Optimal. Leaf size=77 \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{11} a^2 B x^{11/2}+\frac{4}{13} a A c x^{13/2}+\frac{4}{15} a B c x^{15/2}+\frac{2}{17} A c^2 x^{17/2}+\frac{2}{19} B c^2 x^{19/2} \]
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Rubi [A] time = 0.0245772, 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} \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{11} a^2 B x^{11/2}+\frac{4}{13} a A c x^{13/2}+\frac{4}{15} a B c x^{15/2}+\frac{2}{17} A c^2 x^{17/2}+\frac{2}{19} B c^2 x^{19/2} \]
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin{align*} \int x^{7/2} (A+B x) \left (a+c x^2\right )^2 \, dx &=\int \left (a^2 A x^{7/2}+a^2 B x^{9/2}+2 a A c x^{11/2}+2 a B c x^{13/2}+A c^2 x^{15/2}+B c^2 x^{17/2}\right ) \, dx\\ &=\frac{2}{9} a^2 A x^{9/2}+\frac{2}{11} a^2 B x^{11/2}+\frac{4}{13} a A c x^{13/2}+\frac{4}{15} a B c x^{15/2}+\frac{2}{17} A c^2 x^{17/2}+\frac{2}{19} B c^2 x^{19/2}\\ \end{align*}
Mathematica [A] time = 0.0233011, size = 60, normalized size = 0.78 \[ \frac{2}{99} a^2 x^{9/2} (11 A+9 B x)+\frac{4}{195} a c x^{13/2} (15 A+13 B x)+\frac{2}{323} c^2 x^{17/2} (19 A+17 B x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 54, normalized size = 0.7 \begin{align*}{\frac{218790\,B{c}^{2}{x}^{5}+244530\,A{c}^{2}{x}^{4}+554268\,aBc{x}^{3}+639540\,aAc{x}^{2}+377910\,{a}^{2}Bx+461890\,A{a}^{2}}{2078505}{x}^{{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.963717, size = 72, normalized size = 0.94 \begin{align*} \frac{2}{19} \, B c^{2} x^{\frac{19}{2}} + \frac{2}{17} \, A c^{2} x^{\frac{17}{2}} + \frac{4}{15} \, B a c x^{\frac{15}{2}} + \frac{4}{13} \, A a c x^{\frac{13}{2}} + \frac{2}{11} \, B a^{2} x^{\frac{11}{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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Fricas [A] time = 1.35608, size = 180, normalized size = 2.34 \begin{align*} \frac{2}{2078505} \,{\left (109395 \, B c^{2} x^{9} + 122265 \, A c^{2} x^{8} + 277134 \, B a c x^{7} + 319770 \, A a c x^{6} + 188955 \, B a^{2} x^{5} + 230945 \, A a^{2} x^{4}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 21.1513, size = 80, normalized size = 1.04 \begin{align*} \frac{2 A a^{2} x^{\frac{9}{2}}}{9} + \frac{4 A a c x^{\frac{13}{2}}}{13} + \frac{2 A c^{2} x^{\frac{17}{2}}}{17} + \frac{2 B a^{2} x^{\frac{11}{2}}}{11} + \frac{4 B a c x^{\frac{15}{2}}}{15} + \frac{2 B c^{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 = 1.16654, size = 72, normalized size = 0.94 \begin{align*} \frac{2}{19} \, B c^{2} x^{\frac{19}{2}} + \frac{2}{17} \, A c^{2} x^{\frac{17}{2}} + \frac{4}{15} \, B a c x^{\frac{15}{2}} + \frac{4}{13} \, A a c x^{\frac{13}{2}} + \frac{2}{11} \, B a^{2} x^{\frac{11}{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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