Optimal. Leaf size=120 \[ \frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a+3 x^2}}{\sqrt{a}}\right )}{\sqrt{3} x \sqrt [4]{a+3 x^2}}\right )}{3 \sqrt{3} \sqrt [4]{a}}-\frac{\tan ^{-1}\left (\frac{a^{3/4} \left (\frac{\sqrt{a+3 x^2}}{\sqrt{a}}+1\right )}{\sqrt{3} x \sqrt [4]{a+3 x^2}}\right )}{3 \sqrt{3} \sqrt [4]{a}} \]
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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.
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Rule 441
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.
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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.
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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.
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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.
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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.
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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.
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