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

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))

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Rubi [A]  time = 0.0282203, 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, Rules used = {441} \[ \frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a+3 x^2}}{\sqrt{a}}\right )}{\sqrt{3} x \sqrt [4]{a+3 x^2}}\right )}{3 \sqrt{3} \sqrt [4]{a}}-\frac{\tan ^{-1}\left (\frac{a^{3/4} \left (\frac{\sqrt{a+3 x^2}}{\sqrt{a}}+1\right )}{\sqrt{3} x \sqrt [4]{a+3 x^2}}\right )}{3 \sqrt{3} \sqrt [4]{a}} \]

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))

Rule 441

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

Rubi steps

\begin{align*} \int \frac{x^2}{\left (a+3 x^2\right )^{3/4} \left (2 a+3 x^2\right )} \, dx &=-\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 [C]  time = 0.0497414, size = 65, normalized size = 0.54 \[ \frac{x^3 \left (\frac{a+3 x^2}{a}\right )^{3/4} F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right )}{6 a \left (a+3 x^2\right )^{3/4}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{3\,{x}^{2}+2\,a} \left ( 3\,{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}

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)

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

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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.63863, size = 495, normalized size = 4.12 \begin{align*} -\frac{2}{3} \, \left (\frac{1}{36}\right )^{\frac{1}{4}} \left (-\frac{1}{a}\right )^{\frac{1}{4}} \arctan \left (\frac{12 \,{\left (\sqrt{\frac{1}{2}} \left (\frac{1}{36}\right )^{\frac{3}{4}} a x \left (-\frac{1}{a}\right )^{\frac{3}{4}} \sqrt{\frac{3 \, x^{2} \sqrt{-\frac{1}{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 \left (-\frac{1}{a}\right )^{\frac{3}{4}}\right )}}{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} \, \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 ) \end{align*}

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, algorithm="fricas")

[Out]

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

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

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)

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

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, algorithm="giac")

[Out]

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