Optimal. Leaf size=19 \[ \frac{2 \sqrt{x}}{a \sqrt{a+b x}} \]
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Rubi [A] time = 0.001838, antiderivative size = 19, 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} \[ \frac{2 \sqrt{x}}{a \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} (a+b x)^{3/2}} \, dx &=\frac{2 \sqrt{x}}{a \sqrt{a+b x}}\\ \end{align*}
Mathematica [A] time = 0.0044999, size = 19, normalized size = 1. \[ \frac{2 \sqrt{x}}{a \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 16, normalized size = 0.8 \begin{align*} 2\,{\frac{\sqrt{x}}{a\sqrt{bx+a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09507, size = 20, normalized size = 1.05 \begin{align*} \frac{2 \, \sqrt{x}}{\sqrt{b x + a} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81284, size = 53, normalized size = 2.79 \begin{align*} \frac{2 \, \sqrt{b x + a} \sqrt{x}}{a b x + a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.2184, size = 17, normalized size = 0.89 \begin{align*} \frac{2}{a \sqrt{b} \sqrt{\frac{a}{b x} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.05505, size = 61, normalized size = 3.21 \begin{align*} \frac{4 \, b^{\frac{3}{2}}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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