Optimal. Leaf size=34 \[ -\frac {3}{32} \sin ^{-1}\left (\frac {8 x^2}{3}+1\right )-\frac {1}{8} \sqrt {-4 x^4-3 x^2} \]
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Rubi [A] time = 0.06, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2018, 640, 619, 216} \begin {gather*} -\frac {1}{8} \sqrt {-4 x^4-3 x^2}-\frac {3}{32} \sin ^{-1}\left (\frac {8 x^2}{3}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 216
Rule 619
Rule 640
Rule 2018
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {-3 x^2-4 x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\sqrt {-3 x-4 x^2}} \, dx,x,x^2\right )\\ &=-\frac {1}{8} \sqrt {-3 x^2-4 x^4}-\frac {3}{16} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-3 x-4 x^2}} \, dx,x,x^2\right )\\ &=-\frac {1}{8} \sqrt {-3 x^2-4 x^4}+\frac {1}{32} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{9}}} \, dx,x,-3-8 x^2\right )\\ &=-\frac {1}{8} \sqrt {-3 x^2-4 x^4}-\frac {3}{32} \sin ^{-1}\left (1+\frac {8 x^2}{3}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 52, normalized size = 1.53 \begin {gather*} \frac {x \left (8 x^3-3 \sqrt {4 x^2+3} \sinh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )+6 x\right )}{16 \sqrt {-x^2 \left (4 x^2+3\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 51, normalized size = 1.50 \begin {gather*} \frac {3}{16} \tan ^{-1}\left (\frac {2 \sqrt {-4 x^4-3 x^2}}{4 x^2+3}\right )-\frac {1}{8} \sqrt {-4 x^4-3 x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [C] time = 0.91, size = 59, normalized size = 1.74 \begin {gather*} -\frac {1}{8} \, \sqrt {-4 \, x^{2} - 3} x - \frac {3}{32} i \, \log \left (-\frac {8 \, x + 4 i \, \sqrt {-4 \, x^{2} - 3}}{x}\right ) + \frac {3}{32} i \, \log \left (-\frac {8 \, x - 4 i \, \sqrt {-4 \, x^{2} - 3}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 27, normalized size = 0.79 \begin {gather*} -\frac {1}{8} \, \sqrt {4 \, x^{4} + 3 \, x^{2}} i - \frac {3}{32} \, \arcsin \left (\frac {8}{3} \, x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 54, normalized size = 1.59 \begin {gather*} -\frac {\sqrt {-4 x^{2}-3}\, \left (2 \sqrt {-4 x^{2}-3}\, x +3 \arctan \left (\frac {2 x}{\sqrt {-4 x^{2}-3}}\right )\right ) x}{16 \sqrt {-4 x^{4}-3 x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 26, normalized size = 0.76 \begin {gather*} -\frac {1}{8} \, \sqrt {-4 \, x^{4} - 3 \, x^{2}} + \frac {3}{32} \, \arcsin \left (-\frac {8}{3} \, x^{2} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.36, size = 41, normalized size = 1.21 \begin {gather*} -\frac {\sqrt {-4\,x^4-3\,x^2}}{8}+\frac {\ln \left (\frac {\sqrt {4\,x^2+3}\,\sqrt {x^2}}{2}+x^2+\frac {3}{8}\right )\,3{}\mathrm {i}}{32} \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^{3}}{\sqrt {- x^{2} \left (4 x^{2} + 3\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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