Optimal. Leaf size=20 \[ \frac {1}{10} (5 x+3) \sqrt {(5 x+3)^2} \]
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Rubi [A] time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 15
Rule 30
Rule 247
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.62, size = 9, normalized size = 0.45 \[ \frac {5}{2} \, x^{2} + 3 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 26, normalized size = 1.30 \[ \frac {1}{2} \, {\left (5 \, x^{2} + 6 \, x\right )} \mathrm {sgn}\left (5 \, x + 3\right ) + \frac {9}{10} \, \mathrm {sgn}\left (5 \, x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 1.25 \[ \frac {\left (5 x +6\right ) \sqrt {\left (5 x +3\right )^{2}}\, x}{6+10 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 30, normalized size = 1.50 \[ \frac {1}{2} \, \sqrt {25 \, x^{2} + 30 \, x + 9} x + \frac {3}{10} \, \sqrt {25 \, x^{2} + 30 \, x + 9} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 13, normalized size = 0.65 \[ \frac {\left |5\,x+3\right |\,\left (5\,x+3\right )}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 8, normalized size = 0.40 \[ \frac {5 x^{2}}{2} + 3 x \]
Verification of antiderivative is not currently implemented for this CAS.
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