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 = 0.036049, 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 \[ \frac{3 a^{m x}}{m \log (a)}+\frac{3 a^{2 m x}}{2 m \log (a)}+\frac{a^{3 m x}}{3 m \log (a)}+x \]
Antiderivative was successfully verified.
[In] Int[(1 + a^(m*x))^3,x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a^{3 m x}}{3 m \log{\left (a \right )}} + \frac{3 a^{m x}}{m \log{\left (a \right )}} + \frac{\log{\left (a^{m x} \right )}}{m \log{\left (a \right )}} + \frac{3 \int ^{a^{m x}} x\, dx}{m \log{\left (a \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+a**(m*x))**3,x)
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Mathematica [A] time = 0.0194399, size = 41, normalized size = 0.82 \[ \frac{18 a^{m x}+9 a^{2 m x}+2 a^{3 m x}+6 m x \log (a)}{6 m \log (a)} \]
Antiderivative was successfully verified.
[In] Integrate[(1 + a^(m*x))^3,x]
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Maple [A] time = 0.003, size = 62, normalized size = 1.2 \[{\frac{ \left ({a}^{mx} \right ) ^{3}}{3\,m\ln \left ( a \right ) }}+{\frac{3\, \left ({a}^{mx} \right ) ^{2}}{2\,m\ln \left ( a \right ) }}+3\,{\frac{{a}^{mx}}{m\ln \left ( a \right ) }}+{\frac{\ln \left ({a}^{mx} \right ) }{m\ln \left ( a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+a^(m*x))^3,x)
[Out]
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Maxima [A] time = 1.37683, size = 62, normalized size = 1.24 \[ x + \frac{a^{3 \, m x}}{3 \, m \log \left (a\right )} + \frac{3 \, a^{2 \, m x}}{2 \, m \log \left (a\right )} + \frac{3 \, a^{m x}}{m \log \left (a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a^(m*x) + 1)^3,x, algorithm="maxima")
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Fricas [A] time = 0.241489, size = 53, normalized size = 1.06 \[ \frac{6 \, m x \log \left (a\right ) + 2 \, a^{3 \, m x} + 9 \, a^{2 \, m x} + 18 \, a^{m x}}{6 \, m \log \left (a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a^(m*x) + 1)^3,x, algorithm="fricas")
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Sympy [A] time = 0.14797, size = 71, normalized size = 1.42 \[ x + \begin{cases} \frac{2 a^{3 m x} m^{2} \log{\left (a \right )}^{2} + 9 a^{2 m x} m^{2} \log{\left (a \right )}^{2} + 18 a^{m x} m^{2} \log{\left (a \right )}^{2}}{6 m^{3} \log{\left (a \right )}^{3}} & \text{for}\: 6 m^{3} \log{\left (a \right )}^{3} \neq 0 \\7 x & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+a**(m*x))**3,x)
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GIAC/XCAS [A] time = 0.200246, size = 54, normalized size = 1.08 \[ \frac{6 \, m x{\rm ln}\left ({\left | a \right |}\right ) + 2 \, a^{3 \, m x} + 9 \, a^{2 \, m x} + 18 \, a^{m x}}{6 \, m{\rm ln}\left (a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a^(m*x) + 1)^3,x, algorithm="giac")
[Out]