Optimal. Leaf size=20 \[ \frac{1}{10} (5 x+3) \sqrt{(5 x+3)^2} \]
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Rubi [A] time = 0.0048126, antiderivative size = 20, 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, 30} \[ \frac{1}{10} (5 x+3) \sqrt{(5 x+3)^2} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \sqrt{(3+5 x)^2} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \sqrt{x^2} \, dx,x,3+5 x\right )\\ &=\frac{\sqrt{(3+5 x)^2} \operatorname{Subst}(\int x \, dx,x,3+5 x)}{5 (3+5 x)}\\ &=\frac{1}{10} (3+5 x) \sqrt{(3+5 x)^2}\\ \end{align*}
Mathematica [A] time = 0.0081942, size = 25, normalized size = 1.25 \[ \frac{x \sqrt{(5 x+3)^2} (5 x+6)}{10 x+6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 1.3 \begin{align*}{\frac{x \left ( 5\,x+6 \right ) }{6+10\,x}\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 = 1.45134, size = 41, normalized size = 2.05 \begin{align*} \frac{1}{2} \, \sqrt{25 \, x^{2} + 30 \, x + 9} x + \frac{3}{10} \, \sqrt{25 \, x^{2} + 30 \, x + 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24234, size = 20, normalized size = 1. \begin{align*} \frac{5}{2} \, x^{2} + 3 \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.065068, size = 8, normalized size = 0.4 \begin{align*} \frac{5 x^{2}}{2} + 3 x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10181, size = 35, normalized size = 1.75 \begin{align*} \frac{1}{2} \,{\left (5 \, x^{2} + 6 \, x\right )} \mathrm{sgn}\left (5 \, x + 3\right ) + \frac{9}{10} \, \mathrm{sgn}\left (5 \, x + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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