Optimal. Leaf size=39 \[ \frac{2}{13} x^{13/2} (A c+b B)+\frac{2}{11} A b x^{11/2}+\frac{2}{15} B c x^{15/2} \]
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Rubi [A] time = 0.0165459, 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} \[ \frac{2}{13} x^{13/2} (A c+b B)+\frac{2}{11} A b x^{11/2}+\frac{2}{15} B c x^{15/2} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int x^{7/2} (A+B x) \left (b x+c x^2\right ) \, dx &=\int \left (A b x^{9/2}+(b B+A c) x^{11/2}+B c x^{13/2}\right ) \, dx\\ &=\frac{2}{11} A b x^{11/2}+\frac{2}{13} (b B+A c) x^{13/2}+\frac{2}{15} B c x^{15/2}\\ \end{align*}
Mathematica [A] time = 0.009356, size = 33, normalized size = 0.85 \[ \frac{2 x^{11/2} (15 A (13 b+11 c x)+11 B x (15 b+13 c x))}{2145} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 28, normalized size = 0.7 \begin{align*}{\frac{286\,Bc{x}^{2}+330\,Acx+330\,bBx+390\,Ab}{2145}{x}^{{\frac{11}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994658, size = 36, normalized size = 0.92 \begin{align*} \frac{2}{15} \, B c x^{\frac{15}{2}} + \frac{2}{11} \, A b x^{\frac{11}{2}} + \frac{2}{13} \,{\left (B b + A c\right )} x^{\frac{13}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92904, size = 89, normalized size = 2.28 \begin{align*} \frac{2}{2145} \,{\left (143 \, B c x^{7} + 195 \, A b x^{5} + 165 \,{\left (B b + A c\right )} x^{6}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.68265, size = 46, normalized size = 1.18 \begin{align*} \frac{2 A b x^{\frac{11}{2}}}{11} + \frac{2 A c x^{\frac{13}{2}}}{13} + \frac{2 B b x^{\frac{13}{2}}}{13} + \frac{2 B c x^{\frac{15}{2}}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14948, size = 39, normalized size = 1. \begin{align*} \frac{2}{15} \, B c x^{\frac{15}{2}} + \frac{2}{13} \, B b x^{\frac{13}{2}} + \frac{2}{13} \, A c x^{\frac{13}{2}} + \frac{2}{11} \, A b x^{\frac{11}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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