Optimal. Leaf size=26 \[ \frac{(5 x+3) \log (5 x+3)}{10 \sqrt{(5 x+3)^2}} \]
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Rubi [A] time = 0.0050095, 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} \[ \frac{(5 x+3) \log (10 x+6)}{10 \sqrt{(5 x+3)^2}} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 29
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{(6+10 x)^2}} \, dx &=\frac{1}{10} \operatorname{Subst}\left (\int \frac{1}{\sqrt{x^2}} \, dx,x,6+10 x\right )\\ &=\frac{(6+10 x) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,6+10 x\right )}{10 \sqrt{(6+10 x)^2}}\\ &=\frac{(3+5 x) \log (3+5 x)}{10 \sqrt{(3+5 x)^2}}\\ \end{align*}
Mathematica [A] time = 0.0059921, size = 26, normalized size = 1. \[ \frac{(10 x+6) \log (10 x+6)}{10 \sqrt{(10 x+6)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 26, normalized size = 1. \begin{align*}{\frac{ \left ( 3+5\,x \right ) \sqrt{4}\ln \left ( 3+5\,x \right ) }{20}{\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 = 1.42113, size = 8, normalized size = 0.31 \begin{align*} \frac{1}{10} \, \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 = 1.24984, size = 26, normalized size = 1. \begin{align*} \frac{1}{10} \, \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 = 0.070765, size = 7, normalized size = 0.27 \begin{align*} \frac{\log{\left (10 x + 6 \right )}}{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12457, size = 20, normalized size = 0.77 \begin{align*} \frac{1}{10} \, \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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