Optimal. Leaf size=65 \[ -\frac{\sqrt{c x^2} \log (a+b x)}{a^2 x}+\frac{\sqrt{c x^2} \log (x)}{a^2 x}+\frac{\sqrt{c x^2}}{a x (a+b x)} \]
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Rubi [A] time = 0.0179407, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 44} \[ -\frac{\sqrt{c x^2} \log (a+b x)}{a^2 x}+\frac{\sqrt{c x^2} \log (x)}{a^2 x}+\frac{\sqrt{c x^2}}{a x (a+b x)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 44
Rubi steps
\begin{align*} \int \frac{\sqrt{c x^2}}{x^2 (a+b x)^2} \, dx &=\frac{\sqrt{c x^2} \int \frac{1}{x (a+b x)^2} \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (\frac{1}{a^2 x}-\frac{b}{a (a+b x)^2}-\frac{b}{a^2 (a+b x)}\right ) \, dx}{x}\\ &=\frac{\sqrt{c x^2}}{a x (a+b x)}+\frac{\sqrt{c x^2} \log (x)}{a^2 x}-\frac{\sqrt{c x^2} \log (a+b x)}{a^2 x}\\ \end{align*}
Mathematica [A] time = 0.014482, size = 45, normalized size = 0.69 \[ \frac{c x (\log (x) (a+b x)-(a+b x) \log (a+b x)+a)}{a^2 \sqrt{c x^2} (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 52, normalized size = 0.8 \begin{align*}{\frac{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}{{a}^{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 [A] time = 1.03193, size = 51, normalized size = 0.78 \begin{align*} \frac{\sqrt{c}}{a b x + a^{2}} - \frac{\sqrt{c} \log \left (b x + a\right )}{a^{2}} + \frac{\sqrt{c} \log \left (x\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43205, size = 89, normalized size = 1.37 \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 x^{2} + a^{3} 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^{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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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