Optimal. Leaf size=65 \[ \frac{4 a^{m x}}{m \log (a)}+\frac{3 a^{2 m x}}{m \log (a)}+\frac{4 a^{3 m x}}{3 m \log (a)}+\frac{a^{4 m x}}{4 m \log (a)}+x \]
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Rubi [A] time = 0.0191191, antiderivative size = 65, 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} \[ \frac{4 a^{m x}}{m \log (a)}+\frac{3 a^{2 m x}}{m \log (a)}+\frac{4 a^{3 m x}}{3 m \log (a)}+\frac{a^{4 m x}}{4 m \log (a)}+x \]
Antiderivative was successfully verified.
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Rule 2282
Rule 43
Rubi steps
\begin{align*} \int \left (1+a^{m x}\right )^4 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(1+x)^4}{x} \, dx,x,a^{m x}\right )}{m \log (a)}\\ &=\frac{\operatorname{Subst}\left (\int \left (4+\frac{1}{x}+6 x+4 x^2+x^3\right ) \, dx,x,a^{m x}\right )}{m \log (a)}\\ &=x+\frac{4 a^{m x}}{m \log (a)}+\frac{3 a^{2 m x}}{m \log (a)}+\frac{4 a^{3 m x}}{3 m \log (a)}+\frac{a^{4 m x}}{4 m \log (a)}\\ \end{align*}
Mathematica [A] time = 0.0197996, size = 49, normalized size = 0.75 \[ \frac{4 a^{m x}+3 a^{2 m x}+\frac{4}{3} a^{3 m x}+\frac{1}{4} a^{4 m x}+m x \log (a)}{m \log (a)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 78, normalized size = 1.2 \begin{align*}{\frac{ \left ({a}^{mx} \right ) ^{4}}{4\,m\ln \left ( a \right ) }}+{\frac{4\, \left ({a}^{mx} \right ) ^{3}}{3\,m\ln \left ( a \right ) }}+3\,{\frac{ \left ({a}^{mx} \right ) ^{2}}{m\ln \left ( a \right ) }}+4\,{\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 = 0.921888, size = 82, normalized size = 1.26 \begin{align*} x + \frac{a^{4 \, m x}}{4 \, m \log \left (a\right )} + \frac{4 \, a^{3 \, m x}}{3 \, m \log \left (a\right )} + \frac{3 \, a^{2 \, m x}}{m \log \left (a\right )} + \frac{4 \, 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 = 1.67255, size = 122, normalized size = 1.88 \begin{align*} \frac{12 \, m x \log \left (a\right ) + 3 \, a^{4 \, m x} + 16 \, a^{3 \, m x} + 36 \, a^{2 \, m x} + 48 \, a^{m x}}{12 \, m \log \left (a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.155902, size = 88, normalized size = 1.35 \begin{align*} x + \begin{cases} \frac{3 a^{4 m x} m^{3} \log{\left (a \right )}^{3} + 16 a^{3 m x} m^{3} \log{\left (a \right )}^{3} + 36 a^{2 m x} m^{3} \log{\left (a \right )}^{3} + 48 a^{m x} m^{3} \log{\left (a \right )}^{3}}{12 m^{4} \log{\left (a \right )}^{4}} & \text{for}\: 12 m^{4} \log{\left (a \right )}^{4} \neq 0 \\15 x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13746, size = 65, normalized size = 1. \begin{align*} \frac{12 \, m x \log \left ({\left | a \right |}\right ) + 3 \, a^{4 \, m x} + 16 \, a^{3 \, m x} + 36 \, a^{2 \, m x} + 48 \, a^{m x}}{12 \, m \log \left (a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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