Optimal. Leaf size=85 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{3 \sqrt {6} \sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{3 \sqrt {6} \sqrt [4]{a}} \]
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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 = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {442} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{3 \sqrt {6} \sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{3 \sqrt {6} \sqrt [4]{a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 442
Rubi steps
\begin {align*} \int \frac {x^2}{\left (-2 a-3 x^2\right ) \left (-a-3 x^2\right )^{3/4}} \, dx &=\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{3 \sqrt {6} \sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{3 \sqrt {6} \sqrt [4]{a}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 67, normalized size = 0.79 \begin {gather*} -\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}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 2.21, size = 87, normalized size = 1.02 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{a} \sqrt [4]{-a-3 x^2}}\right )}{3 \sqrt {6} \sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}} \sqrt [4]{a} \sqrt [4]{-a-3 x^2}}{x}\right )}{3 \sqrt {6} \sqrt [4]{a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.85, size = 145, normalized size = 1.71 \begin {gather*} \frac {2 \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \arctan \left (\frac {12 \, {\left (\sqrt {\frac {1}{2}} \left (\frac {1}{36}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x \sqrt {\frac {\frac {3 \, x^{2}}{\sqrt {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^{\frac {1}{4}}\right )}}{x}\right )}{3 \, a^{\frac {1}{4}}} - \frac {\left (\frac {1}{36}\right )^{\frac {1}{4}} \log \left (\frac {\frac {3 \, \left (\frac {1}{36}\right )^{\frac {1}{4}} x}{a^{\frac {1}{4}}} + {\left (-3 \, x^{2} - a\right )}^{\frac {1}{4}}}{x}\right )}{6 \, a^{\frac {1}{4}}} + \frac {\left (\frac {1}{36}\right )^{\frac {1}{4}} \log \left (-\frac {\frac {3 \, \left (\frac {1}{36}\right )^{\frac {1}{4}} x}{a^{\frac {1}{4}}} - {\left (-3 \, x^{2} - a\right )}^{\frac {1}{4}}}{x}\right )}{6 \, a^{\frac {1}{4}}} \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 {x^{2}}{{\left (3 \, x^{2} + 2 \, a\right )} {\left (-3 \, x^{2} - a\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (-3 x^{2}-2 a \right ) \left (-3 x^{2}-a \right )^{\frac {3}{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 {x^{2}}{{\left (3 \, x^{2} + 2 \, a\right )} {\left (-3 \, x^{2} - a\right )}^{\frac {3}{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 {x^2}{\left (3\,x^2+2\,a\right )\,{\left (-3\,x^2-a\right )}^{3/4}} \,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 {x^{2}}{2 a \left (- a - 3 x^{2}\right )^{\frac {3}{4}} + 3 x^{2} \left (- a - 3 x^{2}\right )^{\frac {3}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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