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 = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 43
Rule 2282
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*}
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Mathematica [A] time = 0.03, size = 35, normalized size = 0.70 \[ \frac {\left (9 a^{m x}+2 a^{2 m x}+18\right ) a^{m x}}{6 m \log (a)}+x \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (1+a^{m x}\right )^3 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.27, size = 39, normalized size = 0.78 \[ \frac {6 \, m x \log \relax (a) + 2 \, a^{3 \, m x} + 9 \, a^{2 \, m x} + 18 \, a^{m x}}{6 \, m \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 40, normalized size = 0.80 \[ \frac {6 \, m x \log \left ({\left | a \right |}\right ) + 2 \, a^{3 \, m x} + 9 \, a^{2 \, m x} + 18 \, a^{m x}}{6 \, m \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 41, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {a^{3 m x}}{3}+\frac {3 a^{2 m x}}{2}+3 a^{m x}+\ln \left (a^{m x}\right )}{m \ln \relax (a )}\) | \(41\) |
default | \(\frac {\frac {a^{3 m x}}{3}+\frac {3 a^{2 m x}}{2}+3 a^{m x}+\ln \left (a^{m x}\right )}{m \ln \relax (a )}\) | \(41\) |
risch | \(x +\frac {3 a^{m x}}{m \ln \relax (a )}+\frac {3 a^{2 m x}}{2 m \ln \relax (a )}+\frac {a^{3 m x}}{3 m \ln \relax (a )}\) | \(49\) |
norman | \(x +\frac {3 \,{\mathrm e}^{m x \ln \relax (a )}}{m \ln \relax (a )}+\frac {3 \,{\mathrm e}^{2 m x \ln \relax (a )}}{2 m \ln \relax (a )}+\frac {{\mathrm e}^{3 m x \ln \relax (a )}}{3 m \ln \relax (a )}\) | \(52\) |
meijerg | error in int/gbinthm/express: improper op or subscript selector\ | N/A |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 46, normalized size = 0.92 \[ x + \frac {a^{3 \, m x}}{3 \, m \log \relax (a)} + \frac {3 \, a^{2 \, m x}}{2 \, m \log \relax (a)} + \frac {3 \, a^{m x}}{m \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 34, normalized size = 0.68 \[ x+\frac {3\,a^{m\,x}+\frac {3\,a^{2\,m\,x}}{2}+\frac {a^{3\,m\,x}}{3}}{m\,\ln \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 71, normalized size = 1.42 \[ x + \begin {cases} \frac {2 a^{3 m x} m^{2} \log {\relax (a )}^{2} + 9 a^{2 m x} m^{2} \log {\relax (a )}^{2} + 18 a^{m x} m^{2} \log {\relax (a )}^{2}}{6 m^{3} \log {\relax (a )}^{3}} & \text {for}\: 6 m^{3} \log {\relax (a )}^{3} \neq 0 \\7 x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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