Optimal. Leaf size=45 \[ \frac{2}{7} a A x^{7/2}+\frac{2}{9} a B x^{9/2}+\frac{2}{11} A c x^{11/2}+\frac{2}{13} B c x^{13/2} \]
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Rubi [A] time = 0.0123421, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {766} \[ \frac{2}{7} a A x^{7/2}+\frac{2}{9} a B x^{9/2}+\frac{2}{11} A c x^{11/2}+\frac{2}{13} B c x^{13/2} \]
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin{align*} \int x^{5/2} (A+B x) \left (a+c x^2\right ) \, dx &=\int \left (a A x^{5/2}+a B x^{7/2}+A c x^{9/2}+B c x^{11/2}\right ) \, dx\\ &=\frac{2}{7} a A x^{7/2}+\frac{2}{9} a B x^{9/2}+\frac{2}{11} A c x^{11/2}+\frac{2}{13} B c x^{13/2}\\ \end{align*}
Mathematica [A] time = 0.0140148, size = 35, normalized size = 0.78 \[ \frac{2 x^{7/2} \left (143 a (9 A+7 B x)+63 c x^2 (13 A+11 B x)\right )}{9009} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 30, normalized size = 0.7 \begin{align*}{\frac{1386\,Bc{x}^{3}+1638\,Ac{x}^{2}+2002\,aBx+2574\,aA}{9009}{x}^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98342, size = 39, normalized size = 0.87 \begin{align*} \frac{2}{13} \, B c x^{\frac{13}{2}} + \frac{2}{11} \, A c x^{\frac{11}{2}} + \frac{2}{9} \, B a x^{\frac{9}{2}} + \frac{2}{7} \, A a x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38773, size = 100, normalized size = 2.22 \begin{align*} \frac{2}{9009} \,{\left (693 \, B c x^{6} + 819 \, A c x^{5} + 1001 \, B a x^{4} + 1287 \, A a x^{3}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.71777, size = 46, normalized size = 1.02 \begin{align*} \frac{2 A a x^{\frac{7}{2}}}{7} + \frac{2 A c x^{\frac{11}{2}}}{11} + \frac{2 B a x^{\frac{9}{2}}}{9} + \frac{2 B c x^{\frac{13}{2}}}{13} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16391, size = 39, normalized size = 0.87 \begin{align*} \frac{2}{13} \, B c x^{\frac{13}{2}} + \frac{2}{11} \, A c x^{\frac{11}{2}} + \frac{2}{9} \, B a x^{\frac{9}{2}} + \frac{2}{7} \, A a x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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