Optimal. Leaf size=59 \[ -\frac{x \log (a+b x)}{a^2 \sqrt{c x^2}}+\frac{x \log (x)}{a^2 \sqrt{c x^2}}+\frac{x}{a \sqrt{c x^2} (a+b x)} \]
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Rubi [A] time = 0.0164084, antiderivative size = 59, 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, 44} \[ -\frac{x \log (a+b x)}{a^2 \sqrt{c x^2}}+\frac{x \log (x)}{a^2 \sqrt{c x^2}}+\frac{x}{a \sqrt{c x^2} (a+b x)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{c x^2} (a+b x)^2} \, dx &=\frac{x \int \frac{1}{x (a+b x)^2} \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{1}{a^2 x}-\frac{b}{a (a+b x)^2}-\frac{b}{a^2 (a+b x)}\right ) \, dx}{\sqrt{c x^2}}\\ &=\frac{x}{a \sqrt{c x^2} (a+b x)}+\frac{x \log (x)}{a^2 \sqrt{c x^2}}-\frac{x \log (a+b x)}{a^2 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0053642, size = 44, normalized size = 0.75 \[ \frac{x (\log (x) (a+b x)-(a+b x) \log (a+b x)+a)}{a^2 \sqrt{c x^2} (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 50, normalized size = 0.9 \begin{align*}{\frac{x \left ( b\ln \left ( x \right ) x-b\ln \left ( bx+a \right ) x+a\ln \left ( x \right ) -a\ln \left ( bx+a \right ) +a \right ) }{{a}^{2} \left ( bx+a \right ) }{\frac{1}{\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 = 1.38414, size = 95, normalized size = 1.61 \begin{align*} \frac{\sqrt{c x^{2}}{\left ({\left (b x + a\right )} \log \left (\frac{x}{b x + a}\right ) + a\right )}}{a^{2} b c x^{2} + a^{3} c 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}{\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 = 1.12313, size = 116, normalized size = 1.97 \begin{align*} -\frac{\log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{2} \sqrt{c} \mathrm{sgn}\left (-\frac{b}{b x + a} + \frac{a b}{{\left (b x + a\right )}^{2}}\right )} - \frac{1}{{\left (b x + a\right )} a \sqrt{c} \mathrm{sgn}\left (-\frac{b}{b x + a} + \frac{a b}{{\left (b x + a\right )}^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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