Optimal. Leaf size=109 \[ \frac{6}{11} a^2 A c x^{11/2}+\frac{2}{7} a^3 A x^{7/2}+\frac{6}{13} a^2 B c x^{13/2}+\frac{2}{9} a^3 B x^{9/2}+\frac{2}{5} a A c^2 x^{15/2}+\frac{6}{17} a B c^2 x^{17/2}+\frac{2}{19} A c^3 x^{19/2}+\frac{2}{21} B c^3 x^{21/2} \]
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Rubi [A] time = 0.0396323, 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} \[ \frac{6}{11} a^2 A c x^{11/2}+\frac{2}{7} a^3 A x^{7/2}+\frac{6}{13} a^2 B c x^{13/2}+\frac{2}{9} a^3 B x^{9/2}+\frac{2}{5} a A c^2 x^{15/2}+\frac{6}{17} a B c^2 x^{17/2}+\frac{2}{19} A c^3 x^{19/2}+\frac{2}{21} B c^3 x^{21/2} \]
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 )^3 \, dx &=\int \left (a^3 A x^{5/2}+a^3 B x^{7/2}+3 a^2 A c x^{9/2}+3 a^2 B c x^{11/2}+3 a A c^2 x^{13/2}+3 a B c^2 x^{15/2}+A c^3 x^{17/2}+B c^3 x^{19/2}\right ) \, dx\\ &=\frac{2}{7} a^3 A x^{7/2}+\frac{2}{9} a^3 B x^{9/2}+\frac{6}{11} a^2 A c x^{11/2}+\frac{6}{13} a^2 B c x^{13/2}+\frac{2}{5} a A c^2 x^{15/2}+\frac{6}{17} a B c^2 x^{17/2}+\frac{2}{19} A c^3 x^{19/2}+\frac{2}{21} B c^3 x^{21/2}\\ \end{align*}
Mathematica [A] time = 0.0389749, size = 83, normalized size = 0.76 \[ \frac{6}{143} a^2 c x^{11/2} (13 A+11 B x)+\frac{2}{63} a^3 x^{7/2} (9 A+7 B x)+\frac{2}{85} a c^2 x^{15/2} (17 A+15 B x)+\frac{2}{399} c^3 x^{19/2} (21 A+19 B x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 78, normalized size = 0.7 \begin{align*}{\frac{1385670\,B{c}^{3}{x}^{7}+1531530\,A{c}^{3}{x}^{6}+5135130\,aB{c}^{2}{x}^{5}+5819814\,aA{c}^{2}{x}^{4}+6715170\,{a}^{2}Bc{x}^{3}+7936110\,{a}^{2}Ac{x}^{2}+3233230\,{a}^{3}Bx+4157010\,A{a}^{3}}{14549535}{x}^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0363, size = 104, normalized size = 0.95 \begin{align*} \frac{2}{21} \, B c^{3} x^{\frac{21}{2}} + \frac{2}{19} \, A c^{3} x^{\frac{19}{2}} + \frac{6}{17} \, B a c^{2} x^{\frac{17}{2}} + \frac{2}{5} \, A a c^{2} x^{\frac{15}{2}} + \frac{6}{13} \, B a^{2} c x^{\frac{13}{2}} + \frac{6}{11} \, A a^{2} c x^{\frac{11}{2}} + \frac{2}{9} \, B a^{3} x^{\frac{9}{2}} + \frac{2}{7} \, A a^{3} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28519, size = 252, normalized size = 2.31 \begin{align*} \frac{2}{14549535} \,{\left (692835 \, B c^{3} x^{10} + 765765 \, A c^{3} x^{9} + 2567565 \, B a c^{2} x^{8} + 2909907 \, A a c^{2} x^{7} + 3357585 \, B a^{2} c x^{6} + 3968055 \, A a^{2} c x^{5} + 1616615 \, B a^{3} x^{4} + 2078505 \, A a^{3} x^{3}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 18.5394, size = 114, normalized size = 1.05 \begin{align*} \frac{2 A a^{3} x^{\frac{7}{2}}}{7} + \frac{6 A a^{2} c x^{\frac{11}{2}}}{11} + \frac{2 A a c^{2} x^{\frac{15}{2}}}{5} + \frac{2 A c^{3} x^{\frac{19}{2}}}{19} + \frac{2 B a^{3} x^{\frac{9}{2}}}{9} + \frac{6 B a^{2} c x^{\frac{13}{2}}}{13} + \frac{6 B a c^{2} x^{\frac{17}{2}}}{17} + \frac{2 B c^{3} x^{\frac{21}{2}}}{21} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18619, size = 104, normalized size = 0.95 \begin{align*} \frac{2}{21} \, B c^{3} x^{\frac{21}{2}} + \frac{2}{19} \, A c^{3} x^{\frac{19}{2}} + \frac{6}{17} \, B a c^{2} x^{\frac{17}{2}} + \frac{2}{5} \, A a c^{2} x^{\frac{15}{2}} + \frac{6}{13} \, B a^{2} c x^{\frac{13}{2}} + \frac{6}{11} \, A a^{2} c x^{\frac{11}{2}} + \frac{2}{9} \, B a^{3} x^{\frac{9}{2}} + \frac{2}{7} \, A a^{3} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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