3.1053 \(\int \frac{x^2}{(2+3 x^2)^{3/4} (4+3 x^2)} \, dx\)

Optimal. Leaf size=129 \[ \frac{\tanh ^{-1}\left (\frac{2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt{3 x^2+2}}{2 \sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{3 \sqrt [4]{2} \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [4]{2} \sqrt{3 x^2+2}+2\ 2^{3/4}}{2 \sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{3 \sqrt [4]{2} \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0226888, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {441} \[ \frac{\tanh ^{-1}\left (\frac{2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt{3 x^2+2}}{2 \sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{3 \sqrt [4]{2} \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [4]{2} \sqrt{3 x^2+2}+2\ 2^{3/4}}{2 \sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{3 \sqrt [4]{2} \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [C]  time = 0.0380499, size = 37, normalized size = 0.29 \[ \frac{x^3 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right )}{12\ 2^{3/4}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x^3*AppellF1[3/2, 3/4, 1, 5/2, (-3*x^2)/2, (-3*x^2)/4])/(12*2^(3/4))

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Maple [F]  time = 0.032, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{3\,{x}^{2}+4} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [B]  time = 1.86533, size = 857, normalized size = 6.64 \begin{align*} \frac{1}{216} \cdot 72^{\frac{3}{4}} \sqrt{2} \arctan \left (\frac{72^{\frac{1}{4}} \sqrt{6} \sqrt{2} x \sqrt{\frac{72^{\frac{3}{4}} \sqrt{2}{\left (3 \, x^{2} + 2\right )}^{\frac{1}{4}} x + 18 \, \sqrt{2} x^{2} + 24 \, \sqrt{3 \, x^{2} + 2}}{x^{2}}} - 12 \cdot 72^{\frac{1}{4}} \sqrt{2}{\left (3 \, x^{2} + 2\right )}^{\frac{1}{4}} - 36 \, x}{36 \, x}\right ) + \frac{1}{216} \cdot 72^{\frac{3}{4}} \sqrt{2} \arctan \left (\frac{72^{\frac{1}{4}} \sqrt{6} \sqrt{2} x \sqrt{-\frac{72^{\frac{3}{4}} \sqrt{2}{\left (3 \, x^{2} + 2\right )}^{\frac{1}{4}} x - 18 \, \sqrt{2} x^{2} - 24 \, \sqrt{3 \, x^{2} + 2}}{x^{2}}} - 12 \cdot 72^{\frac{1}{4}} \sqrt{2}{\left (3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 36 \, x}{36 \, x}\right ) - \frac{1}{864} \cdot 72^{\frac{3}{4}} \sqrt{2} \log \left (\frac{96 \,{\left (72^{\frac{3}{4}} \sqrt{2}{\left (3 \, x^{2} + 2\right )}^{\frac{1}{4}} x + 18 \, \sqrt{2} x^{2} + 24 \, \sqrt{3 \, x^{2} + 2}\right )}}{x^{2}}\right ) + \frac{1}{864} \cdot 72^{\frac{3}{4}} \sqrt{2} \log \left (-\frac{96 \,{\left (72^{\frac{3}{4}} \sqrt{2}{\left (3 \, x^{2} + 2\right )}^{\frac{1}{4}} x - 18 \, \sqrt{2} x^{2} - 24 \, \sqrt{3 \, x^{2} + 2}\right )}}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/216*72^(3/4)*sqrt(2)*arctan(1/36*(72^(1/4)*sqrt(6)*sqrt(2)*x*sqrt((72^(3/4)*sqrt(2)*(3*x^2 + 2)^(1/4)*x + 18
*sqrt(2)*x^2 + 24*sqrt(3*x^2 + 2))/x^2) - 12*72^(1/4)*sqrt(2)*(3*x^2 + 2)^(1/4) - 36*x)/x) + 1/216*72^(3/4)*sq
rt(2)*arctan(1/36*(72^(1/4)*sqrt(6)*sqrt(2)*x*sqrt(-(72^(3/4)*sqrt(2)*(3*x^2 + 2)^(1/4)*x - 18*sqrt(2)*x^2 - 2
4*sqrt(3*x^2 + 2))/x^2) - 12*72^(1/4)*sqrt(2)*(3*x^2 + 2)^(1/4) + 36*x)/x) - 1/864*72^(3/4)*sqrt(2)*log(96*(72
^(3/4)*sqrt(2)*(3*x^2 + 2)^(1/4)*x + 18*sqrt(2)*x^2 + 24*sqrt(3*x^2 + 2))/x^2) + 1/864*72^(3/4)*sqrt(2)*log(-9
6*(72^(3/4)*sqrt(2)*(3*x^2 + 2)^(1/4)*x - 18*sqrt(2)*x^2 - 24*sqrt(3*x^2 + 2))/x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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