Optimal. Leaf size=39 \[ \frac{2}{5} x^{5/2} (A c+b B)+\frac{2}{3} A b x^{3/2}+\frac{2}{7} B c x^{7/2} \]
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Rubi [A] time = 0.0156641, antiderivative size = 39, 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 = {765} \[ \frac{2}{5} x^{5/2} (A c+b B)+\frac{2}{3} A b x^{3/2}+\frac{2}{7} B c x^{7/2} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )}{\sqrt{x}} \, dx &=\int \left (A b \sqrt{x}+(b B+A c) x^{3/2}+B c x^{5/2}\right ) \, dx\\ &=\frac{2}{3} A b x^{3/2}+\frac{2}{5} (b B+A c) x^{5/2}+\frac{2}{7} B c x^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0106106, size = 33, normalized size = 0.85 \[ \frac{2}{105} x^{3/2} (7 A (5 b+3 c x)+3 B x (7 b+5 c x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 28, normalized size = 0.7 \begin{align*}{\frac{30\,Bc{x}^{2}+42\,Acx+42\,bBx+70\,Ab}{105}{x}^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.996198, size = 36, normalized size = 0.92 \begin{align*} \frac{2}{7} \, B c x^{\frac{7}{2}} + \frac{2}{3} \, A b x^{\frac{3}{2}} + \frac{2}{5} \,{\left (B b + A c\right )} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86162, size = 81, normalized size = 2.08 \begin{align*} \frac{2}{105} \,{\left (15 \, B c x^{3} + 35 \, A b x + 21 \,{\left (B b + A c\right )} x^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.466973, size = 46, normalized size = 1.18 \begin{align*} \frac{2 A b x^{\frac{3}{2}}}{3} + \frac{2 A c x^{\frac{5}{2}}}{5} + \frac{2 B b x^{\frac{5}{2}}}{5} + \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 = 1.0977, size = 39, normalized size = 1. \begin{align*} \frac{2}{7} \, B c x^{\frac{7}{2}} + \frac{2}{5} \, B b x^{\frac{5}{2}} + \frac{2}{5} \, A c x^{\frac{5}{2}} + \frac{2}{3} \, A b x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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