Optimal. Leaf size=51 \[ \frac{2}{3} a^2 b x^{9/2}+\frac{2}{7} a^3 x^{7/2}+\frac{6}{11} a b^2 x^{11/2}+\frac{2}{13} b^3 x^{13/2} \]
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Rubi [A] time = 0.0106187, antiderivative size = 51, 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} \[ \frac{2}{3} a^2 b x^{9/2}+\frac{2}{7} a^3 x^{7/2}+\frac{6}{11} a b^2 x^{11/2}+\frac{2}{13} b^3 x^{13/2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int x^{5/2} (a+b x)^3 \, dx &=\int \left (a^3 x^{5/2}+3 a^2 b x^{7/2}+3 a b^2 x^{9/2}+b^3 x^{11/2}\right ) \, dx\\ &=\frac{2}{7} a^3 x^{7/2}+\frac{2}{3} a^2 b x^{9/2}+\frac{6}{11} a b^2 x^{11/2}+\frac{2}{13} b^3 x^{13/2}\\ \end{align*}
Mathematica [A] time = 0.0112686, size = 39, normalized size = 0.76 \[ \frac{2 x^{7/2} \left (1001 a^2 b x+429 a^3+819 a b^2 x^2+231 b^3 x^3\right )}{3003} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 36, normalized size = 0.7 \begin{align*}{\frac{462\,{b}^{3}{x}^{3}+1638\,a{b}^{2}{x}^{2}+2002\,{a}^{2}bx+858\,{a}^{3}}{3003}{x}^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07651, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{13} \, b^{3} x^{\frac{13}{2}} + \frac{6}{11} \, a b^{2} x^{\frac{11}{2}} + \frac{2}{3} \, a^{2} b x^{\frac{9}{2}} + \frac{2}{7} \, a^{3} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53442, size = 104, normalized size = 2.04 \begin{align*} \frac{2}{3003} \,{\left (231 \, b^{3} x^{6} + 819 \, a b^{2} x^{5} + 1001 \, a^{2} b x^{4} + 429 \, a^{3} x^{3}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.68782, size = 49, normalized size = 0.96 \begin{align*} \frac{2 a^{3} x^{\frac{7}{2}}}{7} + \frac{2 a^{2} b x^{\frac{9}{2}}}{3} + \frac{6 a b^{2} x^{\frac{11}{2}}}{11} + \frac{2 b^{3} x^{\frac{13}{2}}}{13} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28215, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{13} \, b^{3} x^{\frac{13}{2}} + \frac{6}{11} \, a b^{2} x^{\frac{11}{2}} + \frac{2}{3} \, a^{2} b x^{\frac{9}{2}} + \frac{2}{7} \, a^{3} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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