Optimal. Leaf size=129 \[ \frac{\tanh ^{-1}\left (\frac{2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt{3 x^2+2}}{2 \sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{3 \sqrt [4]{2} \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [4]{2} \sqrt{3 x^2+2}+2\ 2^{3/4}}{2 \sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{3 \sqrt [4]{2} \sqrt{3}} \]
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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.
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Rule 441
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.
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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.
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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.
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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.
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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.
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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.
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