Optimal. Leaf size=24 \[ -\frac{\sqrt{c x^2}}{b x (a+b x)} \]
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Rubi [A] time = 0.0036378, antiderivative size = 24, 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 = {15, 32} \[ -\frac{\sqrt{c x^2}}{b x (a+b x)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 32
Rubi steps
\begin{align*} \int \frac{\sqrt{c x^2}}{x (a+b x)^2} \, dx &=\frac{\sqrt{c x^2} \int \frac{1}{(a+b x)^2} \, dx}{x}\\ &=-\frac{\sqrt{c x^2}}{b x (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0061281, size = 23, normalized size = 0.96 \[ -\frac{c x}{b \sqrt{c x^2} (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 23, normalized size = 1. \begin{align*} -{\frac{1}{bx \left ( bx+a \right ) }\sqrt{c{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03162, size = 22, normalized size = 0.92 \begin{align*} -\frac{\sqrt{c}}{b^{2} x + a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26492, size = 43, normalized size = 1.79 \begin{align*} -\frac{\sqrt{c x^{2}}}{b^{2} x^{2} + a b x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.801927, size = 39, normalized size = 1.62 \begin{align*} \begin{cases} - \frac{\sqrt{c} \sqrt{x^{2}}}{a b x + b^{2} x^{2}} & \text{for}\: b \neq 0 \\\frac{\sqrt{c} \sqrt{x^{2}}}{a^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06232, size = 39, normalized size = 1.62 \begin{align*} -\sqrt{c}{\left (\frac{\mathrm{sgn}\left (x\right )}{{\left (b x + a\right )} b} - \frac{\mathrm{sgn}\left (x\right )}{a b}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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