Optimal. Leaf size=63 \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{17} b x^{17/2} (2 a B+A b)+\frac{2}{13} a x^{13/2} (a B+2 A b)+\frac{2}{21} b^2 B x^{21/2} \]
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Rubi [A] time = 0.0299327, 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} \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{17} b x^{17/2} (2 a B+A b)+\frac{2}{13} a x^{13/2} (a B+2 A b)+\frac{2}{21} b^2 B x^{21/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int x^{7/2} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx &=\int \left (a^2 A x^{7/2}+a (2 A b+a B) x^{11/2}+b (A b+2 a B) x^{15/2}+b^2 B x^{19/2}\right ) \, dx\\ &=\frac{2}{9} a^2 A x^{9/2}+\frac{2}{13} a (2 A b+a B) x^{13/2}+\frac{2}{17} b (A b+2 a B) x^{17/2}+\frac{2}{21} b^2 B x^{21/2}\\ \end{align*}
Mathematica [A] time = 0.0310451, size = 53, normalized size = 0.84 \[ \frac{2 x^{9/2} \left (1547 a^2 A+819 b x^4 (2 a B+A b)+1071 a x^2 (a B+2 A b)+663 b^2 B x^6\right )}{13923} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 56, normalized size = 0.9 \begin{align*}{\frac{1326\,B{b}^{2}{x}^{6}+1638\,A{b}^{2}{x}^{4}+3276\,B{x}^{4}ab+4284\,aAb{x}^{2}+2142\,B{x}^{2}{a}^{2}+3094\,{a}^{2}A}{13923}{x}^{{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05195, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{21} \, B b^{2} x^{\frac{21}{2}} + \frac{2}{17} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{17}{2}} + \frac{2}{9} \, A a^{2} x^{\frac{9}{2}} + \frac{2}{13} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{13}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.797779, size = 146, normalized size = 2.32 \begin{align*} \frac{2}{13923} \,{\left (663 \, B b^{2} x^{10} + 819 \,{\left (2 \, B a b + A b^{2}\right )} x^{8} + 1547 \, A a^{2} x^{4} + 1071 \,{\left (B a^{2} + 2 \, A a b\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 = 20.9895, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A a^{2} x^{\frac{9}{2}}}{9} + \frac{4 A a b x^{\frac{13}{2}}}{13} + \frac{2 A b^{2} x^{\frac{17}{2}}}{17} + \frac{2 B a^{2} x^{\frac{13}{2}}}{13} + \frac{4 B a b x^{\frac{17}{2}}}{17} + \frac{2 B b^{2} x^{\frac{21}{2}}}{21} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12682, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{21} \, B b^{2} x^{\frac{21}{2}} + \frac{4}{17} \, B a b x^{\frac{17}{2}} + \frac{2}{17} \, A b^{2} x^{\frac{17}{2}} + \frac{2}{13} \, B a^{2} x^{\frac{13}{2}} + \frac{4}{13} \, A a b x^{\frac{13}{2}} + \frac{2}{9} \, A a^{2} x^{\frac{9}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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