Optimal. Leaf size=37 \[ \frac {e^{n x} \left (e^x\right )^{-n} \left (a+b \left (e^x\right )^n\right )^{p+1}}{b n (p+1)} \]
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Rubi [A] time = 0.06, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2247, 2246, 32} \[ \frac {e^{n x} \left (e^x\right )^{-n} \left (a+b \left (e^x\right )^n\right )^{p+1}}{b n (p+1)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2246
Rule 2247
Rubi steps
\begin {align*} \int e^{n x} \left (a+b \left (e^x\right )^n\right )^p \, dx &=\left (e^{n x} \left (e^x\right )^{-n}\right ) \int \left (e^x\right )^n \left (a+b \left (e^x\right )^n\right )^p \, dx\\ &=\frac {\left (e^{n x} \left (e^x\right )^{-n}\right ) \operatorname {Subst}\left (\int (a+b x)^p \, dx,x,\left (e^x\right )^n\right )}{n}\\ &=\frac {e^{n x} \left (e^x\right )^{-n} \left (a+b \left (e^x\right )^n\right )^{1+p}}{b n (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 36, normalized size = 0.97 \[ \frac {e^{n x} \left (e^x\right )^{-n} \left (a+b \left (e^x\right )^n\right )^{p+1}}{b n p+b n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 29, normalized size = 0.78 \[ \frac {{\left (b e^{\left (n x\right )} + a\right )} {\left (b e^{\left (n x\right )} + a\right )}^{p}}{b n p + b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 30, normalized size = 0.81 \[ \frac {{\left (b e^{\left (n x\right )} + a\right )} {\left (b e^{\left (n x\right )} + a\right )}^{p}}{b n {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 52, normalized size = 1.41 \[ \frac {{\mathrm e}^{n x} {\mathrm e}^{p \ln \left (b \,{\mathrm e}^{n x}+a \right )}}{\left (p +1\right ) n}+\frac {a \,{\mathrm e}^{p \ln \left (b \,{\mathrm e}^{n x}+a \right )}}{\left (p +1\right ) b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 24, normalized size = 0.65 \[ \frac {{\left (b e^{\left (n x\right )} + a\right )}^{p + 1}}{b n {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.44, size = 24, normalized size = 0.65 \[ \frac {{\left (a+b\,{\mathrm {e}}^{n\,x}\right )}^{p+1}}{b\,n\,\left (p+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \left (e^{x}\right )^{n}\right )^{p} e^{n x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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