Optimal. Leaf size=39 \[ \frac{2}{9} x^{9/2} (A c+b B)+\frac{2}{7} A b x^{7/2}+\frac{2}{11} B c x^{11/2} \]
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Rubi [A] time = 0.0163165, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ \frac{2}{9} x^{9/2} (A c+b B)+\frac{2}{7} A b x^{7/2}+\frac{2}{11} B c x^{11/2} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int x^{3/2} (A+B x) \left (b x+c x^2\right ) \, dx &=\int \left (A b x^{5/2}+(b B+A c) x^{7/2}+B c x^{9/2}\right ) \, dx\\ &=\frac{2}{7} A b x^{7/2}+\frac{2}{9} (b B+A c) x^{9/2}+\frac{2}{11} B c x^{11/2}\\ \end{align*}
Mathematica [A] time = 0.0090838, size = 33, normalized size = 0.85 \[ \frac{2}{693} x^{7/2} (11 A (9 b+7 c x)+7 B x (11 b+9 c x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 28, normalized size = 0.7 \begin{align*}{\frac{126\,Bc{x}^{2}+154\,Acx+154\,bBx+198\,Ab}{693}{x}^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02077, size = 36, normalized size = 0.92 \begin{align*} \frac{2}{11} \, B c x^{\frac{11}{2}} + \frac{2}{7} \, A b x^{\frac{7}{2}} + \frac{2}{9} \,{\left (B b + A c\right )} x^{\frac{9}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7995, size = 84, normalized size = 2.15 \begin{align*} \frac{2}{693} \,{\left (63 \, B c x^{5} + 99 \, A b x^{3} + 77 \,{\left (B b + A c\right )} x^{4}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.81431, size = 46, normalized size = 1.18 \begin{align*} \frac{2 A b x^{\frac{7}{2}}}{7} + \frac{2 A c x^{\frac{9}{2}}}{9} + \frac{2 B b x^{\frac{9}{2}}}{9} + \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 = 1.17017, size = 39, normalized size = 1. \begin{align*} \frac{2}{11} \, B c x^{\frac{11}{2}} + \frac{2}{9} \, B b x^{\frac{9}{2}} + \frac{2}{9} \, A c x^{\frac{9}{2}} + \frac{2}{7} \, A b x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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