Integrand size = 9, antiderivative size = 50 \[ \int \left (1+a^{m x}\right )^3 \, dx=x+\frac {3 a^{m x}}{m \log (a)}+\frac {3 a^{2 m x}}{2 m \log (a)}+\frac {a^{3 m x}}{3 m \log (a)} \]
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Time = 0.01 (sec) , antiderivative size = 50, 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} \[ \int \left (1+a^{m x}\right )^3 \, dx=\frac {3 a^{m x}}{m \log (a)}+\frac {3 a^{2 m x}}{2 m \log (a)}+\frac {a^{3 m x}}{3 m \log (a)}+x \]
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Rule 45
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(1+x)^3}{x} \, dx,x,a^{m x}\right )}{m \log (a)} \\ & = \frac {\text {Subst}\left (\int \left (3+\frac {1}{x}+3 x+x^2\right ) \, dx,x,a^{m x}\right )}{m \log (a)} \\ & = x+\frac {3 a^{m x}}{m \log (a)}+\frac {3 a^{2 m x}}{2 m \log (a)}+\frac {a^{3 m x}}{3 m \log (a)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.90 \[ \int \left (1+a^{m x}\right )^3 \, dx=\frac {\frac {a^{m x} \left (18+9 a^{m x}+2 a^{2 m x}\right )}{6 m}+\frac {\log \left (a^{m x}\right )}{m}}{\log (a)} \]
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Time = 0.04 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {a^{3 m x}}{3}+\frac {3 a^{2 m x}}{2}+3 a^{m x}+\ln \left (a^{m x}\right )}{m \ln \left (a \right )}\) | \(41\) |
default | \(\frac {\frac {a^{3 m x}}{3}+\frac {3 a^{2 m x}}{2}+3 a^{m x}+\ln \left (a^{m x}\right )}{m \ln \left (a \right )}\) | \(41\) |
parallelrisch | \(\frac {2 a^{3 m x}+6 m x \ln \left (a \right )+9 a^{2 m x}+18 a^{m x}}{6 \ln \left (a \right ) m}\) | \(42\) |
risch | \(x +\frac {3 a^{m x}}{m \ln \left (a \right )}+\frac {3 a^{2 m x}}{2 m \ln \left (a \right )}+\frac {a^{3 m x}}{3 m \ln \left (a \right )}\) | \(49\) |
norman | \(x +\frac {3 \,{\mathrm e}^{m x \ln \left (a \right )}}{m \ln \left (a \right )}+\frac {3 \,{\mathrm e}^{2 m x \ln \left (a \right )}}{2 m \ln \left (a \right )}+\frac {{\mathrm e}^{3 m x \ln \left (a \right )}}{3 m \ln \left (a \right )}\) | \(52\) |
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Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.78 \[ \int \left (1+a^{m x}\right )^3 \, dx=\frac {6 \, m x \log \left (a\right ) + 2 \, a^{3 \, m x} + 9 \, a^{2 \, m x} + 18 \, a^{m x}}{6 \, m \log \left (a\right )} \]
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Time = 0.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.40 \[ \int \left (1+a^{m x}\right )^3 \, dx=x + \begin {cases} \frac {2 a^{3 m x} m^{2} \log {\left (a \right )}^{2} + 9 a^{2 m x} m^{2} \log {\left (a \right )}^{2} + 18 a^{m x} m^{2} \log {\left (a \right )}^{2}}{6 m^{3} \log {\left (a \right )}^{3}} & \text {for}\: m^{3} \log {\left (a \right )}^{3} \neq 0 \\7 x & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.92 \[ \int \left (1+a^{m x}\right )^3 \, dx=x + \frac {a^{3 \, m x}}{3 \, m \log \left (a\right )} + \frac {3 \, a^{2 \, m x}}{2 \, m \log \left (a\right )} + \frac {3 \, a^{m x}}{m \log \left (a\right )} \]
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Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.80 \[ \int \left (1+a^{m x}\right )^3 \, dx=\frac {6 \, m x \log \left ({\left | a \right |}\right ) + 2 \, a^{3 \, m x} + 9 \, a^{2 \, m x} + 18 \, a^{m x}}{6 \, m \log \left (a\right )} \]
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Time = 0.36 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.68 \[ \int \left (1+a^{m x}\right )^3 \, dx=x+\frac {3\,a^{m\,x}+\frac {3\,a^{2\,m\,x}}{2}+\frac {a^{3\,m\,x}}{3}}{m\,\ln \left (a\right )} \]
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