Optimal. Leaf size=37 \[ \frac{2}{3} x^{3/2} (A c+b B)+2 A b \sqrt{x}+\frac{2}{5} B c x^{5/2} \]
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Rubi [A] time = 0.0149419, antiderivative size = 37, 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} \[ \frac{2}{3} x^{3/2} (A c+b B)+2 A b \sqrt{x}+\frac{2}{5} B c x^{5/2} \]
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 )}{x^{3/2}} \, dx &=\int \left (\frac{A b}{\sqrt{x}}+(b B+A c) \sqrt{x}+B c x^{3/2}\right ) \, dx\\ &=2 A b \sqrt{x}+\frac{2}{3} (b B+A c) x^{3/2}+\frac{2}{5} B c x^{5/2}\\ \end{align*}
Mathematica [A] time = 0.010796, size = 31, normalized size = 0.84 \[ \frac{2}{15} \sqrt{x} (5 A (3 b+c x)+B x (5 b+3 c x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 28, normalized size = 0.8 \begin{align*}{\frac{6\,Bc{x}^{2}+10\,Acx+10\,bBx+30\,Ab}{15}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02592, size = 36, normalized size = 0.97 \begin{align*} \frac{2}{5} \, B c x^{\frac{5}{2}} + 2 \, A b \sqrt{x} + \frac{2}{3} \,{\left (B b + A c\right )} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83261, size = 72, normalized size = 1.95 \begin{align*} \frac{2}{15} \,{\left (3 \, B c x^{2} + 15 \, A b + 5 \,{\left (B b + A c\right )} x\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.637288, size = 44, normalized size = 1.19 \begin{align*} 2 A b \sqrt{x} + \frac{2 A c x^{\frac{3}{2}}}{3} + \frac{2 B b x^{\frac{3}{2}}}{3} + \frac{2 B c x^{\frac{5}{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12281, size = 39, normalized size = 1.05 \begin{align*} \frac{2}{5} \, B c x^{\frac{5}{2}} + \frac{2}{3} \, B b x^{\frac{3}{2}} + \frac{2}{3} \, A c x^{\frac{3}{2}} + 2 \, A b \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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