Optimal. Leaf size=59 \[ -\frac {s \left (a+b e^{n x}\right )^{\frac {r+s}{s}} \, _2F_1\left (1,\frac {r+s}{s};\frac {r}{s}+2;\frac {e^{n x} b}{a}+1\right )}{a n (r+s)} \]
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Rubi [A] time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2282, 65} \[ -\frac {s \left (a+b e^{n x}\right )^{\frac {r+s}{s}} \text {Hypergeometric2F1}\left (1,\frac {r+s}{s},\frac {r}{s}+2,\frac {b e^{n x}}{a}+1\right )}{a n (r+s)} \]
Antiderivative was successfully verified.
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Rule 65
Rule 2282
Rubi steps
\begin {align*} \int \left (a+b e^{n x}\right )^{r/s} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^{r/s}}{x} \, dx,x,e^{n x}\right )}{n}\\ &=-\frac {\left (a+b e^{n x}\right )^{\frac {r+s}{s}} s \, _2F_1\left (1,\frac {r+s}{s};2+\frac {r}{s};1+\frac {b e^{n x}}{a}\right )}{a n (r+s)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 59, normalized size = 1.00 \[ -\frac {s \left (a+b e^{n x}\right )^{\frac {r+s}{s}} \, _2F_1\left (1,\frac {r+s}{s};\frac {r}{s}+2;\frac {e^{n x} b}{a}+1\right )}{a n (r+s)} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a+b e^{n x}\right )^{r/s} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b e^{\left (n x\right )} + a\right )}^{\frac {r}{s}}, x\right ) \]
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 \[ \int {\left (b e^{\left (n x\right )} + a\right )}^{\frac {r}{s}}\,{d x} \]
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 \[\int \left (a +b \,{\mathrm e}^{n x}\right )^{\frac {r}{s}}\, dx\]
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 \[ \int {\left (b e^{\left (n x\right )} + a\right )}^{\frac {r}{s}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 75, normalized size = 1.27 \[ \frac {s\,{\left (a+b\,{\mathrm {e}}^{n\,x}\right )}^{r/s}\,{{}}_2{\mathrm {F}}_1\left (-\frac {r}{s},-\frac {r}{s};\ 1-\frac {r}{s};\ -\frac {a\,{\mathrm {e}}^{-n\,x}}{b}\right )}{n\,r\,{\left (\frac {a\,{\mathrm {e}}^{-n\,x}}{b}+1\right )}^{r/s}} \]
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 \[ \int \left (a + b e^{n x}\right )^{\frac {r}{s}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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