\(\int \frac {x^2}{(a-3 x^2)^{3/4} (2 a-3 x^2)} \, dx\) [1058]

Optimal result

Integrand size = 26, antiderivative size = 120 \[ \int \frac {x^2}{\left (a-3 x^2\right )^{3/4} \left (2 a-3 x^2\right )} \, dx=\frac {\arctan \left (\frac {a^{3/4} \left (1-\frac {\sqrt {a-3 x^2}}{\sqrt {a}}\right )}{\sqrt {3} x \sqrt [4]{a-3 x^2}}\right )}{3 \sqrt {3} \sqrt [4]{a}}-\frac {\text {arctanh}\left (\frac {a^{3/4} \left (1+\frac {\sqrt {a-3 x^2}}{\sqrt {a}}\right )}{\sqrt {3} x \sqrt [4]{a-3 x^2}}\right )}{3 \sqrt {3} \sqrt [4]{a}} \]

[Out]

1/9*arctan(1/3*a^(3/4)*(1-(-3*x^2+a)^(1/2)/a^(1/2))/x/(-3*x^2+a)^(1/4)*3^(1/2))/a^(1/4)*3^(1/2)-1/9*arctanh(1/
3*a^(3/4)*(1+(-3*x^2+a)^(1/2)/a^(1/2))/x/(-3*x^2+a)^(1/4)*3^(1/2))/a^(1/4)*3^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {452} \[ \int \frac {x^2}{\left (a-3 x^2\right )^{3/4} \left (2 a-3 x^2\right )} \, dx=\frac {\arctan \left (\frac {a^{3/4} \left (1-\frac {\sqrt {a-3 x^2}}{\sqrt {a}}\right )}{\sqrt {3} x \sqrt [4]{a-3 x^2}}\right )}{3 \sqrt {3} \sqrt [4]{a}}-\frac {\text {arctanh}\left (\frac {a^{3/4} \left (\frac {\sqrt {a-3 x^2}}{\sqrt {a}}+1\right )}{\sqrt {3} x \sqrt [4]{a-3 x^2}}\right )}{3 \sqrt {3} \sqrt [4]{a}} \]

[In]

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

[Out]

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

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 {a^{3/4} \left (1-\frac {\sqrt {a-3 x^2}}{\sqrt {a}}\right )}{\sqrt {3} x \sqrt [4]{a-3 x^2}}\right )}{3 \sqrt {3} \sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {a^{3/4} \left (1+\frac {\sqrt {a-3 x^2}}{\sqrt {a}}\right )}{\sqrt {3} x \sqrt [4]{a-3 x^2}}\right )}{3 \sqrt {3} \sqrt [4]{a}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.79 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.99 \[ \int \frac {x^2}{\left (a-3 x^2\right )^{3/4} \left (2 a-3 x^2\right )} \, dx=-\frac {\arctan \left (\frac {-3 x^2+2 \sqrt {a} \sqrt {a-3 x^2}}{2 \sqrt {3} \sqrt [4]{a} x \sqrt [4]{a-3 x^2}}\right )+\text {arctanh}\left (\frac {2 \sqrt {3} \sqrt [4]{a} x \sqrt [4]{a-3 x^2}}{3 x^2+2 \sqrt {a} \sqrt {a-3 x^2}}\right )}{6 \sqrt {3} \sqrt [4]{a}} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {x^{2}}{\left (-3 x^{2}+a \right )^{\frac {3}{4}} \left (-3 x^{2}+2 a \right )}d x\]

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.37 \[ \int \frac {x^2}{\left (a-3 x^2\right )^{3/4} \left (2 a-3 x^2\right )} \, dx=-\frac {1}{6} \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \left (-\frac {1}{a}\right )^{\frac {1}{4}} \log \left (\frac {3 \, \left (\frac {1}{36}\right )^{\frac {1}{4}} x \left (-\frac {1}{a}\right )^{\frac {1}{4}} + {\left (-3 \, x^{2} + a\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{6} \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \left (-\frac {1}{a}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, \left (\frac {1}{36}\right )^{\frac {1}{4}} x \left (-\frac {1}{a}\right )^{\frac {1}{4}} - {\left (-3 \, x^{2} + a\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{6} i \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \left (-\frac {1}{a}\right )^{\frac {1}{4}} \log \left (\frac {3 i \, \left (\frac {1}{36}\right )^{\frac {1}{4}} x \left (-\frac {1}{a}\right )^{\frac {1}{4}} + {\left (-3 \, x^{2} + a\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{6} i \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \left (-\frac {1}{a}\right )^{\frac {1}{4}} \log \left (\frac {-3 i \, \left (\frac {1}{36}\right )^{\frac {1}{4}} x \left (-\frac {1}{a}\right )^{\frac {1}{4}} + {\left (-3 \, x^{2} + a\right )}^{\frac {1}{4}}}{x}\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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