Optimal. Leaf size=21 \[ -\frac{2 \sqrt{b x+c x^2}}{b x} \]
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Rubi [A] time = 0.0073153, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {650} \[ -\frac{2 \sqrt{b x+c x^2}}{b x} \]
Antiderivative was successfully verified.
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Rule 650
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{b x+c x^2}} \, dx &=-\frac{2 \sqrt{b x+c x^2}}{b x}\\ \end{align*}
Mathematica [A] time = 0.006572, size = 21, normalized size = 1. \[ -\frac{2 (b+c x)}{b \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 22, normalized size = 1.1 \begin{align*} -2\,{\frac{cx+b}{b\sqrt{c{x}^{2}+bx}}} \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 = 1.7453, size = 38, normalized size = 1.81 \begin{align*} -\frac{2 \, \sqrt{c x^{2} + b x}}{b x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{x \left (b + c x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19682, size = 31, normalized size = 1.48 \begin{align*} \frac{2}{\sqrt{c} x - \sqrt{c x^{2} + b x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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