Optimal. Leaf size=52 \[ -\frac {a^3 e^{-n x}}{n}+3 a^2 b x+\frac {3 a b^2 e^{n x}}{n}+\frac {b^3 e^{2 n x}}{2 n} \]
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Rubi [A] time = 0.04, antiderivative size = 52, normalized size of antiderivative = 1.00, 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 43
Rule 2248
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*}
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Mathematica [A] time = 0.03, size = 48, normalized size = 0.92 \[ \frac {a^3 \left (-e^{-n x}\right )+3 a^2 b n x+3 a b^2 e^{n x}+\frac {1}{2} b^3 e^{2 n x}}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 48, normalized size = 0.92 \[ \frac {{\left (6 \, a^{2} b n x e^{\left (n x\right )} + b^{3} e^{\left (3 \, n x\right )} + 6 \, a b^{2} e^{\left (2 \, n x\right )} - 2 \, a^{3}\right )} e^{\left (-n x\right )}}{2 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 47, normalized size = 0.90 \[ 3 \, a^{2} b x + \frac {b^{3} e^{\left (2 \, n x\right )}}{2 \, n} + \frac {3 \, a b^{2} e^{\left (n x\right )}}{n} - \frac {a^{3} e^{\left (-n x\right )}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 57, normalized size = 1.10 \[ -\frac {a^{3} {\mathrm e}^{-n x}}{n}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{n x}\right )}{n}+\frac {3 a \,b^{2} {\mathrm e}^{n x}}{n}+\frac {b^{3} {\mathrm e}^{2 n x}}{2 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 47, normalized size = 0.90 \[ 3 \, a^{2} b x + \frac {b^{3} e^{\left (2 \, n x\right )}}{2 \, n} + \frac {3 \, a b^{2} e^{\left (n x\right )}}{n} - \frac {a^{3} e^{\left (-n x\right )}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.52, size = 44, normalized size = 0.85 \[ \frac {{\mathrm {e}}^{-n\,x}\,\left (-2\,a^3+6\,{\mathrm {e}}^{2\,n\,x}\,a\,b^2+{\mathrm {e}}^{3\,n\,x}\,b^3\right )}{2\,n}+3\,a^2\,b\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 73, normalized size = 1.40 \[ 3 a^{2} b x + \begin {cases} \frac {- 2 a^{3} n^{2} e^{- n x} + 6 a b^{2} n^{2} e^{n x} + b^{3} n^{2} e^{2 n x}}{2 n^{3}} & \text {for}\: 2 n^{3} \neq 0 \\x \left (a^{3} + 3 a b^{2} + b^{3}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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