Integrand size = 17, antiderivative size = 25 \[ \int \frac {1}{(2+3 x) \sqrt {1+5 x}} \, dx=\frac {2 \arctan \left (\sqrt {\frac {3}{7}} \sqrt {1+5 x}\right )}{\sqrt {21}} \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {65, 209} \[ \int \frac {1}{(2+3 x) \sqrt {1+5 x}} \, dx=\frac {2 \arctan \left (\sqrt {\frac {3}{7}} \sqrt {5 x+1}\right )}{\sqrt {21}} \]
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Rule 65
Rule 209
Rubi steps \begin{align*} \text {integral}& = \frac {2}{5} \text {Subst}\left (\int \frac {1}{\frac {7}{5}+\frac {3 x^2}{5}} \, dx,x,\sqrt {1+5 x}\right ) \\ & = \frac {2 \tan ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1+5 x}\right )}{\sqrt {21}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(2+3 x) \sqrt {1+5 x}} \, dx=\frac {2 \arctan \left (\sqrt {\frac {3}{7}} \sqrt {1+5 x}\right )}{\sqrt {21}} \]
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Time = 0.56 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(\frac {2 \arctan \left (\frac {\sqrt {21}\, \sqrt {1+5 x}}{7}\right ) \sqrt {21}}{21}\) | \(19\) |
default | \(\frac {2 \arctan \left (\frac {\sqrt {21}\, \sqrt {1+5 x}}{7}\right ) \sqrt {21}}{21}\) | \(19\) |
pseudoelliptic | \(\frac {2 \arctan \left (\frac {\sqrt {21}\, \sqrt {1+5 x}}{7}\right ) \sqrt {21}}{21}\) | \(19\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+21\right ) \ln \left (\frac {-15 \operatorname {RootOf}\left (\textit {\_Z}^{2}+21\right ) x +42 \sqrt {1+5 x}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+21\right )}{2+3 x}\right )}{21}\) | \(45\) |
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none
Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(2+3 x) \sqrt {1+5 x}} \, dx=\frac {2}{21} \, \sqrt {21} \arctan \left (\frac {1}{7} \, \sqrt {21} \sqrt {5 \, x + 1}\right ) \]
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Result contains complex when optimal does not.
Time = 0.59 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.44 \[ \int \frac {1}{(2+3 x) \sqrt {1+5 x}} \, dx=\begin {cases} \frac {2 \sqrt {21} i \operatorname {acosh}{\left (\frac {\sqrt {105}}{15 \sqrt {x + \frac {2}{3}}} \right )}}{21} & \text {for}\: \frac {1}{\left |{x + \frac {2}{3}}\right |} > \frac {15}{7} \\- \frac {2 \sqrt {21} \operatorname {asin}{\left (\frac {\sqrt {105}}{15 \sqrt {x + \frac {2}{3}}} \right )}}{21} & \text {otherwise} \end {cases} \]
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Time = 0.31 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(2+3 x) \sqrt {1+5 x}} \, dx=\frac {2}{21} \, \sqrt {21} \arctan \left (\frac {1}{7} \, \sqrt {21} \sqrt {5 \, x + 1}\right ) \]
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Time = 0.31 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(2+3 x) \sqrt {1+5 x}} \, dx=\frac {2}{21} \, \sqrt {21} \arctan \left (\frac {1}{7} \, \sqrt {21} \sqrt {5 \, x + 1}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.60 \[ \int \frac {1}{(2+3 x) \sqrt {1+5 x}} \, dx=\frac {2\,\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {105\,x+21}}{7}\right )}{21} \]
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