Optimal. Leaf size=20 \[ \frac {\left (a+b e^x\right )^{n+1}}{b (n+1)} \]
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Rubi [A] time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2246, 32} \[ \frac {\left (a+b e^x\right )^{n+1}}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2246
Rubi steps
\begin {align*} \int e^x \left (a+b e^x\right )^n \, dx &=\operatorname {Subst}\left (\int (a+b x)^n \, dx,x,e^x\right )\\ &=\frac {\left (a+b e^x\right )^{1+n}}{b (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 19, normalized size = 0.95 \[ \frac {\left (a+b e^x\right )^{n+1}}{b n+b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 22, normalized size = 1.10 \[ \frac {{\left (b e^{x} + a\right )} {\left (b e^{x} + a\right )}^{n}}{b n + b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 19, normalized size = 0.95 \[ \frac {{\left (b e^{x} + a\right )}^{n + 1}}{b {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 20, normalized size = 1.00 \[ \frac {\left (b \,{\mathrm e}^{x}+a \right )^{n +1}}{\left (n +1\right ) b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 19, normalized size = 0.95 \[ \frac {{\left (b e^{x} + a\right )}^{n + 1}}{b {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.50, size = 19, normalized size = 0.95 \[ \frac {{\left (a+b\,{\mathrm {e}}^x\right )}^{n+1}}{b\,\left (n+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.92, size = 56, normalized size = 2.80 \[ \begin {cases} \frac {e^{x}}{a} & \text {for}\: b = 0 \wedge n = -1 \\a^{n} e^{x} & \text {for}\: b = 0 \\\frac {\log {\left (\frac {a}{b} + e^{x} \right )}}{b} & \text {for}\: n = -1 \\\frac {a \left (a + b e^{x}\right )^{n}}{b n + b} + \frac {b \left (a + b e^{x}\right )^{n} e^{x}}{b n + b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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