Optimal. Leaf size=52 \[ 3 a^2 b x-\frac{a^3 e^{-n x}}{n}+\frac{3 a b^2 e^{n x}}{n}+\frac{b^3 e^{2 n x}}{2 n} \]
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Rubi [A] time = 0.0416476, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2248, 43} \[ 3 a^2 b x-\frac{a^3 e^{-n x}}{n}+\frac{3 a b^2 e^{n x}}{n}+\frac{b^3 e^{2 n x}}{2 n} \]
Antiderivative was successfully verified.
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Rule 2248
Rule 43
Rubi steps
\begin{align*} \int e^{-n x} \left (a+b e^{n x}\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^2} \, dx,x,e^{n x}\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (3 a b^2+\frac{a^3}{x^2}+\frac{3 a^2 b}{x}+b^3 x\right ) \, dx,x,e^{n x}\right )}{n}\\ &=-\frac{a^3 e^{-n x}}{n}+\frac{3 a b^2 e^{n x}}{n}+\frac{b^3 e^{2 n x}}{2 n}+3 a^2 b x\\ \end{align*}
Mathematica [A] time = 0.0249187, size = 48, normalized size = 0.92 \[ \frac{3 a^2 b n x+a^3 \left (-e^{-n x}\right )+3 a b^2 e^{n x}+\frac{1}{2} b^3 e^{2 n x}}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 57, normalized size = 1.1 \begin{align*}{\frac{{b}^{3} \left ({{\rm e}^{nx}} \right ) ^{2}}{2\,n}}+3\,{\frac{a{b}^{2}{{\rm e}^{nx}}}{n}}+3\,{\frac{{a}^{2}b\ln \left ({{\rm e}^{nx}} \right ) }{n}}-{\frac{{a}^{3}}{n{{\rm e}^{nx}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15863, size = 63, normalized size = 1.21 \begin{align*} 3 \, a^{2} b x + \frac{b^{3} e^{\left (2 \, n x\right )}}{2 \, n} + \frac{3 \, a b^{2} e^{\left (n x\right )}}{n} - \frac{a^{3} e^{\left (-n x\right )}}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49152, size = 111, normalized size = 2.13 \begin{align*} \frac{{\left (6 \, a^{2} b n x e^{\left (n x\right )} + b^{3} e^{\left (3 \, n x\right )} + 6 \, a b^{2} e^{\left (2 \, n x\right )} - 2 \, a^{3}\right )} e^{\left (-n x\right )}}{2 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.190626, size = 73, normalized size = 1.4 \begin{align*} 3 a^{2} b x + \begin{cases} \frac{- 2 a^{3} n^{2} e^{- n x} + 6 a b^{2} n^{2} e^{n x} + b^{3} n^{2} e^{2 n x}}{2 n^{3}} & \text{for}\: 2 n^{3} \neq 0 \\x \left (a^{3} + 3 a b^{2} + b^{3}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30001, size = 63, normalized size = 1.21 \begin{align*} 3 \, a^{2} b x + \frac{b^{3} e^{\left (2 \, n x\right )}}{2 \, n} + \frac{3 \, a b^{2} e^{\left (n x\right )}}{n} - \frac{a^{3} e^{\left (-n x\right )}}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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