Optimal. Leaf size=39 \[ \frac{2}{9} x^{9/2} (a B+A b)+\frac{2}{5} a A x^{5/2}+\frac{2}{13} b B x^{13/2} \]
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Rubi [A] time = 0.015995, 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 = {448} \[ \frac{2}{9} x^{9/2} (a B+A b)+\frac{2}{5} a A x^{5/2}+\frac{2}{13} b B x^{13/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int x^{3/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx &=\int \left (a A x^{3/2}+(A b+a B) x^{7/2}+b B x^{11/2}\right ) \, dx\\ &=\frac{2}{5} a A x^{5/2}+\frac{2}{9} (A b+a B) x^{9/2}+\frac{2}{13} b B x^{13/2}\\ \end{align*}
Mathematica [A] time = 0.0133352, size = 33, normalized size = 0.85 \[ \frac{2}{585} x^{5/2} \left (65 x^2 (a B+A b)+117 a A+45 b B x^4\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 32, normalized size = 0.8 \begin{align*}{\frac{90\,bB{x}^{4}+130\,A{x}^{2}b+130\,B{x}^{2}a+234\,Aa}{585}{x}^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05933, size = 36, normalized size = 0.92 \begin{align*} \frac{2}{13} \, B b x^{\frac{13}{2}} + \frac{2}{9} \,{\left (B a + A b\right )} x^{\frac{9}{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 = 0.881758, size = 85, normalized size = 2.18 \begin{align*} \frac{2}{585} \,{\left (45 \, B b x^{6} + 65 \,{\left (B a + A b\right )} x^{4} + 117 \, 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 = 2.56887, size = 46, normalized size = 1.18 \begin{align*} \frac{2 A a x^{\frac{5}{2}}}{5} + \frac{2 A b x^{\frac{9}{2}}}{9} + \frac{2 B a x^{\frac{9}{2}}}{9} + \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 = 1.21152, size = 39, normalized size = 1. \begin{align*} \frac{2}{13} \, B b x^{\frac{13}{2}} + \frac{2}{9} \, B a x^{\frac{9}{2}} + \frac{2}{9} \, A b x^{\frac{9}{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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