Optimal. Leaf size=36 \[ \frac{2}{7} a^2 x^{7/2}+\frac{4}{9} a b x^{9/2}+\frac{2}{11} b^2 x^{11/2} \]
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Rubi [A] time = 0.0071863, 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} \[ \frac{2}{7} a^2 x^{7/2}+\frac{4}{9} a b x^{9/2}+\frac{2}{11} b^2 x^{11/2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int x^{5/2} (a+b x)^2 \, dx &=\int \left (a^2 x^{5/2}+2 a b x^{7/2}+b^2 x^{9/2}\right ) \, dx\\ &=\frac{2}{7} a^2 x^{7/2}+\frac{4}{9} a b x^{9/2}+\frac{2}{11} b^2 x^{11/2}\\ \end{align*}
Mathematica [A] time = 0.0077778, size = 28, normalized size = 0.78 \[ \frac{2}{693} x^{7/2} \left (99 a^2+154 a b x+63 b^2 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 0.7 \begin{align*}{\frac{126\,{b}^{2}{x}^{2}+308\,abx+198\,{a}^{2}}{693}{x}^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991102, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{11} \, b^{2} x^{\frac{11}{2}} + \frac{4}{9} \, a b x^{\frac{9}{2}} + \frac{2}{7} \, a^{2} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44803, size = 74, normalized size = 2.06 \begin{align*} \frac{2}{693} \,{\left (63 \, b^{2} x^{5} + 154 \, a b x^{4} + 99 \, a^{2} x^{3}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.74924, size = 34, normalized size = 0.94 \begin{align*} \frac{2 a^{2} x^{\frac{7}{2}}}{7} + \frac{4 a b x^{\frac{9}{2}}}{9} + \frac{2 b^{2} x^{\frac{11}{2}}}{11} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2254, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{11} \, b^{2} x^{\frac{11}{2}} + \frac{4}{9} \, a b x^{\frac{9}{2}} + \frac{2}{7} \, a^{2} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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