Optimal. Leaf size=40 \[ a^3 x+3 a^2 b e^x+\frac {3}{2} a b^2 e^{2 x}+\frac {1}{3} b^3 e^{3 x} \]
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Rubi [A] time = 0.02, antiderivative size = 40, 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} \[ 3 a^2 b e^x+a^3 x+\frac {3}{2} a b^2 e^{2 x}+\frac {1}{3} b^3 e^{3 x} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2282
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*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 1.00 \[ a^3 x+3 a^2 b e^x+\frac {3}{2} a b^2 e^{2 x}+\frac {1}{3} b^3 e^{3 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 33, normalized size = 0.82 \[ a^{3} x + \frac {1}{3} \, b^{3} e^{\left (3 \, x\right )} + \frac {3}{2} \, a b^{2} e^{\left (2 \, x\right )} + 3 \, a^{2} b e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 33, normalized size = 0.82 \[ a^{3} x + \frac {1}{3} \, b^{3} e^{\left (3 \, x\right )} + \frac {3}{2} \, a b^{2} e^{\left (2 \, x\right )} + 3 \, a^{2} b e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 36, normalized size = 0.90 \[ a^{3} \ln \left ({\mathrm e}^{x}\right )+3 a^{2} b \,{\mathrm e}^{x}+\frac {3 a \,b^{2} {\mathrm e}^{2 x}}{2}+\frac {b^{3} {\mathrm e}^{3 x}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 33, normalized size = 0.82 \[ a^{3} x + \frac {1}{3} \, b^{3} e^{\left (3 \, x\right )} + \frac {3}{2} \, a b^{2} e^{\left (2 \, x\right )} + 3 \, a^{2} b e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 33, normalized size = 0.82 \[ x\,a^3+3\,{\mathrm {e}}^x\,a^2\,b+\frac {3\,{\mathrm {e}}^{2\,x}\,a\,b^2}{2}+\frac {{\mathrm {e}}^{3\,x}\,b^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 37, normalized size = 0.92 \[ a^{3} x + 3 a^{2} b e^{x} + \frac {3 a b^{2} e^{2 x}}{2} + \frac {b^{3} e^{3 x}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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