Optimal. Leaf size=51 \[ \frac{6}{7} a^2 b x^{7/2}+\frac{2}{5} a^3 x^{5/2}+\frac{2}{3} a b^2 x^{9/2}+\frac{2}{11} b^3 x^{11/2} \]
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Rubi [A] time = 0.0115775, 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} \[ \frac{6}{7} a^2 b x^{7/2}+\frac{2}{5} a^3 x^{5/2}+\frac{2}{3} a b^2 x^{9/2}+\frac{2}{11} b^3 x^{11/2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int x^{3/2} (a+b x)^3 \, dx &=\int \left (a^3 x^{3/2}+3 a^2 b x^{5/2}+3 a b^2 x^{7/2}+b^3 x^{9/2}\right ) \, dx\\ &=\frac{2}{5} a^3 x^{5/2}+\frac{6}{7} a^2 b x^{7/2}+\frac{2}{3} a b^2 x^{9/2}+\frac{2}{11} b^3 x^{11/2}\\ \end{align*}
Mathematica [A] time = 0.0099204, size = 39, normalized size = 0.76 \[ \frac{2 x^{5/2} \left (495 a^2 b x+231 a^3+385 a b^2 x^2+105 b^3 x^3\right )}{1155} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 36, normalized size = 0.7 \begin{align*}{\frac{210\,{b}^{3}{x}^{3}+770\,a{b}^{2}{x}^{2}+990\,{a}^{2}bx+462\,{a}^{3}}{1155}{x}^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04363, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{11} \, b^{3} x^{\frac{11}{2}} + \frac{2}{3} \, a b^{2} x^{\frac{9}{2}} + \frac{6}{7} \, a^{2} b x^{\frac{7}{2}} + \frac{2}{5} \, a^{3} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64924, size = 103, normalized size = 2.02 \begin{align*} \frac{2}{1155} \,{\left (105 \, b^{3} x^{5} + 385 \, a b^{2} x^{4} + 495 \, a^{2} b x^{3} + 231 \, a^{3} x^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.27367, size = 49, normalized size = 0.96 \begin{align*} \frac{2 a^{3} x^{\frac{5}{2}}}{5} + \frac{6 a^{2} b x^{\frac{7}{2}}}{7} + \frac{2 a b^{2} x^{\frac{9}{2}}}{3} + \frac{2 b^{3} x^{\frac{11}{2}}}{11} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21508, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{11} \, b^{3} x^{\frac{11}{2}} + \frac{2}{3} \, a b^{2} x^{\frac{9}{2}} + \frac{6}{7} \, a^{2} b x^{\frac{7}{2}} + \frac{2}{5} \, a^{3} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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