Optimal. Leaf size=50 \[ \frac{3 a^{m x}}{m \log (a)}+\frac{3 a^{2 m x}}{2 m \log (a)}+\frac{a^{3 m x}}{3 m \log (a)}+x \]
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Rubi [A] time = 0.0157746, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2282, 43} \[ \frac{3 a^{m x}}{m \log (a)}+\frac{3 a^{2 m x}}{2 m \log (a)}+\frac{a^{3 m x}}{3 m \log (a)}+x \]
Antiderivative was successfully verified.
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Rule 2282
Rule 43
Rubi steps
\begin{align*} \int \left (1+a^{m x}\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(1+x)^3}{x} \, dx,x,a^{m x}\right )}{m \log (a)}\\ &=\frac{\operatorname{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*}
Mathematica [A] time = 0.0257468, size = 35, normalized size = 0.7 \[ \frac{\left (9 a^{m x}+2 a^{2 m x}+18\right ) a^{m x}}{6 m \log (a)}+x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 62, normalized size = 1.2 \begin{align*}{\frac{ \left ({a}^{mx} \right ) ^{3}}{3\,m\ln \left ( a \right ) }}+{\frac{3\, \left ({a}^{mx} \right ) ^{2}}{2\,m\ln \left ( a \right ) }}+3\,{\frac{{a}^{mx}}{m\ln \left ( a \right ) }}+{\frac{\ln \left ({a}^{mx} \right ) }{m\ln \left ( a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.934788, size = 62, normalized size = 1.24 \begin{align*} 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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80834, size = 97, normalized size = 1.94 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.133762, size = 71, normalized size = 1.42 \begin{align*} 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}\: 6 m^{3} \log{\left (a \right )}^{3} \neq 0 \\7 x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08985, size = 54, normalized size = 1.08 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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