Optimal. Leaf size=65 \[ x+\frac {4 a^{m x}}{m \log (a)}+\frac {3 a^{2 m x}}{m \log (a)}+\frac {4 a^{3 m x}}{3 m \log (a)}+\frac {a^{4 m x}}{4 m \log (a)} \]
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Rubi [A]
time = 0.01, antiderivative size = 65, 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 = {2320, 45}
\begin {gather*} \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 \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2320
Rubi steps
\begin {align*} \int \left (1+a^{m x}\right )^4 \, dx &=\frac {\text {Subst}\left (\int \frac {(1+x)^4}{x} \, dx,x,a^{m x}\right )}{m \log (a)}\\ &=\frac {\text {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*}
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Mathematica [A]
time = 0.02, size = 53, normalized size = 0.82 \begin {gather*} \frac {\frac {a^{m x} \left (48+36 a^{m x}+16 a^{2 m x}+3 a^{3 m x}\right )}{12 m}+\frac {\log \left (a^{m x}\right )}{m}}{\log (a)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 50, normalized size = 0.77
method | result | size |
derivativedivides | \(\frac {\frac {a^{4 m x}}{4}+\frac {4 a^{3 m x}}{3}+3 a^{2 m x}+4 a^{m x}+\ln \left (a^{m x}\right )}{m \ln \left (a \right )}\) | \(50\) |
default | \(\frac {\frac {a^{4 m x}}{4}+\frac {4 a^{3 m x}}{3}+3 a^{2 m x}+4 a^{m x}+\ln \left (a^{m x}\right )}{m \ln \left (a \right )}\) | \(50\) |
risch | \(x +\frac {4 a^{m x}}{m \ln \left (a \right )}+\frac {3 a^{2 m x}}{m \ln \left (a \right )}+\frac {4 a^{3 m x}}{3 m \ln \left (a \right )}+\frac {a^{4 m x}}{4 m \ln \left (a \right )}\) | \(65\) |
norman | \(x +\frac {4 \,{\mathrm e}^{m x \ln \left (a \right )}}{m \ln \left (a \right )}+\frac {3 \,{\mathrm e}^{2 m x \ln \left (a \right )}}{m \ln \left (a \right )}+\frac {4 \,{\mathrm e}^{3 m x \ln \left (a \right )}}{3 m \ln \left (a \right )}+\frac {{\mathrm e}^{4 m x \ln \left (a \right )}}{4 m \ln \left (a \right )}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.07, size = 61, normalized size = 0.94 \begin {gather*} 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.63, size = 47, normalized size = 0.72 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 87, normalized size = 1.34 \begin {gather*} 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}\: m^{4} \log {\left (a \right )}^{4} \neq 0 \\15 x & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.99, size = 48, normalized size = 0.74 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.33, size = 42, normalized size = 0.65 \begin {gather*} x+\frac {4\,a^{m\,x}+3\,a^{2\,m\,x}+\frac {4\,a^{3\,m\,x}}{3}+\frac {a^{4\,m\,x}}{4}}{m\,\ln \left (a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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