Optimal. Leaf size=47 \[ \frac{a \sqrt{c x^2}}{b^2 x (a+b x)}+\frac{\sqrt{c x^2} \log (a+b x)}{b^2 x} \]
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Rubi [A] time = 0.0162818, antiderivative size = 47, 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} \[ \frac{a \sqrt{c x^2}}{b^2 x (a+b x)}+\frac{\sqrt{c x^2} \log (a+b x)}{b^2 x} \]
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)^2} \, dx &=\frac{\sqrt{c x^2} \int \frac{x}{(a+b x)^2} \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (-\frac{a}{b (a+b x)^2}+\frac{1}{b (a+b x)}\right ) \, dx}{x}\\ &=\frac{a \sqrt{c x^2}}{b^2 x (a+b x)}+\frac{\sqrt{c x^2} \log (a+b x)}{b^2 x}\\ \end{align*}
Mathematica [A] time = 0.0116019, size = 36, normalized size = 0.77 \[ \frac{c x ((a+b x) \log (a+b x)+a)}{b^2 \sqrt{c x^2} (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 41, normalized size = 0.9 \begin{align*}{\frac{b\ln \left ( bx+a \right ) x+a\ln \left ( bx+a \right ) +a}{{b}^{2}x \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 [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 = 1.51192, size = 84, normalized size = 1.79 \begin{align*} \frac{\sqrt{c x^{2}}{\left ({\left (b x + a\right )} \log \left (b x + a\right ) + a\right )}}{b^{3} x^{2} + a 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}}}{\left (a + b x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07896, size = 62, normalized size = 1.32 \begin{align*} -\sqrt{c}{\left (\frac{{\left (\log \left ({\left | a \right |}\right ) + 1\right )} \mathrm{sgn}\left (x\right )}{b^{2}} - \frac{\log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{2}} - \frac{a \mathrm{sgn}\left (x\right )}{{\left (b x + a\right )} b^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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