3.11.69 \(\int \frac {x^2}{(2-3 x^2)^{3/4} (4-3 x^2)} \, dx\) [1069]

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

[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 = {452} \begin {gather*} \frac {\text {ArcTan}\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}} \end {gather*}

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 452

Int[(x_)^2/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Simp[(-b/(a*d*Rt[b^2/a, 4]^3))*Ar
cTan[(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))], x] + Simp[(b/(a*d*Rt[b^2/a, 4
]^3))*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))], 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 [A]
time = 0.04, size = 109, normalized size = 0.91 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {-3 \sqrt {2} x^2+4 \sqrt {2-3 x^2}}{2\ 2^{3/4} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )+\tanh ^{-1}\left (\frac {2 \sqrt {3} x \sqrt [4]{4-6 x^2}}{3 x^2+2 \sqrt {4-6 x^2}}\right )}{6 \sqrt [4]{2} \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.00, size = 186, normalized size = 1.55

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{4}+18\right ) \ln \left (\frac {\left (-3 x^{2}+2\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{4}+18\right )^{3}+3 \RootOf \left (\textit {\_Z}^{4}+18\right )^{2} x +9 \sqrt {-3 x^{2}+2}\, x -6 \RootOf \left (\textit {\_Z}^{4}+18\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}}{3 x^{2}-4}\right )}{18}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+18\right )^{2}\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+18\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+18\right )^{2}\right ) \left (-3 x^{2}+2\right )^{\frac {3}{4}}+3 \RootOf \left (\textit {\_Z}^{4}+18\right )^{2} x -9 \sqrt {-3 x^{2}+2}\, x +6 \left (-3 x^{2}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+18\right )^{2}\right )}{3 x^{2}-4}\right )}{18}\)	\(186\)

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,method=_RETURNVERBOSE)

[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*RootO
f(_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*(-3*x^2+2)^(1/4)*RootOf(_Z^2
+RootOf(_Z^4+18)^2))/(3*x^2-4))

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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)^(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] Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (89) = 178\).
time = 0.96, size = 282, normalized size = 2.35 \begin {gather*} \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 {gather*}

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

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

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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