Optimal. Leaf size=59 \[ -\frac {\left (a+b e^{n x}\right )^{\frac {r+s}{s}} s \, _2F_1\left (1,\frac {r+s}{s};2+\frac {r}{s};1+\frac {b e^{n x}}{a}\right )}{a n (r+s)} \]
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Rubi [A]
time = 0.02, 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 = {2320, 67}
\begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 2320
Rubi steps
\begin {align*} \int \left (a+b e^{n x}\right )^{r/s} \, dx &=\frac {\text {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 = 0.04, size = 59, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, 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 \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 + b e^{n x}\right )^{\frac {r}{s}}\, 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 = 0.40, size = 75, normalized size = 1.27 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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