Optimal. Leaf size=77 \[ \frac{2}{5} a^2 A x^{5/2}+\frac{2}{7} a^2 B x^{7/2}+\frac{4}{9} a A c x^{9/2}+\frac{4}{11} a B c x^{11/2}+\frac{2}{13} A c^2 x^{13/2}+\frac{2}{15} B c^2 x^{15/2} \]
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Rubi [A] time = 0.0238589, 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} \[ \frac{2}{5} a^2 A x^{5/2}+\frac{2}{7} a^2 B x^{7/2}+\frac{4}{9} a A c x^{9/2}+\frac{4}{11} a B c x^{11/2}+\frac{2}{13} A c^2 x^{13/2}+\frac{2}{15} B c^2 x^{15/2} \]
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin{align*} \int x^{3/2} (A+B x) \left (a+c x^2\right )^2 \, dx &=\int \left (a^2 A x^{3/2}+a^2 B x^{5/2}+2 a A c x^{7/2}+2 a B c x^{9/2}+A c^2 x^{11/2}+B c^2 x^{13/2}\right ) \, dx\\ &=\frac{2}{5} a^2 A x^{5/2}+\frac{2}{7} a^2 B x^{7/2}+\frac{4}{9} a A c x^{9/2}+\frac{4}{11} a B c x^{11/2}+\frac{2}{13} A c^2 x^{13/2}+\frac{2}{15} B c^2 x^{15/2}\\ \end{align*}
Mathematica [A] time = 0.0210009, size = 54, normalized size = 0.7 \[ \frac{2 x^{5/2} \left (1287 a^2 (7 A+5 B x)+910 a c x^2 (11 A+9 B x)+231 c^2 x^4 (15 A+13 B x)\right )}{45045} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 54, normalized size = 0.7 \begin{align*}{\frac{6006\,B{c}^{2}{x}^{5}+6930\,A{c}^{2}{x}^{4}+16380\,aBc{x}^{3}+20020\,aAc{x}^{2}+12870\,{a}^{2}Bx+18018\,A{a}^{2}}{45045}{x}^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980072, size = 72, normalized size = 0.94 \begin{align*} \frac{2}{15} \, B c^{2} x^{\frac{15}{2}} + \frac{2}{13} \, A c^{2} x^{\frac{13}{2}} + \frac{4}{11} \, B a c x^{\frac{11}{2}} + \frac{4}{9} \, A a c x^{\frac{9}{2}} + \frac{2}{7} \, B a^{2} x^{\frac{7}{2}} + \frac{2}{5} \, A a^{2} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44443, size = 162, normalized size = 2.1 \begin{align*} \frac{2}{45045} \,{\left (3003 \, B c^{2} x^{7} + 3465 \, A c^{2} x^{6} + 8190 \, B a c x^{5} + 10010 \, A a c x^{4} + 6435 \, B a^{2} x^{3} + 9009 \, A a^{2} x^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.89759, size = 80, normalized size = 1.04 \begin{align*} \frac{2 A a^{2} x^{\frac{5}{2}}}{5} + \frac{4 A a c x^{\frac{9}{2}}}{9} + \frac{2 A c^{2} x^{\frac{13}{2}}}{13} + \frac{2 B a^{2} x^{\frac{7}{2}}}{7} + \frac{4 B a c x^{\frac{11}{2}}}{11} + \frac{2 B c^{2} x^{\frac{15}{2}}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26828, size = 72, normalized size = 0.94 \begin{align*} \frac{2}{15} \, B c^{2} x^{\frac{15}{2}} + \frac{2}{13} \, A c^{2} x^{\frac{13}{2}} + \frac{4}{11} \, B a c x^{\frac{11}{2}} + \frac{4}{9} \, A a c x^{\frac{9}{2}} + \frac{2}{7} \, B a^{2} x^{\frac{7}{2}} + \frac{2}{5} \, A a^{2} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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