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 = 0.02, antiderivative size = 120, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}
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*}
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Mathematica [C] time = 0.15, size = 155, normalized size = 1.29 \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 = 0.25, size = 136, normalized size = 1.13 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {2 \sqrt {3} \sqrt [4]{a} x \sqrt [4]{a+3 x^2}}{2 \sqrt {a} \sqrt {a+3 x^2}+3 x^2}\right )}{4 \sqrt {3} a^{3/4}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [4]{a} \sqrt {a+3 x^2}}{\sqrt {3}}-\frac {\sqrt {3} x^2}{2 \sqrt [4]{a}}}{x \sqrt [4]{a+3 x^2}}\right )}{4 \sqrt {3} a^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 24.63, size = 286, normalized size = 2.38 \begin {gather*} \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 {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 = 0.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (3 x^{2}+a \right )^{\frac {1}{4}} \left (3 x^{2}+2 a \right )}\, 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}{\sqrt [4]{a + 3 x^{2}} \left (2 a + 3 x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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