Optimal. Leaf size=40 \[ -\frac {\left (a^{m x}+1\right )^{n+1} \, _2F_1\left (1,n+1;n+2;a^{m x}+1\right )}{m (n+1) \log (a)} \]
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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 = {2282, 65} \[ -\frac {\left (a^{m x}+1\right )^{n+1} \text {Hypergeometric2F1}\left (1,n+1,n+2,a^{m x}+1\right )}{m (n+1) \log (a)} \]
Antiderivative was successfully verified.
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Rule 65
Rule 2282
Rubi steps
\begin {align*} \int \left (1+a^{m x}\right )^n \, dx &=\frac {\operatorname {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 = 0.01, size = 40, normalized size = 1.00 \[ -\frac {\left (a^{m x}+1\right )^{n+1} \, _2F_1\left (1,n+1;n+2;a^{m x}+1\right )}{m (n+1) \log (a)} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (1+a^{m x}\right )^n \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{m x} + 1\right )}^{n}, x\right ) \]
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 \[ \int {\left (a^{m x} + 1\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, 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 \[ \int {\left (a^{m x} + 1\right )}^{n}\,{d x} \]
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 \[ \frac {{\left (a^{m\,x}+1\right )}^n\,{{}}_2{\mathrm {F}}_1\left (-n,-n;\ 1-n;\ -\frac {1}{a^{m\,x}}\right )}{m\,n\,\ln \relax (a)\,{\left (\frac {1}{a^{m\,x}}+1\right )}^n} \]
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 \[ \int \left (a^{m x} + 1\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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