Optimal. Leaf size=26 \[ \frac {(3+5 x) \log (3+5 x)}{5 \sqrt {(3+5 x)^2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 26, 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 = {253, 15, 29}
\begin {gather*} \frac {(5 x+3) \log (5 x+3)}{5 \sqrt {(5 x+3)^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 29
Rule 253
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {(3+5 x)^2}} \, dx &=\frac {1}{5} \text {Subst}\left (\int \frac {1}{\sqrt {x^2}} \, dx,x,3+5 x\right )\\ &=\frac {(3+5 x) \text {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*}
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Mathematica [A]
time = 0.01, size = 26, normalized size = 1.00 \begin {gather*} \frac {(3+5 x) \log (3+5 x)}{5 \sqrt {(3+5 x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 23, normalized size = 0.88
method | result | size |
default | \(\frac {\left (3+5 x \right ) \ln \left (3+5 x \right )}{5 \sqrt {\left (3+5 x \right )^{2}}}\) | \(23\) |
risch | \(\frac {\sqrt {\left (3+5 x \right )^{2}}\, \ln \left (3+5 x \right )}{15+25 x}\) | \(25\) |
meijerg | \(\frac {3 \ln \left (1+\frac {5 x}{3}\right )}{5 \sqrt {\left (3+5 x \right )^{2}}}+\frac {x \ln \left (1+\frac {5 x}{3}\right )}{\sqrt {\left (3+5 x \right )^{2}}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 6, normalized size = 0.23 \begin {gather*} \frac {1}{5} \, \log \left (x + \frac {3}{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 8, normalized size = 0.31 \begin {gather*} \frac {1}{5} \, \log \left (5 \, x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 7, normalized size = 0.27 \begin {gather*} \frac {\log {\left (5 x + 3 \right )}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.39, size = 15, normalized size = 0.58 \begin {gather*} \frac {1}{5} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \mathrm {sgn}\left (5 \, x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.38, size = 14, normalized size = 0.54 \begin {gather*} \frac {\ln \left (5\,x+3\right )\,\mathrm {sign}\left (5\,x+3\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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