Optimal. Leaf size=63 \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{11} b x^{11/2} (2 a B+A b)+\frac{2}{9} a x^{9/2} (a B+2 A b)+\frac{2}{13} b^2 B x^{13/2} \]
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Rubi [A] time = 0.0273691, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {76} \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{11} b x^{11/2} (2 a B+A b)+\frac{2}{9} a x^{9/2} (a B+2 A b)+\frac{2}{13} b^2 B x^{13/2} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int x^{5/2} (a+b x)^2 (A+B x) \, dx &=\int \left (a^2 A x^{5/2}+a (2 A b+a B) x^{7/2}+b (A b+2 a B) x^{9/2}+b^2 B x^{11/2}\right ) \, dx\\ &=\frac{2}{7} a^2 A x^{7/2}+\frac{2}{9} a (2 A b+a B) x^{9/2}+\frac{2}{11} b (A b+2 a B) x^{11/2}+\frac{2}{13} b^2 B x^{13/2}\\ \end{align*}
Mathematica [A] time = 0.0160576, size = 52, normalized size = 0.83 \[ \frac{2 x^{7/2} \left (143 a^2 (9 A+7 B x)+182 a b x (11 A+9 B x)+63 b^2 x^2 (13 A+11 B x)\right )}{9009} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 52, normalized size = 0.8 \begin{align*}{\frac{1386\,B{b}^{2}{x}^{3}+1638\,A{b}^{2}{x}^{2}+3276\,B{x}^{2}ab+4004\,aAbx+2002\,{a}^{2}Bx+2574\,{a}^{2}A}{9009}{x}^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10745, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{2}{7} \, A a^{2} x^{\frac{7}{2}} + \frac{2}{11} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{11}{2}} + \frac{2}{9} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{9}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.23043, size = 143, normalized size = 2.27 \begin{align*} \frac{2}{9009} \,{\left (693 \, B b^{2} x^{6} + 1287 \, A a^{2} x^{3} + 819 \,{\left (2 \, B a b + A b^{2}\right )} x^{5} + 1001 \,{\left (B a^{2} + 2 \, A a b\right )} x^{4}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.75096, 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{9}{2}}}{9} + \frac{2 A b^{2} x^{\frac{11}{2}}}{11} + \frac{2 B a^{2} x^{\frac{9}{2}}}{9} + \frac{4 B a b x^{\frac{11}{2}}}{11} + \frac{2 B b^{2} x^{\frac{13}{2}}}{13} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21738, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{4}{11} \, B a b x^{\frac{11}{2}} + \frac{2}{11} \, A b^{2} x^{\frac{11}{2}} + \frac{2}{9} \, B a^{2} x^{\frac{9}{2}} + \frac{4}{9} \, A a b x^{\frac{9}{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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