Integrand size = 9, antiderivative size = 33 \[ \int \left (1+a^{m x}\right )^2 \, dx=x+\frac {2 a^{m x}}{m \log (a)}+\frac {a^{2 m x}}{2 m \log (a)} \]
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Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2320, 45} \[ \int \left (1+a^{m x}\right )^2 \, dx=\frac {2 a^{m x}}{m \log (a)}+\frac {a^{2 m x}}{2 m \log (a)}+x \]
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Rule 45
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(1+x)^2}{x} \, dx,x,a^{m x}\right )}{m \log (a)} \\ & = \frac {\text {Subst}\left (\int \left (2+\frac {1}{x}+x\right ) \, dx,x,a^{m x}\right )}{m \log (a)} \\ & = x+\frac {2 a^{m x}}{m \log (a)}+\frac {a^{2 m x}}{2 m \log (a)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \left (1+a^{m x}\right )^2 \, dx=\frac {\frac {a^{m x} \left (4+a^{m x}\right )}{2 m}+\frac {\log \left (a^{m x}\right )}{m}}{\log (a)} \]
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Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94
method | result | size |
parallelrisch | \(\frac {2 m x \ln \left (a \right )+a^{2 m x}+4 a^{m x}}{2 \ln \left (a \right ) m}\) | \(31\) |
derivativedivides | \(\frac {\frac {a^{2 m x}}{2}+2 a^{m x}+\ln \left (a^{m x}\right )}{m \ln \left (a \right )}\) | \(32\) |
default | \(\frac {\frac {a^{2 m x}}{2}+2 a^{m x}+\ln \left (a^{m x}\right )}{m \ln \left (a \right )}\) | \(32\) |
risch | \(x +\frac {2 a^{m x}}{m \ln \left (a \right )}+\frac {a^{2 m x}}{2 m \ln \left (a \right )}\) | \(33\) |
norman | \(x +\frac {2 \,{\mathrm e}^{m x \ln \left (a \right )}}{m \ln \left (a \right )}+\frac {{\mathrm e}^{2 m x \ln \left (a \right )}}{2 m \ln \left (a \right )}\) | \(35\) |
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Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \left (1+a^{m x}\right )^2 \, dx=\frac {2 \, m x \log \left (a\right ) + a^{2 \, m x} + 4 \, a^{m x}}{2 \, m \log \left (a\right )} \]
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Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \left (1+a^{m x}\right )^2 \, dx=x + \begin {cases} \frac {a^{2 m x} m \log {\left (a \right )} + 4 a^{m x} m \log {\left (a \right )}}{2 m^{2} \log {\left (a \right )}^{2}} & \text {for}\: m^{2} \log {\left (a \right )}^{2} \neq 0 \\3 x & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \left (1+a^{m x}\right )^2 \, dx=x + \frac {a^{2 \, m x}}{2 \, m \log \left (a\right )} + \frac {2 \, a^{m x}}{m \log \left (a\right )} \]
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Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \left (1+a^{m x}\right )^2 \, dx=\frac {2 \, m x \log \left ({\left | a \right |}\right ) + a^{2 \, m x} + 4 \, a^{m x}}{2 \, m \log \left (a\right )} \]
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Time = 0.35 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \left (1+a^{m x}\right )^2 \, dx=x+\frac {2\,a^{m\,x}+\frac {a^{2\,m\,x}}{2}}{m\,\ln \left (a\right )} \]
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