Optimal. Leaf size=43 \[ 2 a A \sqrt{x}+\frac{2}{3} a B x^{3/2}+\frac{2}{5} A c x^{5/2}+\frac{2}{7} B c x^{7/2} \]
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Rubi [A] time = 0.0121786, antiderivative size = 43, 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} \[ 2 a A \sqrt{x}+\frac{2}{3} a B x^{3/2}+\frac{2}{5} A c x^{5/2}+\frac{2}{7} B c x^{7/2} \]
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )}{\sqrt{x}} \, dx &=\int \left (\frac{a A}{\sqrt{x}}+a B \sqrt{x}+A c x^{3/2}+B c x^{5/2}\right ) \, dx\\ &=2 a A \sqrt{x}+\frac{2}{3} a B x^{3/2}+\frac{2}{5} A c x^{5/2}+\frac{2}{7} B c x^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0127366, size = 34, normalized size = 0.79 \[ \frac{2}{105} \sqrt{x} \left (35 a (3 A+B x)+3 c x^2 (7 A+5 B x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 30, normalized size = 0.7 \begin{align*}{\frac{30\,Bc{x}^{3}+42\,Ac{x}^{2}+70\,aBx+210\,aA}{105}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983477, size = 39, normalized size = 0.91 \begin{align*} \frac{2}{7} \, B c x^{\frac{7}{2}} + \frac{2}{5} \, A c x^{\frac{5}{2}} + \frac{2}{3} \, B a x^{\frac{3}{2}} + 2 \, A a \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27349, size = 84, normalized size = 1.95 \begin{align*} \frac{2}{105} \,{\left (15 \, B c x^{3} + 21 \, A c x^{2} + 35 \, B a x + 105 \, A a\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.695626, size = 44, normalized size = 1.02 \begin{align*} 2 A a \sqrt{x} + \frac{2 A c x^{\frac{5}{2}}}{5} + \frac{2 B a x^{\frac{3}{2}}}{3} + \frac{2 B c x^{\frac{7}{2}}}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16405, size = 39, normalized size = 0.91 \begin{align*} \frac{2}{7} \, B c x^{\frac{7}{2}} + \frac{2}{5} \, A c x^{\frac{5}{2}} + \frac{2}{3} \, B a x^{\frac{3}{2}} + 2 \, A a \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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