Optimal. Leaf size=40 \[ -\frac {\left (1+a^{m x}\right )^{1+n} \, _2F_1\left (1,1+n;2+n;1+a^{m x}\right )}{m (1+n) \log (a)} \]
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Rubi [A]
time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2320, 67}
\begin {gather*} -\frac {\left (a^{m x}+1\right )^{n+1} \text {Hypergeometric2F1}\left (1,n+1,n+2,a^{m x}+1\right )}{m (n+1) \log (a)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 2320
Rubi steps
\begin {align*} \int \left (1+a^{m x}\right )^n \, dx &=\frac {\text {Subst}\left (\int \frac {(1+x)^n}{x} \, dx,x,a^{m x}\right )}{m \log (a)}\\ &=-\frac {\left (1+a^{m x}\right )^{1+n} \, _2F_1\left (1,1+n;2+n;1+a^{m x}\right )}{m (1+n) \log (a)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 40, normalized size = 1.00 \begin {gather*} -\frac {\left (1+a^{m x}\right )^{1+n} \, _2F_1\left (1,1+n;2+n;1+a^{m x}\right )}{m (1+n) \log (a)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (1+a^{m x}\right )^{n}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a^{m x} + 1\right )^{n}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 55, normalized size = 1.38 \begin {gather*} \frac {{\left (a^{m\,x}+1\right )}^n\,{{}}_2{\mathrm {F}}_1\left (-n,-n;\ 1-n;\ -\frac {1}{a^{m\,x}}\right )}{m\,n\,\ln \left (a\right )\,{\left (\frac {1}{a^{m\,x}}+1\right )}^n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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