Optimal. Leaf size=22 \[ \frac{\sqrt{c x^2} \log (a+b x)}{b x} \]
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Rubi [A] time = 0.0032198, antiderivative size = 22, 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, 31} \[ \frac{\sqrt{c x^2} \log (a+b x)}{b x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 31
Rubi steps
\begin{align*} \int \frac{\sqrt{c x^2}}{x (a+b x)} \, dx &=\frac{\sqrt{c x^2} \int \frac{1}{a+b x} \, dx}{x}\\ &=\frac{\sqrt{c x^2} \log (a+b x)}{b x}\\ \end{align*}
Mathematica [A] time = 0.0036721, size = 21, normalized size = 0.95 \[ \frac{c x \log (a+b x)}{b \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 21, normalized size = 1. \begin{align*}{\frac{\ln \left ( bx+a \right ) }{bx}\sqrt{c{x}^{2}}} \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 = 1.46506, size = 43, normalized size = 1.95 \begin{align*} \frac{\sqrt{c x^{2}} \log \left (b x + a\right )}{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{\sqrt{c x^{2}}}{x \left (a + b x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05504, size = 38, normalized size = 1.73 \begin{align*} \sqrt{c}{\left (\frac{\log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (x\right )}{b} - \frac{\log \left ({\left | a \right |}\right ) \mathrm{sgn}\left (x\right )}{b}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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