Optimal. Leaf size=37 \[ \frac{\sqrt{x}}{3 \sqrt{b x+2}}+\frac{\sqrt{x}}{3 (b x+2)^{3/2}} \]
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Rubi [A] time = 0.0034164, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ \frac{\sqrt{x}}{3 \sqrt{b x+2}}+\frac{\sqrt{x}}{3 (b x+2)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} (2+b x)^{5/2}} \, dx &=\frac{\sqrt{x}}{3 (2+b x)^{3/2}}+\frac{1}{3} \int \frac{1}{\sqrt{x} (2+b x)^{3/2}} \, dx\\ &=\frac{\sqrt{x}}{3 (2+b x)^{3/2}}+\frac{\sqrt{x}}{3 \sqrt{2+b x}}\\ \end{align*}
Mathematica [A] time = 0.0061133, size = 23, normalized size = 0.62 \[ \frac{\sqrt{x} (b x+3)}{3 (b x+2)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 18, normalized size = 0.5 \begin{align*}{\frac{bx+3}{3}\sqrt{x} \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 = 1.00239, size = 32, normalized size = 0.86 \begin{align*} -\frac{{\left (b - \frac{3 \,{\left (b x + 2\right )}}{x}\right )} x^{\frac{3}{2}}}{6 \,{\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 [A] time = 1.4751, size = 80, normalized size = 2.16 \begin{align*} \frac{{\left (b x + 3\right )} \sqrt{b x + 2} \sqrt{x}}{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 [B] time = 3.98838, size = 75, normalized size = 2.03 \begin{align*} \frac{b x}{3 b^{\frac{3}{2}} x \sqrt{1 + \frac{2}{b x}} + 6 \sqrt{b} \sqrt{1 + \frac{2}{b x}}} + \frac{3}{3 b^{\frac{3}{2}} x \sqrt{1 + \frac{2}{b x}} + 6 \sqrt{b} \sqrt{1 + \frac{2}{b x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.09351, size = 107, normalized size = 2.89 \begin{align*} \frac{8 \,{\left (3 \,{\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )} b^{\frac{5}{2}}}{3 \,{\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}^{3}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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