Optimal. Leaf size=26 \[ \frac{(5 x+3) \log (5 x+3)}{5 \sqrt{(5 x+3)^2}} \]
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Rubi [A] time = 0.0048987, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {247, 15, 29} \[ \frac{(5 x+3) \log (5 x+3)}{5 \sqrt{(5 x+3)^2}} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 29
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{(3+5 x)^2}} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{\sqrt{x^2}} \, dx,x,3+5 x\right )\\ &=\frac{(3+5 x) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,3+5 x\right )}{5 \sqrt{(3+5 x)^2}}\\ &=\frac{(3+5 x) \log (3+5 x)}{5 \sqrt{(3+5 x)^2}}\\ \end{align*}
Mathematica [A] time = 0.0066036, size = 26, normalized size = 1. \[ \frac{(5 x+3) \log (5 x+3)}{5 \sqrt{(5 x+3)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 23, normalized size = 0.9 \begin{align*}{\frac{ \left ( 3+5\,x \right ) \ln \left ( 3+5\,x \right ) }{5}{\frac{1}{\sqrt{ \left ( 3+5\,x \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44625, size = 8, normalized size = 0.31 \begin{align*} \frac{1}{5} \, \log \left (x + \frac{3}{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.19683, size = 24, normalized size = 0.92 \begin{align*} \frac{1}{5} \, \log \left (5 \, x + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.073199, size = 7, normalized size = 0.27 \begin{align*} \frac{\log{\left (5 x + 3 \right )}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10842, size = 20, normalized size = 0.77 \begin{align*} \frac{1}{5} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \mathrm{sgn}\left (5 \, x + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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