Optimal. Leaf size=77 \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{9} a^2 B x^{9/2}+\frac{4}{11} a A c x^{11/2}+\frac{4}{13} a B c x^{13/2}+\frac{2}{15} A c^2 x^{15/2}+\frac{2}{17} B c^2 x^{17/2} \]
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Rubi [A] time = 0.0257879, antiderivative size = 77, 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 = {766} \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{9} a^2 B x^{9/2}+\frac{4}{11} a A c x^{11/2}+\frac{4}{13} a B c x^{13/2}+\frac{2}{15} A c^2 x^{15/2}+\frac{2}{17} B c^2 x^{17/2} \]
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin{align*} \int x^{5/2} (A+B x) \left (a+c x^2\right )^2 \, dx &=\int \left (a^2 A x^{5/2}+a^2 B x^{7/2}+2 a A c x^{9/2}+2 a B c x^{11/2}+A c^2 x^{13/2}+B c^2 x^{15/2}\right ) \, dx\\ &=\frac{2}{7} a^2 A x^{7/2}+\frac{2}{9} a^2 B x^{9/2}+\frac{4}{11} a A c x^{11/2}+\frac{4}{13} a B c x^{13/2}+\frac{2}{15} A c^2 x^{15/2}+\frac{2}{17} B c^2 x^{17/2}\\ \end{align*}
Mathematica [A] time = 0.023185, size = 60, normalized size = 0.78 \[ \frac{2}{63} a^2 x^{7/2} (9 A+7 B x)+\frac{4}{143} a c x^{11/2} (13 A+11 B x)+\frac{2}{255} c^2 x^{15/2} (17 A+15 B x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 54, normalized size = 0.7 \begin{align*}{\frac{90090\,B{c}^{2}{x}^{5}+102102\,A{c}^{2}{x}^{4}+235620\,aBc{x}^{3}+278460\,aAc{x}^{2}+170170\,{a}^{2}Bx+218790\,A{a}^{2}}{765765}{x}^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994453, size = 72, normalized size = 0.94 \begin{align*} \frac{2}{17} \, B c^{2} x^{\frac{17}{2}} + \frac{2}{15} \, A c^{2} x^{\frac{15}{2}} + \frac{4}{13} \, B a c x^{\frac{13}{2}} + \frac{4}{11} \, A a c x^{\frac{11}{2}} + \frac{2}{9} \, B a^{2} 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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Fricas [A] time = 1.27833, size = 174, normalized size = 2.26 \begin{align*} \frac{2}{765765} \,{\left (45045 \, B c^{2} x^{8} + 51051 \, A c^{2} x^{7} + 117810 \, B a c x^{6} + 139230 \, A a c x^{5} + 85085 \, B a^{2} x^{4} + 109395 \, A a^{2} x^{3}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.5877, size = 80, normalized size = 1.04 \begin{align*} \frac{2 A a^{2} x^{\frac{7}{2}}}{7} + \frac{4 A a c x^{\frac{11}{2}}}{11} + \frac{2 A c^{2} x^{\frac{15}{2}}}{15} + \frac{2 B a^{2} x^{\frac{9}{2}}}{9} + \frac{4 B a c x^{\frac{13}{2}}}{13} + \frac{2 B c^{2} x^{\frac{17}{2}}}{17} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20202, size = 72, normalized size = 0.94 \begin{align*} \frac{2}{17} \, B c^{2} x^{\frac{17}{2}} + \frac{2}{15} \, A c^{2} x^{\frac{15}{2}} + \frac{4}{13} \, B a c x^{\frac{13}{2}} + \frac{4}{11} \, A a c x^{\frac{11}{2}} + \frac{2}{9} \, B a^{2} 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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