Optimal. Leaf size=45 \[ \frac{2}{3} a A x^{3/2}+\frac{2}{5} a B x^{5/2}+\frac{2}{7} A c x^{7/2}+\frac{2}{9} B c x^{9/2} \]
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Rubi [A] time = 0.0123035, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {766} \[ \frac{2}{3} a A x^{3/2}+\frac{2}{5} a B x^{5/2}+\frac{2}{7} A c x^{7/2}+\frac{2}{9} B c x^{9/2} \]
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin{align*} \int \sqrt{x} (A+B x) \left (a+c x^2\right ) \, dx &=\int \left (a A \sqrt{x}+a B x^{3/2}+A c x^{5/2}+B c x^{7/2}\right ) \, dx\\ &=\frac{2}{3} a A x^{3/2}+\frac{2}{5} a B x^{5/2}+\frac{2}{7} A c x^{7/2}+\frac{2}{9} B c x^{9/2}\\ \end{align*}
Mathematica [A] time = 0.0123757, size = 35, normalized size = 0.78 \[ \frac{2}{315} x^{3/2} \left (21 a (5 A+3 B x)+5 c x^2 (9 A+7 B x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 30, normalized size = 0.7 \begin{align*}{\frac{70\,Bc{x}^{3}+90\,Ac{x}^{2}+126\,aBx+210\,aA}{315}{x}^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98034, size = 39, normalized size = 0.87 \begin{align*} \frac{2}{9} \, B c x^{\frac{9}{2}} + \frac{2}{7} \, A c x^{\frac{7}{2}} + \frac{2}{5} \, B a x^{\frac{5}{2}} + \frac{2}{3} \, A a x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33899, size = 89, normalized size = 1.98 \begin{align*} \frac{2}{315} \,{\left (35 \, B c x^{4} + 45 \, A c x^{3} + 63 \, B a x^{2} + 105 \, A a x\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.15829, size = 46, normalized size = 1.02 \begin{align*} \frac{2 A a x^{\frac{3}{2}}}{3} + \frac{2 A c x^{\frac{7}{2}}}{7} + \frac{2 B a x^{\frac{5}{2}}}{5} + \frac{2 B c x^{\frac{9}{2}}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16384, size = 39, normalized size = 0.87 \begin{align*} \frac{2}{9} \, B c x^{\frac{9}{2}} + \frac{2}{7} \, A c x^{\frac{7}{2}} + \frac{2}{5} \, B a x^{\frac{5}{2}} + \frac{2}{3} \, A a x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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