\(\int \frac {x^2}{(-2-3 x^2) (-1-3 x^2)^{3/4}} \, dx\) [1074]

Optimal result

Integrand size = 24, antiderivative size = 61 \[ \int \frac {x^2}{\left (-2-3 x^2\right ) \left (-1-3 x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{3 \sqrt {6}}-\frac {\text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{3 \sqrt {6}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 61, 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 = {453} \[ \int \frac {x^2}{\left (-2-3 x^2\right ) \left (-1-3 x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{3 \sqrt {6}}-\frac {\text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{3 \sqrt {6}} \]

[In]

Int[x^2/((-2 - 3*x^2)*(-1 - 3*x^2)^(3/4)),x]

[Out]

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

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*} \text {integral}& = \frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{3 \sqrt {6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{3 \sqrt {6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.62 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89 \[ \int \frac {x^2}{\left (-2-3 x^2\right ) \left (-1-3 x^2\right )^{3/4}} \, dx=-\frac {-\arctan \left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )+\text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{3 \sqrt {6}} \]

[In]

Integrate[x^2/((-2 - 3*x^2)*(-1 - 3*x^2)^(3/4)),x]

[Out]

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

Maple [C] (verified)

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

Time = 3.29 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.26

method	result	size
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) \left (-3 x^{2}-1\right )^{\frac {3}{4}}+3 \sqrt {-3 x^{2}-1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) \left (-3 x^{2}-1\right )^{\frac {1}{4}}-3 x}{3 x^{2}+2}\right )}{18}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+6\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+6\right ) \left (-3 x^{2}-1\right )^{\frac {3}{4}}-3 \sqrt {-3 x^{2}-1}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+6\right ) \left (-3 x^{2}-1\right )^{\frac {1}{4}}-3 x}{3 x^{2}+2}\right )}{18}\)	\(138\)

[In]

int(x^2/(-3*x^2-2)/(-3*x^2-1)^(3/4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.89 \[ \int \frac {x^2}{\left (-2-3 x^2\right ) \left (-1-3 x^2\right )^{3/4}} \, dx=-\frac {1}{36} \, \sqrt {6} \log \left (\frac {\sqrt {6} x + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) + \frac {1}{36} \, \sqrt {6} \log \left (-\frac {\sqrt {6} x - 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {1}{36} i \, \sqrt {6} \log \left (\frac {i \, \sqrt {6} x + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) + \frac {1}{36} i \, \sqrt {6} \log \left (\frac {-i \, \sqrt {6} x + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) \]

[In]

integrate(x^2/(-3*x^2-2)/(-3*x^2-1)^(3/4),x, algorithm="fricas")

[Out]

-1/36*sqrt(6)*log(1/2*(sqrt(6)*x + 2*(-3*x^2 - 1)^(1/4))/x) + 1/36*sqrt(6)*log(-1/2*(sqrt(6)*x - 2*(-3*x^2 - 1
)^(1/4))/x) - 1/36*I*sqrt(6)*log(1/2*(I*sqrt(6)*x + 2*(-3*x^2 - 1)^(1/4))/x) + 1/36*I*sqrt(6)*log(1/2*(-I*sqrt
(6)*x + 2*(-3*x^2 - 1)^(1/4))/x)

Sympy [F]

\[ \int \frac {x^2}{\left (-2-3 x^2\right ) \left (-1-3 x^2\right )^{3/4}} \, dx=- \int \frac {x^{2}}{3 x^{2} \left (- 3 x^{2} - 1\right )^{\frac {3}{4}} + 2 \left (- 3 x^{2} - 1\right )^{\frac {3}{4}}}\, dx \]

[In]

integrate(x**2/(-3*x**2-2)/(-3*x**2-1)**(3/4),x)

[Out]

-Integral(x**2/(3*x**2*(-3*x**2 - 1)**(3/4) + 2*(-3*x**2 - 1)**(3/4)), x)

Maxima [F]

\[ \int \frac {x^2}{\left (-2-3 x^2\right ) \left (-1-3 x^2\right )^{3/4}} \, dx=\int { -\frac {x^{2}}{{\left (3 \, x^{2} + 2\right )} {\left (-3 \, x^{2} - 1\right )}^{\frac {3}{4}}} \,d x } \]

[In]

integrate(x^2/(-3*x^2-2)/(-3*x^2-1)^(3/4),x, algorithm="maxima")

[Out]

-integrate(x^2/((3*x^2 + 2)*(-3*x^2 - 1)^(3/4)), x)

Giac [F]

\[ \int \frac {x^2}{\left (-2-3 x^2\right ) \left (-1-3 x^2\right )^{3/4}} \, dx=\int { -\frac {x^{2}}{{\left (3 \, x^{2} + 2\right )} {\left (-3 \, x^{2} - 1\right )}^{\frac {3}{4}}} \,d x } \]

[In]

integrate(x^2/(-3*x^2-2)/(-3*x^2-1)^(3/4),x, algorithm="giac")

[Out]

integrate(-x^2/((3*x^2 + 2)*(-3*x^2 - 1)^(3/4)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (-2-3 x^2\right ) \left (-1-3 x^2\right )^{3/4}} \, dx=-\int \frac {x^2}{{\left (-3\,x^2-1\right )}^{3/4}\,\left (3\,x^2+2\right )} \,d x \]

[In]

int(-x^2/((- 3*x^2 - 1)^(3/4)*(3*x^2 + 2)),x)

[Out]

-int(x^2/((- 3*x^2 - 1)^(3/4)*(3*x^2 + 2)), x)