Optimal. Leaf size=21 \[ -\frac{2 \sqrt{a+b x}}{a \sqrt{c x}} \]
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Rubi [A] time = 0.0047832, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {16, 37} \[ -\frac{2 \sqrt{a+b x}}{a \sqrt{c x}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{c x} \sqrt{a+b x}} \, dx &=c \int \frac{1}{(c x)^{3/2} \sqrt{a+b x}} \, dx\\ &=-\frac{2 \sqrt{a+b x}}{a \sqrt{c x}}\\ \end{align*}
Mathematica [A] time = 0.0092778, size = 23, normalized size = 1.1 \[ -\frac{2 c x \sqrt{a+b x}}{a (c x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 18, normalized size = 0.9 \begin{align*} -2\,{\frac{\sqrt{bx+a}}{a\sqrt{cx}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95213, size = 49, normalized size = 2.33 \begin{align*} -\frac{2 \, \sqrt{b x + a} \sqrt{c x}}{a c x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.71165, size = 24, normalized size = 1.14 \begin{align*} - \frac{2 \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{a \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.38457, size = 47, normalized size = 2.24 \begin{align*} -\frac{2 \, \sqrt{b x + a} b^{2}}{\sqrt{{\left (b x + a\right )} b c - a b c} a{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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