Optimal. Leaf size=39 \[ \frac{2}{13} x^{13/2} (A c+b B)+\frac{2}{9} A b x^{9/2}+\frac{2}{17} B c x^{17/2} \]
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Rubi [A] time = 0.0229939, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1584, 448} \[ \frac{2}{13} x^{13/2} (A c+b B)+\frac{2}{9} A b x^{9/2}+\frac{2}{17} B c x^{17/2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 448
Rubi steps
\begin{align*} \int x^{3/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right ) \, dx &=\int x^{7/2} \left (A+B x^2\right ) \left (b+c x^2\right ) \, dx\\ &=\int \left (A b x^{7/2}+(b B+A c) x^{11/2}+B c x^{15/2}\right ) \, dx\\ &=\frac{2}{9} A b x^{9/2}+\frac{2}{13} (b B+A c) x^{13/2}+\frac{2}{17} B c x^{17/2}\\ \end{align*}
Mathematica [A] time = 0.0155499, size = 33, normalized size = 0.85 \[ \frac{2 x^{9/2} \left (153 x^2 (A c+b B)+221 A b+117 B c x^4\right )}{1989} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 32, normalized size = 0.8 \begin{align*}{\frac{234\,Bc{x}^{4}+306\,A{x}^{2}c+306\,B{x}^{2}b+442\,Ab}{1989}{x}^{{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06539, size = 36, normalized size = 0.92 \begin{align*} \frac{2}{17} \, B c x^{\frac{17}{2}} + \frac{2}{13} \,{\left (B b + A c\right )} x^{\frac{13}{2}} + \frac{2}{9} \, A b x^{\frac{9}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55594, size = 89, normalized size = 2.28 \begin{align*} \frac{2}{1989} \,{\left (117 \, B c x^{8} + 153 \,{\left (B b + A c\right )} x^{6} + 221 \, A b x^{4}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.2908, size = 46, normalized size = 1.18 \begin{align*} \frac{2 A b x^{\frac{9}{2}}}{9} + \frac{2 A c x^{\frac{13}{2}}}{13} + \frac{2 B b x^{\frac{13}{2}}}{13} + \frac{2 B c x^{\frac{17}{2}}}{17} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13722, size = 39, normalized size = 1. \begin{align*} \frac{2}{17} \, B c x^{\frac{17}{2}} + \frac{2}{13} \, B b x^{\frac{13}{2}} + \frac{2}{13} \, A c x^{\frac{13}{2}} + \frac{2}{9} \, A b x^{\frac{9}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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