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