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 = 0.0812827, 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 \[ -\frac{a^3 e^{-n x}}{n}+3 a^2 b x+\frac{3 a b^2 e^{n x}}{n}+\frac{b^3 e^{2 n x}}{2 n} \]
Antiderivative was successfully verified.
[In] Int[(a + b*E^(n*x))^3/E^(n*x),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{3} e^{- n x}}{n} + \frac{3 a^{2} b \log{\left (e^{n x} \right )}}{n} + \frac{3 a b^{2} e^{n x}}{n} + \frac{b^{3} \int ^{e^{n x}} x\, dx}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*exp(n*x))**3/exp(n*x),x)
[Out]
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Mathematica [A] time = 0.0264373, size = 48, normalized size = 0.92 \[ \frac{-2 a^3 e^{-n x}+6 a^2 b n x+6 a b^2 e^{n x}+b^3 e^{2 n x}}{2 n} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*E^(n*x))^3/E^(n*x),x]
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Maple [A] time = 0.01, size = 57, normalized size = 1.1 \[{\frac{{b}^{3} \left ({{\rm e}^{nx}} \right ) ^{2}}{2\,n}}+3\,{\frac{a{b}^{2}{{\rm e}^{nx}}}{n}}-{\frac{{a}^{3}}{n{{\rm e}^{nx}}}}+3\,{\frac{{a}^{2}b\ln \left ({{\rm e}^{nx}} \right ) }{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*exp(n*x))^3/exp(n*x),x)
[Out]
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Maxima [A] time = 0.805785, size = 63, normalized size = 1.21 \[ 3 \, a^{2} b x + \frac{b^{3} e^{\left (2 \, n x\right )}}{2 \, n} + \frac{3 \, a b^{2} e^{\left (n x\right )}}{n} - \frac{a^{3} e^{\left (-n x\right )}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*e^(n*x) + a)^3*e^(-n*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.244733, size = 65, normalized size = 1.25 \[ \frac{{\left (6 \, a^{2} b n x e^{\left (n x\right )} + b^{3} e^{\left (3 \, n x\right )} + 6 \, a b^{2} e^{\left (2 \, n x\right )} - 2 \, a^{3}\right )} e^{\left (-n x\right )}}{2 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*e^(n*x) + a)^3*e^(-n*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.37586, size = 73, normalized size = 1.4 \[ 3 a^{2} b x + \begin{cases} \frac{- 2 a^{3} n^{2} e^{- n x} + 6 a b^{2} n^{2} e^{n x} + b^{3} n^{2} e^{2 n x}}{2 n^{3}} & \text{for}\: 2 n^{3} \neq 0 \\x \left (a^{3} + 3 a b^{2} + b^{3}\right ) & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*exp(n*x))**3/exp(n*x),x)
[Out]
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GIAC/XCAS [A] time = 0.224016, size = 63, normalized size = 1.21 \[ 3 \, a^{2} b x + \frac{b^{3} e^{\left (2 \, n x\right )}}{2 \, n} + \frac{3 \, a b^{2} e^{\left (n x\right )}}{n} - \frac{a^{3} e^{\left (-n x\right )}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*e^(n*x) + a)^3*e^(-n*x),x, algorithm="giac")
[Out]