Optimal. Leaf size=109 \[ \frac{2}{3} a^2 A c x^{9/2}+\frac{2}{5} a^3 A x^{5/2}+\frac{6}{11} a^2 B c x^{11/2}+\frac{2}{7} a^3 B x^{7/2}+\frac{6}{13} a A c^2 x^{13/2}+\frac{2}{5} a B c^2 x^{15/2}+\frac{2}{17} A c^3 x^{17/2}+\frac{2}{19} B c^3 x^{19/2} \]
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Rubi [A] time = 0.0368277, antiderivative size = 109, 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} \[ \frac{2}{3} a^2 A c x^{9/2}+\frac{2}{5} a^3 A x^{5/2}+\frac{6}{11} a^2 B c x^{11/2}+\frac{2}{7} a^3 B x^{7/2}+\frac{6}{13} a A c^2 x^{13/2}+\frac{2}{5} a B c^2 x^{15/2}+\frac{2}{17} A c^3 x^{17/2}+\frac{2}{19} B c^3 x^{19/2} \]
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 )^3 \, dx &=\int \left (a^3 A x^{3/2}+a^3 B x^{5/2}+3 a^2 A c x^{7/2}+3 a^2 B c x^{9/2}+3 a A c^2 x^{11/2}+3 a B c^2 x^{13/2}+A c^3 x^{15/2}+B c^3 x^{17/2}\right ) \, dx\\ &=\frac{2}{5} a^3 A x^{5/2}+\frac{2}{7} a^3 B x^{7/2}+\frac{2}{3} a^2 A c x^{9/2}+\frac{6}{11} a^2 B c x^{11/2}+\frac{6}{13} a A c^2 x^{13/2}+\frac{2}{5} a B c^2 x^{15/2}+\frac{2}{17} A c^3 x^{17/2}+\frac{2}{19} B c^3 x^{19/2}\\ \end{align*}
Mathematica [A] time = 0.037367, size = 83, normalized size = 0.76 \[ \frac{2}{33} a^2 c x^{9/2} (11 A+9 B x)+\frac{2}{35} a^3 x^{5/2} (7 A+5 B x)+\frac{2}{65} a c^2 x^{13/2} (15 A+13 B x)+\frac{2}{323} c^3 x^{17/2} (19 A+17 B x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 78, normalized size = 0.7 \begin{align*}{\frac{510510\,B{c}^{3}{x}^{7}+570570\,A{c}^{3}{x}^{6}+1939938\,aB{c}^{2}{x}^{5}+2238390\,aA{c}^{2}{x}^{4}+2645370\,{a}^{2}Bc{x}^{3}+3233230\,{a}^{2}Ac{x}^{2}+1385670\,{a}^{3}Bx+1939938\,A{a}^{3}}{4849845}{x}^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0012, size = 104, normalized size = 0.95 \begin{align*} \frac{2}{19} \, B c^{3} x^{\frac{19}{2}} + \frac{2}{17} \, A c^{3} x^{\frac{17}{2}} + \frac{2}{5} \, B a c^{2} x^{\frac{15}{2}} + \frac{6}{13} \, A a c^{2} x^{\frac{13}{2}} + \frac{6}{11} \, B a^{2} c x^{\frac{11}{2}} + \frac{2}{3} \, A a^{2} c x^{\frac{9}{2}} + \frac{2}{7} \, B a^{3} x^{\frac{7}{2}} + \frac{2}{5} \, A a^{3} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38685, size = 246, normalized size = 2.26 \begin{align*} \frac{2}{4849845} \,{\left (255255 \, B c^{3} x^{9} + 285285 \, A c^{3} x^{8} + 969969 \, B a c^{2} x^{7} + 1119195 \, A a c^{2} x^{6} + 1322685 \, B a^{2} c x^{5} + 1616615 \, A a^{2} c x^{4} + 692835 \, B a^{3} x^{3} + 969969 \, A a^{3} x^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.2576, size = 114, normalized size = 1.05 \begin{align*} \frac{2 A a^{3} x^{\frac{5}{2}}}{5} + \frac{2 A a^{2} c x^{\frac{9}{2}}}{3} + \frac{6 A a c^{2} x^{\frac{13}{2}}}{13} + \frac{2 A c^{3} x^{\frac{17}{2}}}{17} + \frac{2 B a^{3} x^{\frac{7}{2}}}{7} + \frac{6 B a^{2} c x^{\frac{11}{2}}}{11} + \frac{2 B a c^{2} x^{\frac{15}{2}}}{5} + \frac{2 B c^{3} x^{\frac{19}{2}}}{19} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26946, size = 104, normalized size = 0.95 \begin{align*} \frac{2}{19} \, B c^{3} x^{\frac{19}{2}} + \frac{2}{17} \, A c^{3} x^{\frac{17}{2}} + \frac{2}{5} \, B a c^{2} x^{\frac{15}{2}} + \frac{6}{13} \, A a c^{2} x^{\frac{13}{2}} + \frac{6}{11} \, B a^{2} c x^{\frac{11}{2}} + \frac{2}{3} \, A a^{2} c x^{\frac{9}{2}} + \frac{2}{7} \, B a^{3} x^{\frac{7}{2}} + \frac{2}{5} \, A a^{3} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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