Optimal. Leaf size=46 \[ \frac{1}{320} \left (4 x^2-9\right )^{5/2}+\frac{3}{32} \left (4 x^2-9\right )^{3/2}+\frac{81}{64} \sqrt{4 x^2-9} \]
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Rubi [A] time = 0.0187101, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{1}{320} \left (4 x^2-9\right )^{5/2}+\frac{3}{32} \left (4 x^2-9\right )^{3/2}+\frac{81}{64} \sqrt{4 x^2-9} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\sqrt{-9+4 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-9+4 x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{81}{16 \sqrt{-9+4 x}}+\frac{9}{8} \sqrt{-9+4 x}+\frac{1}{16} (-9+4 x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=\frac{81}{64} \sqrt{-9+4 x^2}+\frac{3}{32} \left (-9+4 x^2\right )^{3/2}+\frac{1}{320} \left (-9+4 x^2\right )^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0085672, size = 27, normalized size = 0.59 \[ \frac{1}{40} \sqrt{4 x^2-9} \left (2 x^4+6 x^2+27\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 34, normalized size = 0.7 \begin{align*}{\frac{ \left ( -3+2\,x \right ) \left ( 3+2\,x \right ) \left ( 2\,{x}^{4}+6\,{x}^{2}+27 \right ) }{40}{\frac{1}{\sqrt{4\,{x}^{2}-9}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.52361, size = 54, normalized size = 1.17 \begin{align*} \frac{1}{20} \, \sqrt{4 \, x^{2} - 9} x^{4} + \frac{3}{20} \, \sqrt{4 \, x^{2} - 9} x^{2} + \frac{27}{40} \, \sqrt{4 \, x^{2} - 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32353, size = 58, normalized size = 1.26 \begin{align*} \frac{1}{40} \,{\left (2 \, x^{4} + 6 \, x^{2} + 27\right )} \sqrt{4 \, x^{2} - 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.12322, size = 44, normalized size = 0.96 \begin{align*} \frac{x^{4} \sqrt{4 x^{2} - 9}}{20} + \frac{3 x^{2} \sqrt{4 x^{2} - 9}}{20} + \frac{27 \sqrt{4 x^{2} - 9}}{40} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.41269, size = 46, normalized size = 1. \begin{align*} \frac{1}{320} \,{\left (4 \, x^{2} - 9\right )}^{\frac{5}{2}} + \frac{3}{32} \,{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}} + \frac{81}{64} \, \sqrt{4 \, x^{2} - 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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