Optimal. Leaf size=28 \[ \frac{(a+b x) \sqrt{\frac{c}{(a+b x)^2}} \log (a+b x)}{b} \]
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Rubi [A] time = 0.0096413, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {247, 15, 29} \[ \frac{(a+b x) \sqrt{\frac{c}{(a+b x)^2}} \log (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 29
Rubi steps
\begin{align*} \int \sqrt{\frac{c}{(a+b x)^2}} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{\frac{c}{x^2}} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (\sqrt{\frac{c}{(a+b x)^2}} (a+b x)\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,a+b x\right )}{b}\\ &=\frac{\sqrt{\frac{c}{(a+b x)^2}} (a+b x) \log (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0070481, size = 28, normalized size = 1. \[ \frac{(a+b x) \sqrt{\frac{c}{(a+b x)^2}} \log (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 27, normalized size = 1. \begin{align*}{\frac{ \left ( bx+a \right ) \ln \left ( bx+a \right ) }{b}\sqrt{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.22164, size = 18, normalized size = 0.64 \begin{align*} \frac{\sqrt{c} \log \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31028, size = 81, normalized size = 2.89 \begin{align*} \frac{{\left (b x + a\right )} \sqrt{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}} \log \left (b x + a\right )}{b} \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 \sqrt{\frac{c}{\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 = 1.10624, size = 27, normalized size = 0.96 \begin{align*} \frac{\sqrt{c} \log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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