Optimal. Leaf size=45 \[ \frac{2}{5} a A x^{5/2}+\frac{2}{7} a B x^{7/2}+\frac{2}{9} A c x^{9/2}+\frac{2}{11} B c x^{11/2} \]
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Rubi [A] time = 0.0125184, 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} \[ \frac{2}{5} a A x^{5/2}+\frac{2}{7} a B x^{7/2}+\frac{2}{9} A c x^{9/2}+\frac{2}{11} B c x^{11/2} \]
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin{align*} \int x^{3/2} (A+B x) \left (a+c x^2\right ) \, dx &=\int \left (a A x^{3/2}+a B x^{5/2}+A c x^{7/2}+B c x^{9/2}\right ) \, dx\\ &=\frac{2}{5} a A x^{5/2}+\frac{2}{7} a B x^{7/2}+\frac{2}{9} A c x^{9/2}+\frac{2}{11} B c x^{11/2}\\ \end{align*}
Mathematica [A] time = 0.0169846, size = 37, normalized size = 0.82 \[ \frac{2}{35} a x^{5/2} (7 A+5 B x)+\frac{2}{99} c x^{9/2} (11 A+9 B x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 30, normalized size = 0.7 \begin{align*}{\frac{630\,Bc{x}^{3}+770\,Ac{x}^{2}+990\,aBx+1386\,aA}{3465}{x}^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991566, size = 39, normalized size = 0.87 \begin{align*} \frac{2}{11} \, B c x^{\frac{11}{2}} + \frac{2}{9} \, A c x^{\frac{9}{2}} + \frac{2}{7} \, B a x^{\frac{7}{2}} + \frac{2}{5} \, A a x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28202, size = 97, normalized size = 2.16 \begin{align*} \frac{2}{3465} \,{\left (315 \, B c x^{5} + 385 \, A c x^{4} + 495 \, B a x^{3} + 693 \, A a x^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.02586, size = 46, normalized size = 1.02 \begin{align*} \frac{2 A a x^{\frac{5}{2}}}{5} + \frac{2 A c x^{\frac{9}{2}}}{9} + \frac{2 B a x^{\frac{7}{2}}}{7} + \frac{2 B c x^{\frac{11}{2}}}{11} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14921, size = 39, normalized size = 0.87 \begin{align*} \frac{2}{11} \, B c x^{\frac{11}{2}} + \frac{2}{9} \, A c x^{\frac{9}{2}} + \frac{2}{7} \, B a x^{\frac{7}{2}} + \frac{2}{5} \, A a x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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