Optimal. Leaf size=33 \[ \frac {2 a^{m x}}{m \log (a)}+\frac {a^{2 m x}}{2 m \log (a)}+x \]
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Rubi [A] time = 0.01, antiderivative size = 33, 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} \[ \frac {2 a^{m x}}{m \log (a)}+\frac {a^{2 m x}}{2 m \log (a)}+x \]
Antiderivative was successfully verified.
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Rule 43
Rule 2282
Rubi steps
\begin {align*} \int \left (1+a^{m x}\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(1+x)^2}{x} \, dx,x,a^{m x}\right )}{m \log (a)}\\ &=\frac {\operatorname {Subst}\left (\int \left (2+\frac {1}{x}+x\right ) \, dx,x,a^{m x}\right )}{m \log (a)}\\ &=x+\frac {2 a^{m x}}{m \log (a)}+\frac {a^{2 m x}}{2 m \log (a)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 0.94 \[ \frac {2 a^{m x}+\frac {1}{2} a^{2 m x}+m x \log (a)}{m \log (a)} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (1+a^{m x}\right )^2 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.17, size = 29, normalized size = 0.88 \[ \frac {2 \, m x \log \relax (a) + a^{2 \, m x} + 4 \, a^{m x}}{2 \, m \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.58, size = 30, normalized size = 0.91 \[ \frac {2 \, m x \log \left ({\left | a \right |}\right ) + a^{2 \, m x} + 4 \, a^{m x}}{2 \, m \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 32, normalized size = 0.97
method | result | size |
derivativedivides | \(\frac {\frac {a^{2 m x}}{2}+2 a^{m x}+\ln \left (a^{m x}\right )}{m \ln \relax (a )}\) | \(32\) |
default | \(\frac {\frac {a^{2 m x}}{2}+2 a^{m x}+\ln \left (a^{m x}\right )}{m \ln \relax (a )}\) | \(32\) |
risch | \(x +\frac {2 a^{m x}}{m \ln \relax (a )}+\frac {a^{2 m x}}{2 m \ln \relax (a )}\) | \(33\) |
norman | \(x +\frac {2 \,{\mathrm e}^{m x \ln \relax (a )}}{m \ln \relax (a )}+\frac {{\mathrm e}^{2 m x \ln \relax (a )}}{2 m \ln \relax (a )}\) | \(35\) |
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 = 0.53, size = 31, normalized size = 0.94 \[ x + \frac {a^{2 \, m x}}{2 \, m \log \relax (a)} + \frac {2 \, a^{m x}}{m \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 26, normalized size = 0.79 \[ x+\frac {2\,a^{m\,x}+\frac {a^{2\,m\,x}}{2}}{m\,\ln \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 46, normalized size = 1.39 \[ x + \begin {cases} \frac {a^{2 m x} m \log {\relax (a )} + 4 a^{m x} m \log {\relax (a )}}{2 m^{2} \log {\relax (a )}^{2}} & \text {for}\: 2 m^{2} \log {\relax (a )}^{2} \neq 0 \\3 x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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