Optimal. Leaf size=63 \[ \frac{2}{3} A b^2 x^{3/2}+\frac{2}{7} c x^{7/2} (A c+2 b B)+\frac{2}{5} b x^{5/2} (2 A c+b B)+\frac{2}{9} B c^2 x^{9/2} \]
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Rubi [A] time = 0.0312946, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {765} \[ \frac{2}{3} A b^2 x^{3/2}+\frac{2}{7} c x^{7/2} (A c+2 b B)+\frac{2}{5} b x^{5/2} (2 A c+b B)+\frac{2}{9} B c^2 x^{9/2} \]
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 )^2}{x^{3/2}} \, dx &=\int \left (A b^2 \sqrt{x}+b (b B+2 A c) x^{3/2}+c (2 b B+A c) x^{5/2}+B c^2 x^{7/2}\right ) \, dx\\ &=\frac{2}{3} A b^2 x^{3/2}+\frac{2}{5} b (b B+2 A c) x^{5/2}+\frac{2}{7} c (2 b B+A c) x^{7/2}+\frac{2}{9} B c^2 x^{9/2}\\ \end{align*}
Mathematica [A] time = 0.0166943, size = 54, normalized size = 0.86 \[ \frac{2}{315} x^{3/2} \left (3 A \left (35 b^2+42 b c x+15 c^2 x^2\right )+B x \left (63 b^2+90 b c x+35 c^2 x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 52, normalized size = 0.8 \begin{align*}{\frac{70\,B{c}^{2}{x}^{3}+90\,A{c}^{2}{x}^{2}+180\,B{x}^{2}bc+252\,Abcx+126\,{b}^{2}Bx+210\,A{b}^{2}}{315}{x}^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05357, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{9} \, B c^{2} x^{\frac{9}{2}} + \frac{2}{3} \, A b^{2} x^{\frac{3}{2}} + \frac{2}{7} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{7}{2}} + \frac{2}{5} \,{\left (B b^{2} + 2 \, A b 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 = 1.85461, size = 132, normalized size = 2.1 \begin{align*} \frac{2}{315} \,{\left (35 \, B c^{2} x^{4} + 105 \, A b^{2} x + 45 \,{\left (2 \, B b c + A c^{2}\right )} x^{3} + 63 \,{\left (B b^{2} + 2 \, A b 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 = 1.66067, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A b^{2} x^{\frac{3}{2}}}{3} + \frac{4 A b c x^{\frac{5}{2}}}{5} + \frac{2 A c^{2} x^{\frac{7}{2}}}{7} + \frac{2 B b^{2} x^{\frac{5}{2}}}{5} + \frac{4 B b c x^{\frac{7}{2}}}{7} + \frac{2 B c^{2} x^{\frac{9}{2}}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10609, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{9} \, B c^{2} x^{\frac{9}{2}} + \frac{4}{7} \, B b c x^{\frac{7}{2}} + \frac{2}{7} \, A c^{2} x^{\frac{7}{2}} + \frac{2}{5} \, B b^{2} x^{\frac{5}{2}} + \frac{4}{5} \, A b c x^{\frac{5}{2}} + \frac{2}{3} \, A b^{2} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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