Optimal. Leaf size=109 \[ \frac{6}{13} a^2 A c x^{13/2}+\frac{2}{9} a^3 A x^{9/2}+\frac{2}{5} a^2 B c x^{15/2}+\frac{2}{11} a^3 B x^{11/2}+\frac{6}{17} a A c^2 x^{17/2}+\frac{6}{19} a B c^2 x^{19/2}+\frac{2}{21} A c^3 x^{21/2}+\frac{2}{23} B c^3 x^{23/2} \]
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Rubi [A] time = 0.037606, 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} \[ \frac{6}{13} a^2 A c x^{13/2}+\frac{2}{9} a^3 A x^{9/2}+\frac{2}{5} a^2 B c x^{15/2}+\frac{2}{11} a^3 B x^{11/2}+\frac{6}{17} a A c^2 x^{17/2}+\frac{6}{19} a B c^2 x^{19/2}+\frac{2}{21} A c^3 x^{21/2}+\frac{2}{23} B c^3 x^{23/2} \]
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 )^3 \, dx &=\int \left (a^3 A x^{7/2}+a^3 B x^{9/2}+3 a^2 A c x^{11/2}+3 a^2 B c x^{13/2}+3 a A c^2 x^{15/2}+3 a B c^2 x^{17/2}+A c^3 x^{19/2}+B c^3 x^{21/2}\right ) \, dx\\ &=\frac{2}{9} a^3 A x^{9/2}+\frac{2}{11} a^3 B x^{11/2}+\frac{6}{13} a^2 A c x^{13/2}+\frac{2}{5} a^2 B c x^{15/2}+\frac{6}{17} a A c^2 x^{17/2}+\frac{6}{19} a B c^2 x^{19/2}+\frac{2}{21} A c^3 x^{21/2}+\frac{2}{23} B c^3 x^{23/2}\\ \end{align*}
Mathematica [A] time = 0.0429678, size = 83, normalized size = 0.76 \[ \frac{2}{65} a^2 c x^{13/2} (15 A+13 B x)+\frac{2}{99} a^3 x^{9/2} (11 A+9 B x)+\frac{6}{323} a c^2 x^{17/2} (19 A+17 B x)+\frac{2}{483} c^3 x^{21/2} (23 A+21 B x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 78, normalized size = 0.7 \begin{align*}{\frac{29099070\,B{c}^{3}{x}^{7}+31870410\,A{c}^{3}{x}^{6}+105675570\,aB{c}^{2}{x}^{5}+118107990\,aA{c}^{2}{x}^{4}+133855722\,{a}^{2}Bc{x}^{3}+154448910\,{a}^{2}Ac{x}^{2}+60843510\,{a}^{3}Bx+74364290\,A{a}^{3}}{334639305}{x}^{{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02166, size = 104, normalized size = 0.95 \begin{align*} \frac{2}{23} \, B c^{3} x^{\frac{23}{2}} + \frac{2}{21} \, A c^{3} x^{\frac{21}{2}} + \frac{6}{19} \, B a c^{2} x^{\frac{19}{2}} + \frac{6}{17} \, A a c^{2} x^{\frac{17}{2}} + \frac{2}{5} \, B a^{2} c x^{\frac{15}{2}} + \frac{6}{13} \, A a^{2} c x^{\frac{13}{2}} + \frac{2}{11} \, B a^{3} x^{\frac{11}{2}} + \frac{2}{9} \, A a^{3} x^{\frac{9}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.2504, size = 269, normalized size = 2.47 \begin{align*} \frac{2}{334639305} \,{\left (14549535 \, B c^{3} x^{11} + 15935205 \, A c^{3} x^{10} + 52837785 \, B a c^{2} x^{9} + 59053995 \, A a c^{2} x^{8} + 66927861 \, B a^{2} c x^{7} + 77224455 \, A a^{2} c x^{6} + 30421755 \, B a^{3} x^{5} + 37182145 \, A a^{3} x^{4}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 44.6932, size = 114, normalized size = 1.05 \begin{align*} \frac{2 A a^{3} x^{\frac{9}{2}}}{9} + \frac{6 A a^{2} c x^{\frac{13}{2}}}{13} + \frac{6 A a c^{2} x^{\frac{17}{2}}}{17} + \frac{2 A c^{3} x^{\frac{21}{2}}}{21} + \frac{2 B a^{3} x^{\frac{11}{2}}}{11} + \frac{2 B a^{2} c x^{\frac{15}{2}}}{5} + \frac{6 B a c^{2} x^{\frac{19}{2}}}{19} + \frac{2 B c^{3} x^{\frac{23}{2}}}{23} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21594, size = 104, normalized size = 0.95 \begin{align*} \frac{2}{23} \, B c^{3} x^{\frac{23}{2}} + \frac{2}{21} \, A c^{3} x^{\frac{21}{2}} + \frac{6}{19} \, B a c^{2} x^{\frac{19}{2}} + \frac{6}{17} \, A a c^{2} x^{\frac{17}{2}} + \frac{2}{5} \, B a^{2} c x^{\frac{15}{2}} + \frac{6}{13} \, A a^{2} c x^{\frac{13}{2}} + \frac{2}{11} \, B a^{3} x^{\frac{11}{2}} + \frac{2}{9} \, A a^{3} x^{\frac{9}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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