Optimal. Leaf size=45 \[ 6 a^2 b \sqrt{x}-\frac{2 a^3}{\sqrt{x}}+2 a b^2 x^{3/2}+\frac{2}{5} b^3 x^{5/2} \]
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Rubi [A] time = 0.0111342, antiderivative size = 45, 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} \[ 6 a^2 b \sqrt{x}-\frac{2 a^3}{\sqrt{x}}+2 a b^2 x^{3/2}+\frac{2}{5} b^3 x^{5/2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^3}{x^{3/2}} \, dx &=\int \left (\frac{a^3}{x^{3/2}}+\frac{3 a^2 b}{\sqrt{x}}+3 a b^2 \sqrt{x}+b^3 x^{3/2}\right ) \, dx\\ &=-\frac{2 a^3}{\sqrt{x}}+6 a^2 b \sqrt{x}+2 a b^2 x^{3/2}+\frac{2}{5} b^3 x^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0114963, size = 38, normalized size = 0.84 \[ \frac{2 \left (15 a^2 b x-5 a^3+5 a b^2 x^2+b^3 x^3\right )}{5 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 36, normalized size = 0.8 \begin{align*} -{\frac{-2\,{b}^{3}{x}^{3}-10\,a{b}^{2}{x}^{2}-30\,{a}^{2}bx+10\,{a}^{3}}{5}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09566, size = 47, normalized size = 1.04 \begin{align*} \frac{2}{5} \, b^{3} x^{\frac{5}{2}} + 2 \, a b^{2} x^{\frac{3}{2}} + 6 \, a^{2} b \sqrt{x} - \frac{2 \, a^{3}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51226, size = 78, normalized size = 1.73 \begin{align*} \frac{2 \,{\left (b^{3} x^{3} + 5 \, a b^{2} x^{2} + 15 \, a^{2} b x - 5 \, a^{3}\right )}}{5 \, \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.897178, size = 44, normalized size = 0.98 \begin{align*} - \frac{2 a^{3}}{\sqrt{x}} + 6 a^{2} b \sqrt{x} + 2 a b^{2} x^{\frac{3}{2}} + \frac{2 b^{3} x^{\frac{5}{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22818, size = 47, normalized size = 1.04 \begin{align*} \frac{2}{5} \, b^{3} x^{\frac{5}{2}} + 2 \, a b^{2} x^{\frac{3}{2}} + 6 \, a^{2} b \sqrt{x} - \frac{2 \, a^{3}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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