3.1069 \(\int \frac {x^2}{(2-3 x^2)^{3/4} (4-3 x^2)} \, dx\)

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]

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

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Rubi [A]  time = 0.02, antiderivative size = 120, normalized size of antiderivative = 1.00, 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*}

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Mathematica [C]  time = 0.03, 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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fricas [B]  time = 1.01, size = 282, normalized size = 2.35 \[ \frac {1}{216} \cdot 72^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {72^{\frac {1}{4}} \sqrt {6} \sqrt {2} x \sqrt {\frac {72^{\frac {3}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 18 \, \sqrt {2} x^{2} + 24 \, \sqrt {-3 \, x^{2} + 2}}{x^{2}}} - 12 \cdot 72^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} - 36 \, x}{36 \, x}\right ) + \frac {1}{216} \cdot 72^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {72^{\frac {1}{4}} \sqrt {6} \sqrt {2} x \sqrt {-\frac {72^{\frac {3}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x - 18 \, \sqrt {2} x^{2} - 24 \, \sqrt {-3 \, x^{2} + 2}}{x^{2}}} - 12 \cdot 72^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 36 \, x}{36 \, x}\right ) - \frac {1}{864} \cdot 72^{\frac {3}{4}} \sqrt {2} \log \left (\frac {96 \, {\left (72^{\frac {3}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 18 \, \sqrt {2} x^{2} + 24 \, \sqrt {-3 \, x^{2} + 2}\right )}}{x^{2}}\right ) + \frac {1}{864} \cdot 72^{\frac {3}{4}} \sqrt {2} \log \left (-\frac {96 \, {\left (72^{\frac {3}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x - 18 \, \sqrt {2} x^{2} - 24 \, \sqrt {-3 \, x^{2} + 2}\right )}}{x^{2}}\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x^{2}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}}\,{d x} \]

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)

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maple [C]  time = 0.00, size = 186, normalized size = 1.55 \[ -\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+18\right )^{2}\right ) \ln \left (\frac {-3 x \RootOf \left (\textit {\_Z}^{4}+18\right )^{2}+\left (-3 x^{2}+2\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+18\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+18\right )^{2}+9 \sqrt {-3 x^{2}+2}\, x +6 \left (-3 x^{2}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+18\right )^{2}\right )}{3 x^{2}-4}\right )}{18}-\frac {\RootOf \left (\textit {\_Z}^{4}+18\right ) \ln \left (-\frac {-3 x \RootOf \left (\textit {\_Z}^{4}+18\right )^{2}+\left (-3 x^{2}+2\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{4}+18\right )^{3}-9 \sqrt {-3 x^{2}+2}\, x -6 \left (-3 x^{2}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+18\right )}{3 x^{2}-4}\right )}{18} \]

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]

-1/18*RootOf(_Z^4+18)*ln(-((-3*x^2+2)^(3/4)*RootOf(_Z^4+18)^3-3*RootOf(_Z^4+18)^2*x-9*(-3*x^2+2)^(1/2)*x-6*Roo
tOf(_Z^4+18)*(-3*x^2+2)^(1/4))/(3*x^2-4))-1/18*RootOf(_Z^2+RootOf(_Z^4+18)^2)*ln((RootOf(_Z^4+18)^2*RootOf(_Z^
2+RootOf(_Z^4+18)^2)*(-3*x^2+2)^(3/4)-3*RootOf(_Z^4+18)^2*x+9*(-3*x^2+2)^(1/2)*x+6*RootOf(_Z^2+RootOf(_Z^4+18)
^2)*(-3*x^2+2)^(1/4))/(3*x^2-4))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {x^{2}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {x^2}{{\left (2-3\,x^2\right )}^{3/4}\,\left (3\,x^2-4\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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