Optimal. Leaf size=53 \[ 3 a^2 b^2 e^{2 x}+4 a^3 b e^x+a^4 x+\frac{4}{3} a b^3 e^{3 x}+\frac{1}{4} b^4 e^{4 x} \]
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Rubi [A] time = 0.0264042, antiderivative size = 53, 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} \[ 3 a^2 b^2 e^{2 x}+4 a^3 b e^x+a^4 x+\frac{4}{3} a b^3 e^{3 x}+\frac{1}{4} b^4 e^{4 x} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 43
Rubi steps
\begin{align*} \int \left (a+b e^x\right )^4 \, dx &=\operatorname{Subst}\left (\int \frac{(a+b x)^4}{x} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \left (4 a^3 b+\frac{a^4}{x}+6 a^2 b^2 x+4 a b^3 x^2+b^4 x^3\right ) \, dx,x,e^x\right )\\ &=4 a^3 b e^x+3 a^2 b^2 e^{2 x}+\frac{4}{3} a b^3 e^{3 x}+\frac{1}{4} b^4 e^{4 x}+a^4 x\\ \end{align*}
Mathematica [A] time = 0.0165201, size = 53, normalized size = 1. \[ 3 a^2 b^2 e^{2 x}+4 a^3 b e^x+a^4 x+\frac{4}{3} a b^3 e^{3 x}+\frac{1}{4} b^4 e^{4 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 48, normalized size = 0.9 \begin{align*}{\frac{{b}^{4} \left ({{\rm e}^{x}} \right ) ^{4}}{4}}+{\frac{4\,a{b}^{3} \left ({{\rm e}^{x}} \right ) ^{3}}{3}}+3\,{b}^{2}{a}^{2} \left ({{\rm e}^{x}} \right ) ^{2}+4\,{a}^{3}b{{\rm e}^{x}}+{a}^{4}\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 = 0.9777, size = 61, normalized size = 1.15 \begin{align*} a^{4} x + \frac{1}{4} \, b^{4} e^{\left (4 \, x\right )} + \frac{4}{3} \, a b^{3} e^{\left (3 \, x\right )} + 3 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 4 \, a^{3} b e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.865348, size = 107, normalized size = 2.02 \begin{align*} a^{4} x + \frac{1}{4} \, b^{4} e^{\left (4 \, x\right )} + \frac{4}{3} \, a b^{3} e^{\left (3 \, x\right )} + 3 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 4 \, a^{3} b e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.162058, size = 51, normalized size = 0.96 \begin{align*} a^{4} x + 4 a^{3} b e^{x} + 3 a^{2} b^{2} e^{2 x} + \frac{4 a b^{3} e^{3 x}}{3} + \frac{b^{4} e^{4 x}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.55905, size = 61, normalized size = 1.15 \begin{align*} a^{4} x + \frac{1}{4} \, b^{4} e^{\left (4 \, x\right )} + \frac{4}{3} \, a b^{3} e^{\left (3 \, x\right )} + 3 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 4 \, a^{3} b e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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