Optimal. Leaf size=18 \[ \frac{x^{3/2}}{3 (b x+2)^{3/2}} \]
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Rubi [A] time = 0.0012821, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {37} \[ \frac{x^{3/2}}{3 (b x+2)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int \frac{\sqrt{x}}{(2+b x)^{5/2}} \, dx &=\frac{x^{3/2}}{3 (2+b x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0042477, size = 18, normalized size = 1. \[ \frac{x^{3/2}}{3 (b x+2)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 13, normalized size = 0.7 \begin{align*}{\frac{1}{3}{x}^{{\frac{3}{2}}} \left ( bx+2 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01698, size = 16, normalized size = 0.89 \begin{align*} \frac{x^{\frac{3}{2}}}{3 \,{\left (b x + 2\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.52731, size = 66, normalized size = 3.67 \begin{align*} \frac{\sqrt{b x + 2} x^{\frac{3}{2}}}{3 \,{\left (b^{2} x^{2} + 4 \, b x + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.27685, size = 27, normalized size = 1.5 \begin{align*} \frac{x^{\frac{3}{2}}}{3 b x \sqrt{b x + 2} + 6 \sqrt{b x + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10089, size = 111, normalized size = 6.17 \begin{align*} \frac{4 \,{\left (3 \,{\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{4} \sqrt{b} + 4 \, b^{\frac{5}{2}}\right )}{\left | b \right |}}{3 \,{\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}^{3} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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