Optimal. Leaf size=39 \[ \frac{2}{11} x^{11/2} (a B+A b)+\frac{2}{9} a A x^{9/2}+\frac{2}{13} b B x^{13/2} \]
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Rubi [A] time = 0.014199, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {76} \[ \frac{2}{11} x^{11/2} (a B+A b)+\frac{2}{9} a A x^{9/2}+\frac{2}{13} b B x^{13/2} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int x^{7/2} (a+b x) (A+B x) \, dx &=\int \left (a A x^{7/2}+(A b+a B) x^{9/2}+b B x^{11/2}\right ) \, dx\\ &=\frac{2}{9} a A x^{9/2}+\frac{2}{11} (A b+a B) x^{11/2}+\frac{2}{13} b B x^{13/2}\\ \end{align*}
Mathematica [A] time = 0.0114178, size = 33, normalized size = 0.85 \[ \frac{2 x^{9/2} (13 a (11 A+9 B x)+9 b x (13 A+11 B x))}{1287} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 28, normalized size = 0.7 \begin{align*}{\frac{198\,bB{x}^{2}+234\,Abx+234\,Bax+286\,Aa}{1287}{x}^{{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09306, size = 36, normalized size = 0.92 \begin{align*} \frac{2}{13} \, B b x^{\frac{13}{2}} + \frac{2}{9} \, A a x^{\frac{9}{2}} + \frac{2}{11} \,{\left (B a + A b\right )} x^{\frac{11}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.49648, size = 88, normalized size = 2.26 \begin{align*} \frac{2}{1287} \,{\left (99 \, B b x^{6} + 143 \, A a x^{4} + 117 \,{\left (B a + A b\right )} x^{5}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.87348, size = 46, normalized size = 1.18 \begin{align*} \frac{2 A a x^{\frac{9}{2}}}{9} + \frac{2 A b x^{\frac{11}{2}}}{11} + \frac{2 B a x^{\frac{11}{2}}}{11} + \frac{2 B b x^{\frac{13}{2}}}{13} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21406, size = 39, normalized size = 1. \begin{align*} \frac{2}{13} \, B b x^{\frac{13}{2}} + \frac{2}{11} \, B a x^{\frac{11}{2}} + \frac{2}{11} \, A b x^{\frac{11}{2}} + \frac{2}{9} \, A a x^{\frac{9}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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