Optimal. Leaf size=36 \[ \frac{2}{5} a^2 x^{5/2}+\frac{4}{7} a b x^{7/2}+\frac{2}{9} b^2 x^{9/2} \]
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Rubi [A] time = 0.007302, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{2}{5} a^2 x^{5/2}+\frac{4}{7} a b x^{7/2}+\frac{2}{9} b^2 x^{9/2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int x^{3/2} (a+b x)^2 \, dx &=\int \left (a^2 x^{3/2}+2 a b x^{5/2}+b^2 x^{7/2}\right ) \, dx\\ &=\frac{2}{5} a^2 x^{5/2}+\frac{4}{7} a b x^{7/2}+\frac{2}{9} b^2 x^{9/2}\\ \end{align*}
Mathematica [A] time = 0.0077449, size = 28, normalized size = 0.78 \[ \frac{2}{315} x^{5/2} \left (63 a^2+90 a b x+35 b^2 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 25, normalized size = 0.7 \begin{align*}{\frac{70\,{b}^{2}{x}^{2}+180\,abx+126\,{a}^{2}}{315}{x}^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04069, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{9} \, b^{2} x^{\frac{9}{2}} + \frac{4}{7} \, a b x^{\frac{7}{2}} + \frac{2}{5} \, a^{2} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50846, size = 73, normalized size = 2.03 \begin{align*} \frac{2}{315} \,{\left (35 \, b^{2} x^{4} + 90 \, a b x^{3} + 63 \, a^{2} x^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.31535, size = 34, normalized size = 0.94 \begin{align*} \frac{2 a^{2} x^{\frac{5}{2}}}{5} + \frac{4 a b x^{\frac{7}{2}}}{7} + \frac{2 b^{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 = 1.22786, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{9} \, b^{2} x^{\frac{9}{2}} + \frac{4}{7} \, a b x^{\frac{7}{2}} + \frac{2}{5} \, a^{2} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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