Optimal. Leaf size=55 \[ \frac{2}{7} x^{7/2} (a B+A b)+\frac{2}{5} a A x^{5/2}+\frac{2}{9} x^{9/2} (A c+b B)+\frac{2}{11} B c x^{11/2} \]
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Rubi [A] time = 0.0245567, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {765} \[ \frac{2}{7} x^{7/2} (a B+A b)+\frac{2}{5} a A x^{5/2}+\frac{2}{9} x^{9/2} (A c+b B)+\frac{2}{11} B c x^{11/2} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int x^{3/2} (A+B x) \left (a+b x+c x^2\right ) \, dx &=\int \left (a A x^{3/2}+(A b+a B) x^{5/2}+(b B+A c) x^{7/2}+B c x^{9/2}\right ) \, dx\\ &=\frac{2}{5} a A x^{5/2}+\frac{2}{7} (A b+a B) x^{7/2}+\frac{2}{9} (b B+A c) x^{9/2}+\frac{2}{11} B c x^{11/2}\\ \end{align*}
Mathematica [A] time = 0.0473992, size = 48, normalized size = 0.87 \[ \frac{2 x^{5/2} (99 a (7 A+5 B x)+5 x (11 A (9 b+7 c x)+7 B x (11 b+9 c x)))}{3465} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 42, normalized size = 0.8 \begin{align*}{\frac{630\,Bc{x}^{3}+770\,Ac{x}^{2}+770\,Bb{x}^{2}+990\,Abx+990\,aBx+1386\,aA}{3465}{x}^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08479, size = 53, normalized size = 0.96 \begin{align*} \frac{2}{11} \, B c x^{\frac{11}{2}} + \frac{2}{9} \,{\left (B b + A c\right )} x^{\frac{9}{2}} + \frac{2}{5} \, A a x^{\frac{5}{2}} + \frac{2}{7} \,{\left (B a + A b\right )} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28996, size = 119, normalized size = 2.16 \begin{align*} \frac{2}{3465} \,{\left (315 \, B c x^{5} + 385 \,{\left (B b + A c\right )} x^{4} + 693 \, A a x^{2} + 495 \,{\left (B a + A b\right )} x^{3}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.45297, size = 70, normalized size = 1.27 \begin{align*} \frac{2 A a x^{\frac{5}{2}}}{5} + \frac{2 A b x^{\frac{7}{2}}}{7} + \frac{2 A c x^{\frac{9}{2}}}{9} + \frac{2 B a x^{\frac{7}{2}}}{7} + \frac{2 B b x^{\frac{9}{2}}}{9} + \frac{2 B c x^{\frac{11}{2}}}{11} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19284, size = 58, normalized size = 1.05 \begin{align*} \frac{2}{11} \, B c x^{\frac{11}{2}} + \frac{2}{9} \, B b x^{\frac{9}{2}} + \frac{2}{9} \, A c x^{\frac{9}{2}} + \frac{2}{7} \, B a x^{\frac{7}{2}} + \frac{2}{7} \, A b x^{\frac{7}{2}} + \frac{2}{5} \, A a x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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