Optimal. Leaf size=40 \[ 3 a^2 b e^x+a^3 x+\frac{3}{2} a b^2 e^{2 x}+\frac{1}{3} b^3 e^{3 x} \]
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Rubi [A] time = 0.0195108, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2282, 43} \[ 3 a^2 b e^x+a^3 x+\frac{3}{2} a b^2 e^{2 x}+\frac{1}{3} b^3 e^{3 x} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 43
Rubi steps
\begin{align*} \int \left (a+b e^x\right )^3 \, dx &=\operatorname{Subst}\left (\int \frac{(a+b x)^3}{x} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \left (3 a^2 b+\frac{a^3}{x}+3 a b^2 x+b^3 x^2\right ) \, dx,x,e^x\right )\\ &=3 a^2 b e^x+\frac{3}{2} a b^2 e^{2 x}+\frac{1}{3} b^3 e^{3 x}+a^3 x\\ \end{align*}
Mathematica [A] time = 0.0124967, size = 40, normalized size = 1. \[ 3 a^2 b e^x+a^3 x+\frac{3}{2} a b^2 e^{2 x}+\frac{1}{3} b^3 e^{3 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 36, normalized size = 0.9 \begin{align*}{\frac{{b}^{3} \left ({{\rm e}^{x}} \right ) ^{3}}{3}}+{\frac{3\,a{b}^{2} \left ({{\rm e}^{x}} \right ) ^{2}}{2}}+3\,{a}^{2}b{{\rm e}^{x}}+{a}^{3}\ln \left ({{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.955045, size = 45, normalized size = 1.12 \begin{align*} a^{3} x + \frac{1}{3} \, b^{3} e^{\left (3 \, x\right )} + \frac{3}{2} \, a b^{2} e^{\left (2 \, x\right )} + 3 \, a^{2} b e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.768507, size = 80, normalized size = 2. \begin{align*} a^{3} x + \frac{1}{3} \, b^{3} e^{\left (3 \, x\right )} + \frac{3}{2} \, a b^{2} e^{\left (2 \, x\right )} + 3 \, a^{2} b e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.141257, size = 37, normalized size = 0.92 \begin{align*} a^{3} x + 3 a^{2} b e^{x} + \frac{3 a b^{2} e^{2 x}}{2} + \frac{b^{3} e^{3 x}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2367, size = 45, normalized size = 1.12 \begin{align*} a^{3} x + \frac{1}{3} \, b^{3} e^{\left (3 \, x\right )} + \frac{3}{2} \, a b^{2} e^{\left (2 \, x\right )} + 3 \, a^{2} b e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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