Optimal. Leaf size=30 \[ \frac {\left (a+b e^{n x}\right )^{\frac {r+s}{s}} s}{b n (r+s)} \]
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Rubi [A]
time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2278, 32}
\begin {gather*} \frac {s \left (a+b e^{n x}\right )^{\frac {r+s}{s}}}{b n (r+s)} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 2278
Rubi steps
\begin {align*} \int e^{n x} \left (a+b e^{n x}\right )^{r/s} \, dx &=\frac {\text {Subst}\left (\int (a+b x)^{r/s} \, dx,x,e^{n x}\right )}{n}\\ &=\frac {\left (a+b e^{n x}\right )^{\frac {r+s}{s}} s}{b n (r+s)}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 30, normalized size = 1.00 \begin {gather*} \frac {\left (a+b e^{n x}\right )^{1+\frac {r}{s}} s}{b n r+b n s} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 2.28, size = 144, normalized size = 4.80 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {x}{a},b\text {==}0\text {\&\&}n\text {==}0\text {\&\&}r\text {==}-s\right \},\left \{\frac {E^{n x} a^{\frac {r}{s}}}{n},b\text {==}0\right \},\left \{x \left (a+b\right )^{\frac {r}{s}},n\text {==}0\right \},\left \{\frac {\text {Log}\left [\frac {a}{b}+E^{n x}\right ]}{b n},r\text {==}-s\right \}\right \},\frac {a s {\left (a+b E^{n x}\right )}^{\frac {r}{s}}}{b n r+b n s}+\frac {b s E^{n x} {\left (a+b E^{n x}\right )}^{\frac {r}{s}}}{b n r+b n s}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.03, size = 33, normalized size = 1.10
method | result | size |
derivativedivides | \(\frac {\left (a +b \,{\mathrm e}^{n x}\right )^{\frac {r}{s}+1}}{n b \left (\frac {r}{s}+1\right )}\) | \(33\) |
default | \(\frac {\left (a +b \,{\mathrm e}^{n x}\right )^{\frac {r}{s}+1}}{n b \left (\frac {r}{s}+1\right )}\) | \(33\) |
risch | \(\frac {s \left (a +b \,{\mathrm e}^{n x}\right ) \left (a +b \,{\mathrm e}^{n x}\right )^{\frac {r}{s}}}{b n \left (r +s \right )}\) | \(36\) |
norman | \(\frac {s \,{\mathrm e}^{n x} {\mathrm e}^{\frac {r \ln \left (a +b \,{\mathrm e}^{n x}\right )}{s}}}{n \left (r +s \right )}+\frac {a s \,{\mathrm e}^{\frac {r \ln \left (a +b \,{\mathrm e}^{n x}\right )}{s}}}{b n \left (r +s \right )}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 32, normalized size = 1.07 \begin {gather*} \frac {{\left (b e^{\left (n x\right )} + a\right )}^{\frac {r}{s} + 1}}{b n {\left (\frac {r}{s} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 37, normalized size = 1.23 \begin {gather*} \frac {{\left (b s e^{\left (n x\right )} + a s\right )} {\left (b e^{\left (n x\right )} + a\right )}^{\frac {r}{s}}}{b n r + b n s} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.49, size = 94, normalized size = 3.13 \begin {gather*} \begin {cases} \frac {x}{a} & \text {for}\: b = 0 \wedge n = 0 \wedge r = - s \\\frac {a^{\frac {r}{s}} e^{n x}}{n} & \text {for}\: b = 0 \\x \left (a + b\right )^{\frac {r}{s}} & \text {for}\: n = 0 \\\frac {\log {\left (\frac {a}{b} + e^{n x} \right )}}{b n} & \text {for}\: r = - s \\\frac {a s \left (a + b e^{n x}\right )^{\frac {r}{s}}}{b n r + b n s} + \frac {b s \left (a + b e^{n x}\right )^{\frac {r}{s}} e^{n x}}{b n r + b n s} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 27, normalized size = 0.90 \begin {gather*} \frac {\left (a+b \mathrm {e}^{n x}\right )^{\frac {r}{s}+1}}{b n \left (\frac {r}{s}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.35, size = 29, normalized size = 0.97 \begin {gather*} \frac {s\,{\left (a+b\,{\mathrm {e}}^{n\,x}\right )}^{\frac {r}{s}+1}}{b\,n\,\left (r+s\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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