Optimal. Leaf size=51 \[ \frac{2}{3} a^2 b x^{9/2}+\frac{2}{5} a^3 x^{5/2}+\frac{6}{13} a b^2 x^{13/2}+\frac{2}{17} b^3 x^{17/2} \]
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Rubi [A] time = 0.0125412, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {270} \[ \frac{2}{3} a^2 b x^{9/2}+\frac{2}{5} a^3 x^{5/2}+\frac{6}{13} a b^2 x^{13/2}+\frac{2}{17} b^3 x^{17/2} \]
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin{align*} \int x^{3/2} \left (a+b x^2\right )^3 \, dx &=\int \left (a^3 x^{3/2}+3 a^2 b x^{7/2}+3 a b^2 x^{11/2}+b^3 x^{15/2}\right ) \, dx\\ &=\frac{2}{5} a^3 x^{5/2}+\frac{2}{3} a^2 b x^{9/2}+\frac{6}{13} a b^2 x^{13/2}+\frac{2}{17} b^3 x^{17/2}\\ \end{align*}
Mathematica [A] time = 0.0096036, size = 41, normalized size = 0.8 \[ \frac{2 x^{5/2} \left (1105 a^2 b x^2+663 a^3+765 a b^2 x^4+195 b^3 x^6\right )}{3315} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 38, normalized size = 0.8 \begin{align*}{\frac{390\,{b}^{3}{x}^{6}+1530\,a{b}^{2}{x}^{4}+2210\,{a}^{2}b{x}^{2}+1326\,{a}^{3}}{3315}{x}^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.17352, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{17} \, b^{3} x^{\frac{17}{2}} + \frac{6}{13} \, a b^{2} x^{\frac{13}{2}} + \frac{2}{3} \, a^{2} b x^{\frac{9}{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 = 1.30844, size = 104, normalized size = 2.04 \begin{align*} \frac{2}{3315} \,{\left (195 \, b^{3} x^{8} + 765 \, a b^{2} x^{6} + 1105 \, a^{2} b x^{4} + 663 \, 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 = 5.72263, size = 49, normalized size = 0.96 \begin{align*} \frac{2 a^{3} x^{\frac{5}{2}}}{5} + \frac{2 a^{2} b x^{\frac{9}{2}}}{3} + \frac{6 a b^{2} x^{\frac{13}{2}}}{13} + \frac{2 b^{3} x^{\frac{17}{2}}}{17} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.25685, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{17} \, b^{3} x^{\frac{17}{2}} + \frac{6}{13} \, a b^{2} x^{\frac{13}{2}} + \frac{2}{3} \, a^{2} b x^{\frac{9}{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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