Optimal. Leaf size=63 \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{15} b x^{15/2} (2 a B+A b)+\frac{2}{11} a x^{11/2} (a B+2 A b)+\frac{2}{19} b^2 B x^{19/2} \]
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Rubi [A] time = 0.0299246, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {448} \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{15} b x^{15/2} (2 a B+A b)+\frac{2}{11} a x^{11/2} (a B+2 A b)+\frac{2}{19} b^2 B x^{19/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int x^{5/2} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx &=\int \left (a^2 A x^{5/2}+a (2 A b+a B) x^{9/2}+b (A b+2 a B) x^{13/2}+b^2 B x^{17/2}\right ) \, dx\\ &=\frac{2}{7} a^2 A x^{7/2}+\frac{2}{11} a (2 A b+a B) x^{11/2}+\frac{2}{15} b (A b+2 a B) x^{15/2}+\frac{2}{19} b^2 B x^{19/2}\\ \end{align*}
Mathematica [A] time = 0.0288283, size = 63, normalized size = 1. \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{15} b x^{15/2} (2 a B+A b)+\frac{2}{11} a x^{11/2} (a B+2 A b)+\frac{2}{19} b^2 B x^{19/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 56, normalized size = 0.9 \begin{align*}{\frac{2310\,B{b}^{2}{x}^{6}+2926\,A{b}^{2}{x}^{4}+5852\,B{x}^{4}ab+7980\,aAb{x}^{2}+3990\,B{x}^{2}{a}^{2}+6270\,{a}^{2}A}{21945}{x}^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05328, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{19} \, B b^{2} x^{\frac{19}{2}} + \frac{2}{15} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{15}{2}} + \frac{2}{7} \, A a^{2} x^{\frac{7}{2}} + \frac{2}{11} \,{\left (B a^{2} + 2 \, 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 = 0.815691, size = 147, normalized size = 2.33 \begin{align*} \frac{2}{21945} \,{\left (1155 \, B b^{2} x^{9} + 1463 \,{\left (2 \, B a b + A b^{2}\right )} x^{7} + 3135 \, A a^{2} x^{3} + 1995 \,{\left (B a^{2} + 2 \, 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 = 11.4721, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A a^{2} x^{\frac{7}{2}}}{7} + \frac{4 A a b x^{\frac{11}{2}}}{11} + \frac{2 A b^{2} x^{\frac{15}{2}}}{15} + \frac{2 B a^{2} x^{\frac{11}{2}}}{11} + \frac{4 B a b x^{\frac{15}{2}}}{15} + \frac{2 B b^{2} x^{\frac{19}{2}}}{19} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15334, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{19} \, B b^{2} x^{\frac{19}{2}} + \frac{4}{15} \, B a b x^{\frac{15}{2}} + \frac{2}{15} \, A b^{2} x^{\frac{15}{2}} + \frac{2}{11} \, B a^{2} x^{\frac{11}{2}} + \frac{4}{11} \, A a b x^{\frac{11}{2}} + \frac{2}{7} \, A a^{2} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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