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

Optimal. Leaf size=85 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{3 x^2-a}}\right )}{3 \sqrt {6} \sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{3 x^2-a}}\right )}{3 \sqrt {6} \sqrt [4]{a}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {442} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{3 x^2-a}}\right )}{3 \sqrt {6} \sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{3 x^2-a}}\right )}{3 \sqrt {6} \sqrt [4]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 442

Int[(x_)^2/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> -Simp[(b*ArcTan[(Rt[-(b^2/a), 4]*
x)/(Sqrt[2]*(a + b*x^2)^(1/4))])/(Sqrt[2]*a*d*Rt[-(b^2/a), 4]^3), x] + Simp[(b*ArcTanh[(Rt[-(b^2/a), 4]*x)/(Sq
rt[2]*(a + b*x^2)^(1/4))])/(Sqrt[2]*a*d*Rt[-(b^2/a), 4]^3), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0
] && NegQ[b^2/a]

Rubi steps

\begin {align*} \int \frac {x^2}{\left (-2 a+3 x^2\right ) \left (-a+3 x^2\right )^{3/4}} \, dx &=\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a+3 x^2}}\right )}{3 \sqrt {6} \sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a+3 x^2}}\right )}{3 \sqrt {6} \sqrt [4]{a}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.11, size = 66, normalized size = 0.78 \begin {gather*} -\frac {x^3 \left (1-\frac {3 x^2}{a}\right )^{3/4} F_1\left (\frac {3}{2};\frac {3}{4},1;\frac {5}{2};\frac {3 x^2}{a},\frac {3 x^2}{2 a}\right )}{6 a \left (3 x^2-a\right )^{3/4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/6*(x^3*(1 - (3*x^2)/a)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, (3*x^2)/a, (3*x^2)/(2*a)])/(a*(-a + 3*x^2)^(3/4))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 2.21, size = 87, normalized size = 1.02 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{3 x^2-a}}\right )}{3 \sqrt {6} \sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}} \sqrt [4]{a} \sqrt [4]{3 x^2-a}}{x}\right )}{3 \sqrt {6} \sqrt [4]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.84, size = 145, normalized size = 1.71 \begin {gather*} \frac {2 \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \arctan \left (\frac {12 \, {\left (\sqrt {\frac {1}{2}} \left (\frac {1}{36}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x \sqrt {\frac {\frac {3 \, x^{2}}{\sqrt {a}} + 2 \, \sqrt {3 \, x^{2} - a}}{x^{2}}} - \left (\frac {1}{36}\right )^{\frac {3}{4}} {\left (3 \, x^{2} - a\right )}^{\frac {1}{4}} a^{\frac {1}{4}}\right )}}{x}\right )}{3 \, a^{\frac {1}{4}}} - \frac {\left (\frac {1}{36}\right )^{\frac {1}{4}} \log \left (\frac {\frac {3 \, \left (\frac {1}{36}\right )^{\frac {1}{4}} x}{a^{\frac {1}{4}}} + {\left (3 \, x^{2} - a\right )}^{\frac {1}{4}}}{x}\right )}{6 \, a^{\frac {1}{4}}} + \frac {\left (\frac {1}{36}\right )^{\frac {1}{4}} \log \left (-\frac {\frac {3 \, \left (\frac {1}{36}\right )^{\frac {1}{4}} x}{a^{\frac {1}{4}}} - {\left (3 \, x^{2} - a\right )}^{\frac {1}{4}}}{x}\right )}{6 \, a^{\frac {1}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(1/36)^(1/4)*arctan(12*(sqrt(1/2)*(1/36)^(3/4)*a^(1/4)*x*sqrt((3*x^2/sqrt(a) + 2*sqrt(3*x^2 - a))/x^2) - (
1/36)^(3/4)*(3*x^2 - a)^(1/4)*a^(1/4))/x)/a^(1/4) - 1/6*(1/36)^(1/4)*log((3*(1/36)^(1/4)*x/a^(1/4) + (3*x^2 -
a)^(1/4))/x)/a^(1/4) + 1/6*(1/36)^(1/4)*log(-(3*(1/36)^(1/4)*x/a^(1/4) - (3*x^2 - a)^(1/4))/x)/a^(1/4)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (3 x^{2}-2 a \right ) \left (3 x^{2}-a \right )^{\frac {3}{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (- 2 a + 3 x^{2}\right ) \left (- a + 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*a)/(3*x**2-a)**(3/4),x)

[Out]

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

________________________________________________________________________________________