Optimal. Leaf size=53 \[ a^4 x+4 a^3 b e^x+3 a^2 b^2 e^{2 x}+\frac {4}{3} a b^3 e^{3 x}+\frac {1}{4} b^4 e^{4 x} \]
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Rubi [A] time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, 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 43
Rule 2282
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*}
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Mathematica [A] time = 0.02, size = 53, normalized size = 1.00 \[ a^4 x+4 a^3 b e^x+3 a^2 b^2 e^{2 x}+\frac {4}{3} a b^3 e^{3 x}+\frac {1}{4} b^4 e^{4 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 45, normalized size = 0.85 \[ a^{4} x + \frac {1}{4} \, b^{4} e^{\left (4 \, x\right )} + \frac {4}{3} \, a b^{3} e^{\left (3 \, x\right )} + 3 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 4 \, a^{3} b e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 45, normalized size = 0.85 \[ a^{4} x + \frac {1}{4} \, b^{4} e^{\left (4 \, x\right )} + \frac {4}{3} \, a b^{3} e^{\left (3 \, x\right )} + 3 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 4 \, a^{3} b e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 48, normalized size = 0.91 \[ a^{4} \ln \left ({\mathrm e}^{x}\right )+4 a^{3} b \,{\mathrm e}^{x}+3 a^{2} b^{2} {\mathrm e}^{2 x}+\frac {4 a \,b^{3} {\mathrm e}^{3 x}}{3}+\frac {b^{4} {\mathrm e}^{4 x}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 45, normalized size = 0.85 \[ a^{4} x + \frac {1}{4} \, b^{4} e^{\left (4 \, x\right )} + \frac {4}{3} \, a b^{3} e^{\left (3 \, x\right )} + 3 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 4 \, a^{3} b e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.43, size = 45, normalized size = 0.85 \[ x\,a^4+4\,{\mathrm {e}}^x\,a^3\,b+3\,{\mathrm {e}}^{2\,x}\,a^2\,b^2+\frac {4\,{\mathrm {e}}^{3\,x}\,a\,b^3}{3}+\frac {{\mathrm {e}}^{4\,x}\,b^4}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 51, normalized size = 0.96 \[ a^{4} x + 4 a^{3} b e^{x} + 3 a^{2} b^{2} e^{2 x} + \frac {4 a b^{3} e^{3 x}}{3} + \frac {b^{4} e^{4 x}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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