Optimal. Leaf size=27 \[ \frac {\left (a+b e^{c+d x}\right )^{n+1}}{b d (n+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 27, 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 = {2246, 32} \[ \frac {\left (a+b e^{c+d x}\right )^{n+1}}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2246
Rubi steps
\begin {align*} \int e^{c+d x} \left (a+b e^{c+d x}\right )^n \, dx &=\frac {\operatorname {Subst}\left (\int (a+b x)^n \, dx,x,e^{c+d x}\right )}{d}\\ &=\frac {\left (a+b e^{c+d x}\right )^{1+n}}{b d (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 26, normalized size = 0.96 \[ \frac {\left (a+b e^{c+d x}\right )^{n+1}}{b d n+b d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 33, normalized size = 1.22 \[ \frac {{\left (b e^{\left (d x + c\right )} + a\right )} {\left (b e^{\left (d x + c\right )} + a\right )}^{n}}{b d n + b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 26, normalized size = 0.96 \[ \frac {{\left (b e^{\left (d x + c\right )} + a\right )}^{n + 1}}{b d {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 27, normalized size = 1.00 \[ \frac {\left (b \,{\mathrm e}^{d x +c}+a \right )^{n +1}}{\left (n +1\right ) b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 26, normalized size = 0.96 \[ \frac {{\left (b e^{\left (d x + c\right )} + a\right )}^{n + 1}}{b d {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.48, size = 26, normalized size = 0.96 \[ \frac {{\left (a+b\,{\mathrm {e}}^{c+d\,x}\right )}^{n+1}}{b\,d\,\left (n+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 18.74, size = 107, normalized size = 3.96 \[ \begin {cases} \frac {x e^{c}}{a} & \text {for}\: b = 0 \wedge d = 0 \wedge n = -1 \\\frac {a^{n} e^{c} e^{d x}}{d} & \text {for}\: b = 0 \\x \left (a + b e^{c}\right )^{n} e^{c} & \text {for}\: d = 0 \\\frac {\log {\left (\frac {a e^{- c}}{b} + e^{d x} \right )}}{b d} & \text {for}\: n = -1 \\\frac {a \left (a + b e^{c} e^{d x}\right )^{n}}{b d n + b d} + \frac {b \left (a + b e^{c} e^{d x}\right )^{n} e^{c} e^{d x}}{b d n + b d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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