Optimal. Leaf size=120 \[ -\frac{\tan ^{-1}\left (\frac{a^{3/4} \left (\frac{\sqrt{a+3 x^2}}{\sqrt{a}}+1\right )}{\sqrt{3} x \sqrt [4]{a+3 x^2}}\right )}{2 \sqrt{3} a^{3/4}}-\frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a+3 x^2}}{\sqrt{a}}\right )}{\sqrt{3} x \sqrt [4]{a+3 x^2}}\right )}{2 \sqrt{3} a^{3/4}} \]
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Rubi [A] time = 0.0198623, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {397} \[ -\frac{\tan ^{-1}\left (\frac{a^{3/4} \left (\frac{\sqrt{a+3 x^2}}{\sqrt{a}}+1\right )}{\sqrt{3} x \sqrt [4]{a+3 x^2}}\right )}{2 \sqrt{3} a^{3/4}}-\frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a+3 x^2}}{\sqrt{a}}\right )}{\sqrt{3} x \sqrt [4]{a+3 x^2}}\right )}{2 \sqrt{3} a^{3/4}} \]
Antiderivative was successfully verified.
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Rule 397
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [4]{a+3 x^2} \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 )}{2 \sqrt{3} a^{3/4}}-\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 )}{2 \sqrt{3} a^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.140052, size = 155, normalized size = 1.29 \[ -\frac{2 a x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right )}{\sqrt [4]{a+3 x^2} \left (2 a+3 x^2\right ) \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right )\right )-2 a F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{3 x^2}{a},-\frac{3 x^2}{2 a}\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{3\,{x}^{2}+2\,a}{\frac{1}{\sqrt [4]{3\,{x}^{2}+a}}}}\, 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{1}{{\left (3 \, x^{2} + 2 \, a\right )}{\left (3 \, x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 58.6065, size = 826, normalized size = 6.88 \begin{align*} \left (\frac{1}{36}\right )^{\frac{1}{4}} \left (-\frac{1}{a^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{2 \,{\left (\sqrt{\frac{1}{2}}{\left (6 \, \left (\frac{1}{36}\right )^{\frac{3}{4}} a^{3} \left (-\frac{1}{a^{3}}\right )^{\frac{3}{4}} + \left (\frac{1}{36}\right )^{\frac{1}{4}} \sqrt{3 \, x^{2} + a} a \left (-\frac{1}{a^{3}}\right )^{\frac{1}{4}}\right )} \sqrt{-a \sqrt{-\frac{1}{a^{3}}}} - \left (\frac{1}{36}\right )^{\frac{1}{4}}{\left (3 \, x^{2} + a\right )}^{\frac{1}{4}} a \left (-\frac{1}{a^{3}}\right )^{\frac{1}{4}}\right )}}{x}\right ) - \frac{1}{4} \, \left (\frac{1}{36}\right )^{\frac{1}{4}} \left (-\frac{1}{a^{3}}\right )^{\frac{1}{4}} \log \left (\frac{18 \, \left (\frac{1}{36}\right )^{\frac{3}{4}} \sqrt{3 \, x^{2} + a} a^{2} x \left (-\frac{1}{a^{3}}\right )^{\frac{3}{4}} +{\left (3 \, x^{2} + a\right )}^{\frac{1}{4}} a^{2} \sqrt{-\frac{1}{a^{3}}} - 3 \, \left (\frac{1}{36}\right )^{\frac{1}{4}} a x \left (-\frac{1}{a^{3}}\right )^{\frac{1}{4}} +{\left (3 \, x^{2} + a\right )}^{\frac{3}{4}}}{3 \, x^{2} + 2 \, a}\right ) + \frac{1}{4} \, \left (\frac{1}{36}\right )^{\frac{1}{4}} \left (-\frac{1}{a^{3}}\right )^{\frac{1}{4}} \log \left (-\frac{18 \, \left (\frac{1}{36}\right )^{\frac{3}{4}} \sqrt{3 \, x^{2} + a} a^{2} x \left (-\frac{1}{a^{3}}\right )^{\frac{3}{4}} -{\left (3 \, x^{2} + a\right )}^{\frac{1}{4}} a^{2} \sqrt{-\frac{1}{a^{3}}} - 3 \, \left (\frac{1}{36}\right )^{\frac{1}{4}} a x \left (-\frac{1}{a^{3}}\right )^{\frac{1}{4}} -{\left (3 \, x^{2} + a\right )}^{\frac{3}{4}}}{3 \, x^{2} + 2 \, a}\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{1}{\sqrt [4]{a + 3 x^{2}} \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{1}{{\left (3 \, x^{2} + 2 \, a\right )}{\left (3 \, x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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