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

Optimal result

Integrand size = 24, antiderivative size = 120 \[ \int \frac {x^2}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {\arctan \left (\frac {2-\sqrt {2} \sqrt {2-3 x^2}}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt [4]{2} \sqrt {3}}-\frac {\text {arctanh}\left (\frac {2+\sqrt {2} \sqrt {2-3 x^2}}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt [4]{2} \sqrt {3}} \]

[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)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {452} \[ \int \frac {x^2}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\frac {\arctan \left (\frac {2-\sqrt {2} \sqrt {2-3 x^2}}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt [4]{2} \sqrt {3}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {2-3 x^2}+2}{\sqrt [4]{2} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt [4]{2} \sqrt {3}} \]

[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*} \text {integral}& = \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 [A] (verified)

Time = 0.04 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.91 \[ \int \frac {x^2}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\frac {\arctan \left (\frac {-3 \sqrt {2} x^2+4 \sqrt {2-3 x^2}}{2\ 2^{3/4} \sqrt {3} x \sqrt [4]{2-3 x^2}}\right )+\text {arctanh}\left (\frac {2 \sqrt {3} x \sqrt [4]{4-6 x^2}}{3 x^2+2 \sqrt {4-6 x^2}}\right )}{6 \sqrt [4]{2} \sqrt {3}} \]

[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])

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 4.49 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.56

method	result	size
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{2}\right ) \ln \left (-\frac {\left (-3 x^{2}+2\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{2}\right )-9 \sqrt {-3 x^{2}+2}\, x +3 x \operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{2}+6 \left (-3 x^{2}+2\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{2}\right )}{3 x^{2}-4}\right )}{18}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right ) \ln \left (-\frac {\left (-3 x^{2}+2\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{3}-9 \sqrt {-3 x^{2}+2}\, x -3 x \operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{2}-6 \operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}}{3 x^{2}-4}\right )}{18}\)	\(187\)

[In]

int(x^2/(-3*x^2+2)^(3/4)/(-3*x^2+4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.14 \[ \int \frac {x^2}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\left (\frac {1}{864} i + \frac {1}{864}\right ) \cdot 72^{\frac {3}{4}} \sqrt {2} \log \left (\frac {\left (i + 1\right ) \cdot 72^{\frac {3}{4}} \sqrt {2} x + 48 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}}{x}\right ) + \left (\frac {1}{864} i - \frac {1}{864}\right ) \cdot 72^{\frac {3}{4}} \sqrt {2} \log \left (\frac {-\left (i - 1\right ) \cdot 72^{\frac {3}{4}} \sqrt {2} x + 48 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}}{x}\right ) - \left (\frac {1}{864} i - \frac {1}{864}\right ) \cdot 72^{\frac {3}{4}} \sqrt {2} \log \left (\frac {\left (i - 1\right ) \cdot 72^{\frac {3}{4}} \sqrt {2} x + 48 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}}{x}\right ) + \left (\frac {1}{864} i + \frac {1}{864}\right ) \cdot 72^{\frac {3}{4}} \sqrt {2} \log \left (\frac {-\left (i + 1\right ) \cdot 72^{\frac {3}{4}} \sqrt {2} x + 48 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}}{x}\right ) \]

[In]

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

[Out]

-(1/864*I + 1/864)*72^(3/4)*sqrt(2)*log(((I + 1)*72^(3/4)*sqrt(2)*x + 48*(-3*x^2 + 2)^(1/4))/x) + (1/864*I - 1
/864)*72^(3/4)*sqrt(2)*log((-(I - 1)*72^(3/4)*sqrt(2)*x + 48*(-3*x^2 + 2)^(1/4))/x) - (1/864*I - 1/864)*72^(3/
4)*sqrt(2)*log(((I - 1)*72^(3/4)*sqrt(2)*x + 48*(-3*x^2 + 2)^(1/4))/x) + (1/864*I + 1/864)*72^(3/4)*sqrt(2)*lo
g((-(I + 1)*72^(3/4)*sqrt(2)*x + 48*(-3*x^2 + 2)^(1/4))/x)

Sympy [F]

\[ \int \frac {x^2}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=- \int \frac {x^{2}}{3 x^{2} \left (2 - 3 x^{2}\right )^{\frac {3}{4}} - 4 \left (2 - 3 x^{2}\right )^{\frac {3}{4}}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {x^2}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\int { -\frac {x^{2}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {x^2}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=\int { -\frac {x^{2}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx=-\int \frac {x^2}{{\left (2-3\,x^2\right )}^{3/4}\,\left (3\,x^2-4\right )} \,d x \]

[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)