Optimal. Leaf size=32 \[ \frac {e^{-c} \left (a+b e^{c+d x}\right )^{n+1}}{b d (n+1)} \]
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Rubi [A] time = 0.07, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2247, 2246, 32} \[ \frac {e^{-c} \left (a+b e^{c+d x}\right )^{n+1}}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2246
Rule 2247
Rubi steps
\begin {align*} \int e^{d x} \left (a+b e^{c+d x}\right )^n \, dx &=e^{-c} \int e^{c+d x} \left (a+b e^{c+d x}\right )^n \, dx\\ &=\frac {e^{-c} \operatorname {Subst}\left (\int (a+b x)^n \, dx,x,e^{c+d x}\right )}{d}\\ &=\frac {e^{-c} \left (a+b e^{c+d x}\right )^{1+n}}{b d (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 31, normalized size = 0.97 \[ \frac {e^{-c} \left (a+b e^{c+d x}\right )^{n+1}}{b d n+b d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 36, normalized size = 1.12 \[ \frac {{\left (b e^{\left (d x\right )} + a e^{\left (-c\right )}\right )} {\left (b e^{\left (d x + c\right )} + a\right )}^{n}}{b d n + b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 38, normalized size = 1.19 \[ \frac {{\left (b e^{\left (d x + c\right )} + a\right )} {\left (b e^{\left (d x + c\right )} + a\right )}^{n} e^{\left (-c\right )}}{b d {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 31, normalized size = 0.97 \[ \frac {\left (b \,{\mathrm e}^{c} {\mathrm e}^{d x}+a \right )^{n +1} {\mathrm e}^{-c}}{\left (n +1\right ) b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 30, normalized size = 0.94 \[ \frac {{\left (b e^{\left (d x + c\right )} + a\right )}^{n + 1} e^{\left (-c\right )}}{b d {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.55, size = 44, normalized size = 1.38 \[ {\left (a+b\,{\mathrm {e}}^{c+d\,x}\right )}^n\,\left (\frac {{\mathrm {e}}^{d\,x}}{d\,\left (n+1\right )}+\frac {a\,{\mathrm {e}}^{-c}}{b\,d\,\left (n+1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.33, size = 114, normalized size = 3.56 \[ \begin {cases} \frac {x}{a} & \text {for}\: b = 0 \wedge d = 0 \wedge n = -1 \\\frac {a^{n} e^{d x}}{d} & \text {for}\: b = 0 \\x \left (a + b e^{c}\right )^{n} & \text {for}\: d = 0 \\\frac {e^{- c} \log {\left (\frac {a}{b} + e^{c} e^{d x} \right )}}{b d} & \text {for}\: n = -1 \\\frac {a \left (a + b e^{c} e^{d x}\right )^{n}}{b d n e^{c} + b d e^{c}} + \frac {b \left (a + b e^{c} e^{d x}\right )^{n} e^{c} e^{d x}}{b d n e^{c} + b d e^{c}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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