3.1057 \(\int \frac{x^2}{\left (a+3 x^2\right )^{3/4} \left (2 a+3 x^2\right )} \, dx\)

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

[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))

_______________________________________________________________________________________

Rubi [A]  time = 0.109356, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a+3 x^2}}{\sqrt{a}}\right )}{\sqrt{3} x \sqrt [4]{a+3 x^2}}\right )}{3 \sqrt{3} \sqrt [4]{a}}-\frac{\tan ^{-1}\left (\frac{a^{3/4} \left (\frac{\sqrt{a+3 x^2}}{\sqrt{a}}+1\right )}{\sqrt{3} x \sqrt [4]{a+3 x^2}}\right )}{3 \sqrt{3} \sqrt [4]{a}} \]

Antiderivative was successfully verified.

[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))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.7863, size = 53, normalized size = 0.44 \[ \frac{x^{3} \sqrt [4]{a + 3 x^{2}} \operatorname{appellf_{1}}{\left (\frac{3}{2},\frac{3}{4},1,\frac{5}{2},- \frac{3 x^{2}}{a},- \frac{3 x^{2}}{2 a} \right )}}{6 a^{2} \sqrt [4]{1 + \frac{3 x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(3*x**2+a)**(3/4)/(3*x**2+2*a),x)

[Out]

x**3*(a + 3*x**2)**(1/4)*appellf1(3/2, 3/4, 1, 5/2, -3*x**2/a, -3*x**2/(2*a))/(6
*a**2*(1 + 3*x**2/a)**(1/4))

_______________________________________________________________________________________

Mathematica [C]  time = 0.272825, size = 162, normalized size = 1.35 \[ -\frac{10 a x^3 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right )}{3 \left (a+3 x^2\right )^{3/4} \left (2 a+3 x^2\right ) \left (3 x^2 \left (2 F_1\left (\frac{5}{2};\frac{3}{4},2;\frac{7}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right )+3 F_1\left (\frac{5}{2};\frac{7}{4},1;\frac{7}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right )\right )-10 a F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-10*a*x^3*AppellF1[3/2, 3/4, 1, 5/2, (-3*x^2)/a, (-3*x^2)/(2*a)])/(3*(a + 3*x^2
)^(3/4)*(2*a + 3*x^2)*(-10*a*AppellF1[3/2, 3/4, 1, 5/2, (-3*x^2)/a, (-3*x^2)/(2*
a)] + 3*x^2*(2*AppellF1[5/2, 3/4, 2, 7/2, (-3*x^2)/a, (-3*x^2)/(2*a)] + 3*Appell
F1[5/2, 7/4, 1, 7/2, (-3*x^2)/a, (-3*x^2)/(2*a)])))

_______________________________________________________________________________________

Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2}}{3\,{x}^{2}+2\,a} \left ( 3\,{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (3 \, x^{2} + 2 \, a\right )}{\left (3 \, x^{2} + a\right )}^{\frac{3}{4}}}\,{d x} \]

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 + 2*a)*(3*x^2 + a)^(3/4)), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.241943, size = 212, normalized size = 1.77 \[ \frac{2}{3} \, \left (\frac{1}{36}\right )^{\frac{1}{4}} \left (-\frac{1}{a}\right )^{\frac{1}{4}} \arctan \left (\frac{3 \, \left (\frac{1}{36}\right )^{\frac{1}{4}} x \left (-\frac{1}{a}\right )^{\frac{1}{4}}}{\sqrt{\frac{1}{2}} x \sqrt{\frac{3 \, x^{2} \sqrt{-\frac{1}{a}} + 2 \, \sqrt{3 \, x^{2} + a}}{x^{2}}} +{\left (3 \, x^{2} + a\right )}^{\frac{1}{4}}}\right ) - \frac{1}{6} \, \left (\frac{1}{36}\right )^{\frac{1}{4}} \left (-\frac{1}{a}\right )^{\frac{1}{4}} \log \left (\frac{3 \, \left (\frac{1}{36}\right )^{\frac{1}{4}} x \left (-\frac{1}{a}\right )^{\frac{1}{4}} +{\left (3 \, x^{2} + a\right )}^{\frac{1}{4}}}{x}\right ) + \frac{1}{6} \, \left (\frac{1}{36}\right )^{\frac{1}{4}} \left (-\frac{1}{a}\right )^{\frac{1}{4}} \log \left (-\frac{3 \, \left (\frac{1}{36}\right )^{\frac{1}{4}} x \left (-\frac{1}{a}\right )^{\frac{1}{4}} -{\left (3 \, x^{2} + a\right )}^{\frac{1}{4}}}{x}\right ) \]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (a + 3 x^{2}\right )^{\frac{3}{4}} \left (2 a + 3 x^{2}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(3*x**2+a)**(3/4)/(3*x**2+2*a),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (3 \, x^{2} + 2 \, a\right )}{\left (3 \, x^{2} + a\right )}^{\frac{3}{4}}}\,{d x} \]

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 + 2*a)*(3*x^2 + a)^(3/4)), x)