Optimal. Leaf size=85 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{3 x^2-a}}\right )}{2 \sqrt {6} a^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{3 x^2-a}}\right )}{2 \sqrt {6} a^{3/4}} \]
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Rubi [A] time = 0.02, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {398} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{3 x^2-a}}\right )}{2 \sqrt {6} a^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{3 x^2-a}}\right )}{2 \sqrt {6} a^{3/4}} \]
Antiderivative was successfully verified.
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Rule 398
Rubi steps
\begin {align*} \int \frac {1}{\left (-2 a+3 x^2\right ) \sqrt [4]{-a+3 x^2}} \, dx &=-\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a+3 x^2}}\right )}{2 \sqrt {6} a^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a+3 x^2}}\right )}{2 \sqrt {6} a^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 157, normalized size = 1.85 \[ \frac {2 a x F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};\frac {3 x^2}{a},\frac {3 x^2}{2 a}\right )}{\left (3 x^2-2 a\right ) \sqrt [4]{3 x^2-a} \left (x^2 \left (2 F_1\left (\frac {3}{2};\frac {1}{4},2;\frac {5}{2};\frac {3 x^2}{a},\frac {3 x^2}{2 a}\right )+F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};\frac {3 x^2}{a},\frac {3 x^2}{2 a}\right )\right )+2 a F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};\frac {3 x^2}{a},\frac {3 x^2}{2 a}\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 9.49, size = 276, normalized size = 3.25 \[ -\left (\frac {1}{36}\right )^{\frac {1}{4}} \frac {1}{a^{3}}^{\frac {1}{4}} \arctan \left (\frac {2 \, {\left (\sqrt {\frac {1}{2}} {\left (6 \, \left (\frac {1}{36}\right )^{\frac {3}{4}} a^{3} \frac {1}{a^{3}}^{\frac {3}{4}} + \left (\frac {1}{36}\right )^{\frac {1}{4}} \sqrt {3 \, x^{2} - a} a \frac {1}{a^{3}}^{\frac {1}{4}}\right )} \sqrt {a \sqrt {\frac {1}{a^{3}}}} - \left (\frac {1}{36}\right )^{\frac {1}{4}} {\left (3 \, x^{2} - a\right )}^{\frac {1}{4}} a \frac {1}{a^{3}}^{\frac {1}{4}}\right )}}{x}\right ) - \frac {1}{4} \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (\frac {18 \, \left (\frac {1}{36}\right )^{\frac {3}{4}} \sqrt {3 \, x^{2} - a} a^{2} \frac {1}{a^{3}}^{\frac {3}{4}} x + {\left (3 \, x^{2} - a\right )}^{\frac {1}{4}} a^{2} \sqrt {\frac {1}{a^{3}}} + 3 \, \left (\frac {1}{36}\right )^{\frac {1}{4}} a \frac {1}{a^{3}}^{\frac {1}{4}} x + {\left (3 \, x^{2} - a\right )}^{\frac {3}{4}}}{3 \, x^{2} - 2 \, a}\right ) + \frac {1}{4} \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (-\frac {18 \, \left (\frac {1}{36}\right )^{\frac {3}{4}} \sqrt {3 \, x^{2} - a} a^{2} \frac {1}{a^{3}}^{\frac {3}{4}} x - {\left (3 \, x^{2} - a\right )}^{\frac {1}{4}} a^{2} \sqrt {\frac {1}{a^{3}}} + 3 \, \left (\frac {1}{36}\right )^{\frac {1}{4}} a \frac {1}{a^{3}}^{\frac {1}{4}} x - {\left (3 \, x^{2} - a\right )}^{\frac {3}{4}}}{3 \, x^{2} - 2 \, a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (3 \, x^{2} - a\right )}^{\frac {1}{4}} {\left (3 \, x^{2} - 2 \, a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (3 x^{2}-2 a \right ) \left (3 x^{2}-a \right )^{\frac {1}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (3 \, x^{2} - a\right )}^{\frac {1}{4}} {\left (3 \, x^{2} - 2 \, a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {1}{\left (2\,a-3\,x^2\right )\,{\left (3\,x^2-a\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- 2 a + 3 x^{2}\right ) \sqrt [4]{- a + 3 x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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