Optimal. Leaf size=38 \[ \frac{\sqrt{c x^2}}{b}-\frac{a \sqrt{c x^2} \log (a+b x)}{b^2 x} \]
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Rubi [A] time = 0.0124095, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {15, 43} \[ \frac{\sqrt{c x^2}}{b}-\frac{a \sqrt{c x^2} \log (a+b x)}{b^2 x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{\sqrt{c x^2}}{a+b x} \, dx &=\frac{\sqrt{c x^2} \int \frac{x}{a+b x} \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (\frac{1}{b}-\frac{a}{b (a+b x)}\right ) \, dx}{x}\\ &=\frac{\sqrt{c x^2}}{b}-\frac{a \sqrt{c x^2} \log (a+b x)}{b^2 x}\\ \end{align*}
Mathematica [A] time = 0.0071633, size = 28, normalized size = 0.74 \[ \frac{c x (b x-a \log (a+b x))}{b^2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 29, normalized size = 0.8 \begin{align*} -{\frac{a\ln \left ( bx+a \right ) -bx}{{b}^{2}x}\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 = 1.55409, size = 59, normalized size = 1.55 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x - a \log \left (b x + a\right )\right )}}{b^{2} 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}}}{a + b x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05594, size = 50, normalized size = 1.32 \begin{align*} \sqrt{c}{\left (\frac{x \mathrm{sgn}\left (x\right )}{b} - \frac{a \log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{2}} + \frac{a \log \left ({\left | a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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