Optimal. Leaf size=25 \[ \frac {\left (a+b \left (e^x\right )^n\right )^{p+1}}{b n (p+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2246, 32} \[ \frac {\left (a+b \left (e^x\right )^n\right )^{p+1}}{b n (p+1)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2246
Rubi steps
\begin {align*} \int \left (e^x\right )^n \left (a+b \left (e^x\right )^n\right )^p \, dx &=\frac {\operatorname {Subst}\left (\int (a+b x)^p \, dx,x,\left (e^x\right )^n\right )}{n}\\ &=\frac {\left (a+b \left (e^x\right )^n\right )^{1+p}}{b n (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 24, normalized size = 0.96 \[ \frac {\left (a+b \left (e^x\right )^n\right )^{p+1}}{b n p+b n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 29, normalized size = 1.16 \[ \frac {{\left (b e^{\left (n x\right )} + a\right )} {\left (b e^{\left (n x\right )} + a\right )}^{p}}{b n p + b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 24, normalized size = 0.96 \[ \frac {{\left (b e^{\left (n x\right )} + a\right )}^{p + 1}}{b n {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 1.00 \[ \frac {\left (b \left ({\mathrm e}^{x}\right )^{n}+a \right )^{p +1}}{\left (p +1\right ) b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 24, normalized size = 0.96 \[ \frac {{\left (b e^{\left (n x\right )} + a\right )}^{p + 1}}{b n {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.51, size = 24, normalized size = 0.96 \[ \frac {{\left (a+b\,{\mathrm {e}}^{n\,x}\right )}^{p+1}}{b\,n\,\left (p+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.38, size = 80, normalized size = 3.20 \[ \begin {cases} \frac {x}{a} & \text {for}\: b = 0 \wedge n = 0 \wedge p = -1 \\\frac {a^{p} \left (e^{x}\right )^{n}}{n} & \text {for}\: b = 0 \\x \left (a + b\right )^{p} & \text {for}\: n = 0 \\\frac {\log {\left (\frac {a}{b} + \left (e^{x}\right )^{n} \right )}}{b n} & \text {for}\: p = -1 \\\frac {a \left (a + b \left (e^{x}\right )^{n}\right )^{p}}{b n p + b n} + \frac {b \left (a + b \left (e^{x}\right )^{n}\right )^{p} \left (e^{x}\right )^{n}}{b n p + b n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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