Optimal. Leaf size=47 \[ \frac{6}{5} a^2 b x^{5/2}+\frac{2}{7} a^3 x^{7/2}+2 a b^2 x^{3/2}+2 b^3 \sqrt{x} \]
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Rubi [A] time = 0.0127937, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {263, 43} \[ \frac{6}{5} a^2 b x^{5/2}+\frac{2}{7} a^3 x^{7/2}+2 a b^2 x^{3/2}+2 b^3 \sqrt{x} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x}\right )^3 x^{5/2} \, dx &=\int \frac{(b+a x)^3}{\sqrt{x}} \, dx\\ &=\int \left (\frac{b^3}{\sqrt{x}}+3 a b^2 \sqrt{x}+3 a^2 b x^{3/2}+a^3 x^{5/2}\right ) \, dx\\ &=2 b^3 \sqrt{x}+2 a b^2 x^{3/2}+\frac{6}{5} a^2 b x^{5/2}+\frac{2}{7} a^3 x^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0108839, size = 39, normalized size = 0.83 \[ \frac{2}{35} \sqrt{x} \left (21 a^2 b x^2+5 a^3 x^3+35 a b^2 x+35 b^3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 36, normalized size = 0.8 \begin{align*}{\frac{10\,{a}^{3}{x}^{3}+42\,{a}^{2}b{x}^{2}+70\,xa{b}^{2}+70\,{b}^{3}}{35}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.975012, size = 50, normalized size = 1.06 \begin{align*} \frac{2}{35} \,{\left (5 \, a^{3} + \frac{21 \, a^{2} b}{x} + \frac{35 \, a b^{2}}{x^{2}} + \frac{35 \, b^{3}}{x^{3}}\right )} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7565, size = 85, normalized size = 1.81 \begin{align*} \frac{2}{35} \,{\left (5 \, a^{3} x^{3} + 21 \, a^{2} b x^{2} + 35 \, a b^{2} x + 35 \, b^{3}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.68115, size = 46, normalized size = 0.98 \begin{align*} \frac{2 a^{3} x^{\frac{7}{2}}}{7} + \frac{6 a^{2} b x^{\frac{5}{2}}}{5} + 2 a b^{2} x^{\frac{3}{2}} + 2 b^{3} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13408, size = 47, normalized size = 1. \begin{align*} \frac{2}{7} \, a^{3} x^{\frac{7}{2}} + \frac{6}{5} \, a^{2} b x^{\frac{5}{2}} + 2 \, a b^{2} x^{\frac{3}{2}} + 2 \, b^{3} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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