Optimal. Leaf size=85 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{2 \sqrt {6} a^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{2 \sqrt {6} a^{3/4}} \]
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Rubi [A] time = 0.01, 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} \begin {gather*} -\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 {gather*}
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 = 0.15, size = 157, normalized size = 1.85 \begin {gather*} \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 )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.16, size = 89, normalized size = 1.05 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {\frac {2}{3}} \sqrt [4]{a} \sqrt [4]{-a-3 x^2}}{x}\right )}{2 \sqrt {6} a^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}} \sqrt [4]{a} \sqrt [4]{-a-3 x^2}}{x}\right )}{2 \sqrt {6} a^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 20.76, size = 278, normalized size = 3.27 \begin {gather*} -\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 ) \end {gather*}
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 \begin {gather*} \int -\frac {1}{{\left (3 \, x^{2} + 2 \, a\right )} {\left (-3 \, x^{2} - a\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (-3 x^{2}-2 a \right ) \left (-3 x^{2}-a \right )^{\frac {1}{4}}}\, dx \end {gather*}
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 \begin {gather*} -\int \frac {1}{{\left (3 \, x^{2} + 2 \, a\right )} {\left (-3 \, x^{2} - a\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
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 \begin {gather*} -\int \frac {1}{\left (3\,x^2+2\,a\right )\,{\left (-3\,x^2-a\right )}^{1/4}} \,d x \end {gather*}
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 \begin {gather*} - \int \frac {1}{2 a \sqrt [4]{- a - 3 x^{2}} + 3 x^{2} \sqrt [4]{- a - 3 x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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