Optimal. Leaf size=39 \[ \frac{2}{11} x^{11/2} (a B+A b)+\frac{2}{7} a A x^{7/2}+\frac{2}{15} b B x^{15/2} \]
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Rubi [A] time = 0.0159631, 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} \[ \frac{2}{11} x^{11/2} (a B+A b)+\frac{2}{7} a A x^{7/2}+\frac{2}{15} b B x^{15/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int x^{5/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx &=\int \left (a A x^{5/2}+(A b+a B) x^{9/2}+b B x^{13/2}\right ) \, dx\\ &=\frac{2}{7} a A x^{7/2}+\frac{2}{11} (A b+a B) x^{11/2}+\frac{2}{15} b B x^{15/2}\\ \end{align*}
Mathematica [A] time = 0.0145277, size = 33, normalized size = 0.85 \[ \frac{2 x^{7/2} \left (105 x^2 (a B+A b)+165 a A+77 b B x^4\right )}{1155} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 32, normalized size = 0.8 \begin{align*}{\frac{154\,bB{x}^{4}+210\,A{x}^{2}b+210\,B{x}^{2}a+330\,Aa}{1155}{x}^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04101, size = 36, normalized size = 0.92 \begin{align*} \frac{2}{15} \, B b x^{\frac{15}{2}} + \frac{2}{11} \,{\left (B a + A b\right )} x^{\frac{11}{2}} + \frac{2}{7} \, A a x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.852479, size = 88, normalized size = 2.26 \begin{align*} \frac{2}{1155} \,{\left (77 \, B b x^{7} + 105 \,{\left (B a + A b\right )} x^{5} + 165 \, A a x^{3}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.66925, size = 46, normalized size = 1.18 \begin{align*} \frac{2 A a x^{\frac{7}{2}}}{7} + \frac{2 A b x^{\frac{11}{2}}}{11} + \frac{2 B a x^{\frac{11}{2}}}{11} + \frac{2 B b x^{\frac{15}{2}}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13482, size = 39, normalized size = 1. \begin{align*} \frac{2}{15} \, B b x^{\frac{15}{2}} + \frac{2}{11} \, B a x^{\frac{11}{2}} + \frac{2}{11} \, A b x^{\frac{11}{2}} + \frac{2}{7} \, A a x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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